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Operator Theory: Advances and Applications, vol. 267. </style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2018</style></year></dates><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">221-246</style></pages></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Density of analytic polynomials in abstract Hardy spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Commentationes Mathematicae</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2017</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://wydawnictwa.ptm.org.pl/index.php/commentationes-mathematicae/article/view/4364/5884</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">57</style></volume><pages><style face="normal" font="default" size="100%">131-141</style></pages><issue><style face="normal" font="default" size="100%">2</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino B.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">The index of weighted singular integral operators with shifts and slowly oscillating data</style></title><secondary-title><style face="normal" font="default" size="100%">Journal of Mathematical Analysis and Applications</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2017</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://dx.doi.org/10.1016/j.jmaa.2017.01.052</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">450</style></volume><pages><style face="normal" font="default" size="100%">606-630</style></pages></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>6</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Bini, Dario</style></author><author><style face="normal" font="default" size="100%">Ehrhardt, Torsten</style></author><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Spitkovsky (eds.), Ilya M.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics. The Albrecht Böttcher Anniversary Volume</style></title></titles><dates><year><style  face="normal" font="default" size="100%">2017</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.springer.com/gp/book/9783319491806</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser Basel</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino B.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Necessary Fredholm conditions for weighted singular integral operators with shifts and slowly oscillating data</style></title><secondary-title><style face="normal" font="default" size="100%">Journal of Integral Equations and Applications</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2017</style></year></dates><volume><style face="normal" font="default" size="100%">29</style></volume><pages><style face="normal" font="default" size="100%">365-399</style></pages><issue><style face="normal" font="default" size="100%">3</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Toeplitz operators on abstract Hardy spaces built upon Banach function spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Journal of Function Spaces</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2017</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">https://www.hindawi.com/journals/jfs/2017/9768210/</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">2017</style></volume><pages><style face="normal" font="default" size="100%">Article ID 9768210, 8 pages.</style></pages></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino B.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">The generalized Cauchy index of some semi-almost periodic functions</style></title><secondary-title><style face="normal" font="default" size="100%">Boletín de la Sociedad Matemática Mexicana</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2016</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/article/10.1007/s40590-016-0119-5</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">22</style></volume><pages><style face="normal" font="default" size="100%">473-485</style></pages><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We compute the generalized Cauchy index of some semi-almost periodic functions, which are important&lt;br /&gt;
in the study of the Fredholm index of singular integral operators with shifts and slowly oscillating data.&lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">2</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino B.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On a weighted singular integral operator with shifts and slowly oscillating data</style></title><secondary-title><style face="normal" font="default" size="100%">Complex Analysis and Operator Theory</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2016</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/article/10.1007/s11785-015-0452-0</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">10</style></volume><pages><style face="normal" font="default" size="100%">1101-1131</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(\alpha,\beta\) be orientation-preserving diffeomorphism (shifts) of \(\mathbb{R}_+=(0,\infty)\) onto itself with the only fixed points \(0\) and \(\infty\) and \(U_\alpha,U_\beta\) be the isometric shift operators on \(L^p(\mathbb{R}_+)\) given by \(U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)\), \(U_\beta f=(\beta')^{1/p}(f\circ\beta)\), and \(P_2^\pm=(I\pm S_2)/2\) where \[ (S_2 f)(t):=\frac{1}{\pi i}\int\limits_0^\infty \left(\frac{t}{\tau}\right)^{1/2-1/p}\frac{f(\tau)}{\tau-t}\,d\tau, \quad t\in\mathbb{R}_+, \]&lt;br /&gt;
is the weighted Cauchy singular integral operator. We prove that if \(\alpha',\beta'\) and \(c,d\) are continuous on \(\mathbb{R}_+\) and slowly oscillating at \(0\) and \(\ infty\), and \[ \limsup_{t\to s}|c(t)|&amp;lt;1,\quad \limsup_{t\to s}|d(t)|&amp;lt;1, \quad s\in\{0,\infty\}, \] then the operator \((I-cU_\alpha)P_2^++(I-dU_\beta)P_2^-\) is Fredholm on \(L^p(\mathbb{R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.
&lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">6</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino B.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">One-sided invertibility criteria for binomial functional operators with shift and slowly oscillating data</style></title><secondary-title><style face="normal" font="default" size="100%">Mediterranean Journal of Mathematics</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2016</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/article/10.1007/s00009-016-0753-1</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">13</style></volume><pages><style face="normal" font="default" size="100%">4413–4435</style></pages><issue><style face="normal" font="default" size="100%">6</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Banach algebra of the Fourier multipliers on weighted Banach function spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Concrete Operators</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2015</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.degruyter.com/view/j/conop.2014.2.issue-1/conop-2015-0001/conop-2015-0001.xml?format=INT</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">2</style></volume><pages><style face="normal" font="default" size="100%">27-36</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(\mathcal{M}_{X,w}(\mathbb{R})\) denote the algebra of the Fourier multipliers on a separable weighted Banach function space \(X(\mathbb{R},w)\). We prove that if the Cauchy singular integral operator \(S\) is bounded on \(X(\mathbb{R},w)\), then \(\mathcal{M}_{X,w}(\mathbb{R})\) is continuously embedded into \(L^\infty(\mathbb{R})\).  An important consequence of the continuous embedding \(\mathcal{M}_{X,w}(\mathbb{R})\subset L^\infty(\mathbb{R})\) is that \(\mathcal{M}_{X,w}(\mathbb{R})\) is a Banach algebra.&lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">1</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Commutators of convolution type operators on some Banach function spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Annals of Functional Analysis</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2015</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://projecteuclid.org/euclid.afa/1435764011</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">6</style></volume><pages><style face="normal" font="default" size="100%">191-205</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We study the boundedness of Fourier convolution operators \(W^0(b)\) and the compactness of commutators of \(W^0(b)\) with multiplication operators \(aI\) on some Banach function spaces \(X(\mathbb{R})\) for certain classes of piecewise quasicontinuous functions \(a\in PQC\) and piecewise slowly oscillating Fourier multipliers \(b\in PSO_{X,1}^\diamond\). We suppose that \(X(\mathbb{R})\) is a separable rearrangement-invariant space with nontrivial Boyd indices or a reflexive variable Lebesgue space, in which the Hardy-Littlewood maximal operator is bounded. Our results complement those of Isaac De La Cruz-Rodríguez, Yuri Karlovich, and Iván Loreto Hernández obtained for Lebesgue spaces with Muckenhoupt weights. &lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">4</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Fredholmness and index of simplest weighted singular integral operators with two slowly oscillating shifts</style></title><secondary-title><style face="normal" font="default" size="100%">Banach Journal of Mathematical Analysis</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2015</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://projecteuclid.org/euclid.bjma/1419001701</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">9</style></volume><pages><style face="normal" font="default" size="100%">24-42</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(\alpha\) and \(\beta\) be orientation-preserving diffeomorphism (shifts) of \(\mathbb{R}_+=(0,\infty)\) onto itself with the only fixed points \(0\) and \(\infty\), where the derivatives \(\alpha'\) and \(\beta'\) may have discontinuities of slowly oscillating type at \(0\) and \(\infty\). For \(p\in(1,\infty)\), we consider the weighted shift operators \(U_\alpha\) and \(U_\beta\) given on the Lebesgue space \(L^p(\mathbb{R}_+)\) by \(U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)\) and \(U_\beta f=(\beta')^{1/p}(f\circ\beta)\). For \(i,j\in\mathbb{Z}\) we study the simplest weighted singular integral operators with two shifts \(A_{ij}=U_\alpha^i P_\gamma^++U_\beta^j P_\gamma^-\) on \(L^p(\mathbb{R}_+)\), where \(P_\gamma^\pm=(I\pm S_\gamma)/2\) are operators associated to the weighted Cauchy singular integral operator \[ (S_\gamma f)(t)=\frac{1}{\pi i}\int_{\mathbb{R}_+} \left(\frac{t}{\tau}\right)^\gamma\frac{f(\tau)}{\tau-t}d\tau \] with \(\gamma\in\mathbb{C}\) satisfying \(0&amp;lt;1/p+\Re\gamma&amp;lt;1\). We prove that the operator \(A_{ij}\) is a Fredholm operator on \(L^p(\mathbb{R}_+)\) and has zero index if \[ 0&amp;lt;\frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\inf_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma), \quad \frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\sup_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma)&amp;lt;1, \] where \(\omega_{ij}(t)=\log[\alpha_i(\beta_{-j}(t))/t]\) and \(\alpha_i\), \(\beta_{-j}\) are iterations of \(\alpha\), \(\beta\). This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for \(\gamma=0\).&lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">3</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Krzysztof Jarosz</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Maximally modulated singular integral operators and their applications to pseudodifferential operators on Banach function spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Function Spaces in Analysis. Contemporary Mathematics, 645</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2015</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.ams.org/books/conm/645/12908/conm645-12908.pdf</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">American Mathematical Society</style></publisher><pub-location><style face="normal" font="default" size="100%">Providence, Rhode Island</style></pub-location><pages><style face="normal" font="default" size="100%">165-178</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space \(X(\mathbb{R}^n)\) and on its associate space \(X'(\mathbb{R}^n)\) and a maximally modulated Calderón-Zygmund singular integral operator \(T^{\Phi}\) is of weak type \((r,r)\) for all \(r\in(1,\infty)\), then \(T^{\Phi}\) extends to a bounded operator on \(X(\mathbb{R}^n)\). This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R})\) under natural assumptions on the variable exponent \(p:\mathbb{R}\to(1,\infty)\). Applications of the above result to the boundedness and compactness of pseudodifferential operators with \(L^\infty(\mathbb{R},V(\mathbb{R}))\)-symbols on variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R})\) are considered. Here the Banach algebra \(L^\infty(\mathbb{R},V(\mathbb{R}))\) consists of all bounded measurable \(V(\mathbb{R})\)-valued functions on \(\mathbb{R}\) where \(V(\mathbb{R})\) is the Banach algebra of all functions of bounded total variation.&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">The Stechkin inequality for Fourier multipliers on variable Lebesgue spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Mathematical Inequalities and Applications</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2015</style></year></dates><volume><style face="normal" font="default" size="100%">18</style></volume><pages><style face="normal" font="default" size="100%">1473–1481</style></pages><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We prove the Stechkin inequality for Fourier multipliers on variable Lebesgue spaces under some natural assumptions on variable exponents.&lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">4</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Bastos, Maria Amélia</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino</style></author><author><style face="normal" font="default" size="100%">Samko, Stefan</style></author><author><style face="normal" font="default" size="100%">Spitkovsky, Ilya M.</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Boundedness of pseudodifferential operators on Banach function spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Operator Theory, Operator Algebras and Applications. Operator Theory: Advances and Applications, 242</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2014</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007%2F978-3-0348-0816-3_10</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser/Springer</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">185-195</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space \(X(\mathbb{R}^n)\) and on its associate space \(X'(\mathbb{R}^n)\), then a pseudodifferential operator \(\operatorname{Op}(a)\) is bounded on \(X(\mathbb{R}^n)\) whenever the symbol \(a\) belongs to the Hörmander class \(S_{\rho,\delta}^{n(\rho-1)}\) with \(0&amp;lt;\rho\le 1\), \(0\le\delta&amp;lt;1\) or to the the Miyachi class \(S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)\) with \(0\le\delta\le\rho\le 1\), \(0\le\delta&amp;lt;1\),  and \(\varkappa&amp;gt;0\). This result is applied to the case of variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R}^n)\).&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Spitkovsky, Ilya M.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Manuel Cepedello Boiso</style></author><author><style face="normal" font="default" size="100%">Håkan Hedenmalm</style></author><author><style face="normal" font="default" size="100%">Marinus A. Kaashoek</style></author><author><style face="normal" font="default" size="100%">Alfonso Montes Rodríguez</style></author><author><style face="normal" font="default" size="100%">Sergei Treil</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">The Cauchy singular integral operator on weighted variable Lebesgue spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, 236</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2014</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-3-0348-0648-0_17</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">275-291</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(p:\mathbb{R}\to(1,\infty)\) be a globally log-Hölder continuous variable exponent and \(w:\mathbb{R}\to[0,\infty]\) be a weight. We prove that the Cauchy singular integral operator \(S\) is bounded on the weighted variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R},w)=\{f:fw\in L^{p(\cdot)}(\mathbb{R})\}\) if and only if the weight \(w\) satisfies $$ \sup_{-\infty &amp;lt; a &amp;lt; b &amp;lt; \infty} \frac{1}{b-a} \|w\chi_{(a,b)}\|_{p(\cdot)} \|w^{-1}\chi_{(a,b)}\|_{p'(\cdot)}&amp;lt;\infty \quad (1/p(x)+1/p'(x)=1). $$
&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino B.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Fredholmness and index of simplest singular integral operators with two slowly oscillating shifts</style></title><secondary-title><style face="normal" font="default" size="100%">Operators and Matrices</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2014</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://dx.doi.org/10.7153/oam-08-52</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">8</style></volume><pages><style face="normal" font="default" size="100%">935-955</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(\alpha\) and \(\beta\) be orientation-preserving diffeomorphisms (shifts) of \(\mathbb{R}_+=(0,\infty)\) onto itself with the only fixed points \(0\) and \(\infty\), where the derivatives \(\alpha'\) and \(\beta'\) may have discontinuities of slowly oscillating type at \(0\) and \(\infty\). For \(p\in(1,\infty)\), we consider the weighted shift operators \(U_\alpha\) and \(U_\beta\) given on the Lebesgue space \(L^p(\mathbb{R}_+)\) by \(U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)\) and \(U_\beta f= (\beta')^{1/p}(f\circ\beta)\). We apply the theory of Mellin pseudodifferential operators with symbols of limited smoothness to study the simplest singular integral operators with two shifts \(A_{ij}=U_\alpha^i P_++U_\beta^j P_-\) on the space \(L^p(\mathbb{R}_+)\), where \(P_\pm=(I\pm S)/2\) are operators associated to the Cauchy singular integral operator \(S\), and \(i,j\in\mathbb{Z}\). We prove that all \(A_{ij}\) are Fredholm operators on \(L^p(\mathbb{R}_+)\) and have zero indices.&lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">4</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino B.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On regularization of Mellin PDO's with slowly oscillating symbols of limited smoothness</style></title><secondary-title><style face="normal" font="default" size="100%">Communications in Mathematical Analysis</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2014</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.math-res-pub.org/cma/17/2/regularization-mellin-pdo%E2%80%99s-slowly-oscillating-symbols-limited-smoothness</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">17</style></volume><pages><style face="normal" font="default" size="100%">189-208</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We study Mellin pseudodifferential operators (shortly, Mellin PDO's) with symbols in the algebra \(\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) of slowly oscillating functions of limited  smoothness introduced in [K09]. We show that if \(\mathfrak{a}\in\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) does not degenerate on the ``boundary&quot; of \(\mathbb{R}_+\times\mathbb{R}\) in a certain sense,  then the Mellin PDO \(\operatorname{Op}(\mathfrak{a})\) is Fredholm on the space \(L^p\) for \(p\in(1,\infty)\) and each its regularizer is of the form \(\operatorname{Op}(\mathfrak{b})+K\) where \(K\) is a compact operator on \(L^p\) and \(\mathfrak{b}\) is a certain explicitly  constructed function in the same algebra \(\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) such that  \(\mathfrak{b}=1/\mathfrak{a}\) on the ``boundary&quot; of \(\mathbb{R}_+\times\mathbb{R}\). This result complements the known Fredholm criterion from [K09] for Mellin PDO's with symbols in the closure of \(\widetilde{\mathcal{E}}(\mathbb{R}_+,V(\mathbb{R}))\) and extends the corresponding result by V.S. Rabinovich (see [R98]) on Mellin PDO's with slowly oscillating symbols in \(C^\infty(\mathbb{R}_+\times\mathbb{R})\). &lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">2</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Spitkovsky, Ilya M.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Luigi Rodino</style></author><author><style face="normal" font="default" size="100%">Bernd Silbermann</style></author><author><style face="normal" font="default" size="100%">Spitkovsky, Ilya M.</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Pseudodifferential operators on variable Lebesgue spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Operator Theory: Advances and Applications, 228</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2013</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-3-0348-0537-7_9</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">173-183</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(\mathcal{M}(\mathbb{R}^n)\) be the class of bounded away from one and infinity functions \(p:\mathbb{R}^n\to[1,\infty]\) such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R}^n)\). We show that if \(a\) belongs to the Hörmander class \(S_{\rho,\delta}^{n(\rho-1)}\) with \(0&amp;lt;\rho\le 1\), \(0\le\delta&amp;lt;1\), then the pseudodifferential operator \(\operatorname{Op}(a)\) is bounded on the variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R}^n)\) provided that \(p\in\mathcal{M}(\mathbb{R}^n)\). Let \(\mathcal{M}^*(\mathbb{R}^n)\) be the class of variable exponents \(p\in\mathcal{M}(\mathbb{R}^n)\) represented as \(1/p(x)=\theta/p_0+(1-\theta)/p_1(x)\) where \(p_0\in(1,\infty)\), \(\theta\in(0,1)\), and \(p_1\in\mathcal{M}(\mathbb{R}^n)\). We prove that if \(a\in S_{1,0}^0\) slowly oscillates at infinity in the first variable, then the condition \[ \lim_{R\to\infty}\inf_{|x|+|\xi|\ge R}|a(x,\xi)|&amp;gt;0 \] is sufficient for the Fredholmness of \(\operatorname{Op}(a)\) on \(L^{p(\cdot)}(\mathbb{R}^n)\) whenever \(p\in\mathcal{M}^*(\mathbb{R}^n)\). Both theorems generalize pioneering results by Rabinovich and Samko [RS08] obtained for globally log-Hölder continuous exponents \(p\), constituting a proper subset of \(\mathcal{M}^*(\mathbb{R}^n)\). &lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino B.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Necessary conditions for Fredholmness of singular integral operators with shifts and slowly oscillating data</style></title><secondary-title><style face="normal" font="default" size="100%">Integral Equations and Operator Theory</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Cauchy singular integral operator</style></keyword><keyword><style  face="normal" font="default" size="100%">Fredholmness}</style></keyword><keyword><style  face="normal" font="default" size="100%">limit operator</style></keyword><keyword><style  face="normal" font="default" size="100%">slowly oscillating function</style></keyword><keyword><style  face="normal" font="default" size="100%">{Orientation-preserving non-Carleman shift</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2011</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{SEP}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.springerlink.com/content/h477767839564520/</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{1}</style></number><publisher><style face="normal" font="default" size="100%">{BIRKHAUSER VERLAG AG}</style></publisher><pub-location><style face="normal" font="default" size="100%">{VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND}</style></pub-location><volume><style face="normal" font="default" size="100%">71</style></volume><pages><style face="normal" font="default" size="100%">29-53</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Suppose \(\alpha\) is an orientation-preserving diffeomorphism (shift) of \(\mathbb{R}_+=(0,\infty)\) onto itself with the only fixed points \(0\) and \(\infty\). In [KKL11] we found sufficient conditions for the Fredholmness of the singular integral operator with shift \[(aI-bW_\alpha)P_++(cI-dW_\alpha)P_-\] acting on \(L^p(\mathbb{R}_+)\) with \( 1 &amp;lt; p &amp;lt; \infty\), where \(P_\pm=(I\pm S)/2\), \(S\) is the Cauchy singular integral operator, and \(W_\alpha f=f\circ\alpha\) is the shift operator, under the assumptions that the coefficients \(a,b,c,d\) and the derivative \(\alpha'\) of the shift are bounded and continuous on \(\mathbb{R}_+\) and may admit discontinuities of slowly oscillating type at \(0\) and \(\infty\). Now we prove that those conditions are also necessary. &lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">1</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Mascarenhas, Helena</style></author><author><style face="normal" font="default" size="100%">Santos, Pedro A.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Erratum to: Finite section method for a Banach algebra of convolution type operators on Lp(R) with symbols generated by PC and SO (vol 37, pg 559, 2010)</style></title><secondary-title><style face="normal" font="default" size="100%">Integral Equations and Operator Theory</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2011</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{MAR}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.springerlink.com/content/eu647210k2w95252/</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{3}</style></number><publisher><style face="normal" font="default" size="100%">{BIRKHAUSER VERLAG AG}</style></publisher><pub-location><style face="normal" font="default" size="100%">{VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND}</style></pub-location><volume><style face="normal" font="default" size="100%">69</style></volume><pages><style face="normal" font="default" size="100%">447-449</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We correct Theorem 3.2 and Corollary 3.3 from [KMS]. This correction ammounts to the observation that the proof of the main result in [KMS] contains a gap in Lemma~10.6 for \(p\ne 2\). The results of [KMS] are true for \(p=2\). &lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">3</style></issue><work-type><style face="normal" font="default" size="100%">{Correction}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu</style></author><author><style face="normal" font="default" size="100%">Spitkovsky, Ilya M.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Journal of Mathematical Analysis and Appliactions</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Fredholmness</style></keyword><keyword><style  face="normal" font="default" size="100%">Invertibility}</style></keyword><keyword><style  face="normal" font="default" size="100%">Semi-almost periodic function</style></keyword><keyword><style  face="normal" font="default" size="100%">Singular integral operator</style></keyword><keyword><style  face="normal" font="default" size="100%">slowly oscillating function</style></keyword><keyword><style  face="normal" font="default" size="100%">Variable Lebesgue space</style></keyword><keyword><style  face="normal" font="default" size="100%">{Almost-periodic function</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2011</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{DEC 15}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.sciencedirect.com/science/article/pii/S0022247X11006147</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{2}</style></number><publisher><style face="normal" font="default" size="100%">{ACADEMIC PRESS INC ELSEVIER SCIENCE}</style></publisher><pub-location><style face="normal" font="default" size="100%">{525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA}</style></pub-location><volume><style face="normal" font="default" size="100%">384</style></volume><pages><style face="normal" font="default" size="100%">706-725</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(a\) be a semi-almost periodic matrix function with the almost periodic representatives \(a_l\) and \(a_r\) at \(-\infty\) and \(+\infty\), respectively. Suppose \(p:\mathbb{R}\to(1,\infty)\) is a slowly oscillating exponent such that the Cauchy singular integral operator \(S\) is bounded on the variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R})\). We prove that if the operator \(aP+Q\) with \(P=(I+S)/2\) and \(Q=(I-S)/2\) is Fredholm on the variable Lebesgue space \(L_N^{p(\cdot)}(\mathbb{R})\), then the operators \(a_lP+Q\) and \(a_rP+Q\) are invertible on standard Lebesgue spaces \(L_N^{q_l}(\mathbb{R})\) and \(L_N^{q_r}(\mathbb{R})\) with some exponents \(q_l\) and \(q_r\) lying in the segments between the lower and the upper limits of \(p\) at \(-\infty\) and \(+\infty\), respectively. &lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">2</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino B.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Sufficient conditions for Fredholmness of singular integral operators with shifts and slowly oscillating data</style></title><secondary-title><style face="normal" font="default" size="100%">Integral Equations and Operator Theory</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Cauchy singular integral operator</style></keyword><keyword><style  face="normal" font="default" size="100%">Fredholmness}</style></keyword><keyword><style  face="normal" font="default" size="100%">Mellin pseudodifferential operator</style></keyword><keyword><style  face="normal" font="default" size="100%">slowly oscillating function</style></keyword><keyword><style  face="normal" font="default" size="100%">{Orientation-preserving non-Carleman shift</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2011</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{AUG}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.springerlink.com/content/cg76170280775q2t/</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{4}</style></number><publisher><style face="normal" font="default" size="100%">{BIRKHAUSER VERLAG AG}</style></publisher><pub-location><style face="normal" font="default" size="100%">{VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND}</style></pub-location><volume><style face="normal" font="default" size="100%">70</style></volume><pages><style face="normal" font="default" size="100%">451-483</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Suppose \(\alpha\) is an orientation preserving diffeomorphism (shift) of \(\mathbb{R}_+=(0,\infty)\)  onto itself with the only fixed points \(0\) and \(\infty\). We establish sufficient conditions for the Fredholmness of the singular integral operator with shift \[ (aI-bW_\alpha)P_++(cI-dW_\alpha)P_- \] acting on \(L^p(\mathbb{R}_+)\) with \( 1 &amp;lt; p &amp;lt; \infty \), where \(P_\pm=(I\pm S)/2\), \(S\) is the Cauchy singular integral operator, and \(W_\alpha f=f\circ\alpha\) is the shift operator, under the assumptions that the coefficients \(a,b,c,d\) and the derivative \(\alpha'\) of the shift are bounded and continuous on \(\mathbb{R}_+\) and may admit discontinuities of slowly oscillating type at \(0\) and \(\infty\). &lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">4</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Hudzik, Henryk</style></author><author><style face="normal" font="default" size="100%">Lewicki, Grzegorz</style></author><author><style face="normal" font="default" size="100%">Musielak, Julian</style></author><author><style face="normal" font="default" size="100%">Nowak, Marian</style></author><author><style face="normal" font="default" size="100%">Skrzypczak, Leszek</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Singular integral operators on Nakano spaces with weights having finite sets of discontinuities</style></title><secondary-title><style face="normal" font="default" size="100%">Function spaces IX. Proceedings of the 9th international conference, Kraków, Poland, July 6–11, 2009. Banach Center Publications, 92</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2011</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://journals.impan.pl/cgi-bin/doi?bc92-0-10</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Polish Academy of Sciences, Institute of Mathematics</style></publisher><pub-location><style face="normal" font="default" size="100%">Warszawa</style></pub-location><pages><style face="normal" font="default" size="100%">143-166</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;In 1968, Gohberg and Krupnik found a Fredholm criterion for singular integral operators of the form \(aP+bQ\), where \(a,b\) are piecewise continuous functions and \(P,Q\) are complementary projections associated to the Cauchy singular integral operator, acting on Lebesgue spaces over Lyapunov curves. We extend this result to the case of Nakano spaces (also known as variable Lebesgue spaces) with certain weights having finite sets of discontinuities on arbitrary Carleson curves. &lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Maximal operators on variable Lebesgue spaces with weights related to oscillations of Carleson curves</style></title><secondary-title><style face="normal" font="default" size="100%">Mathematische Nachrichten</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Carleson curve</style></keyword><keyword><style  face="normal" font="default" size="100%">Dini-Lipschitz condition</style></keyword><keyword><style  face="normal" font="default" size="100%">indices of submultiplicative function</style></keyword><keyword><style  face="normal" font="default" size="100%">oscillating weight</style></keyword><keyword><style  face="normal" font="default" size="100%">spirality indices}</style></keyword><keyword><style  face="normal" font="default" size="100%">weighted variable Lebesgue space</style></keyword><keyword><style  face="normal" font="default" size="100%">{Maximal operator</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2010</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{JAN}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://onlinelibrary.wiley.com/doi/10.1002/mana.200810295/abstract</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{1}</style></number><publisher><style face="normal" font="default" size="100%">{WILEY-V C H VERLAG GMBH}</style></publisher><pub-location><style face="normal" font="default" size="100%">{PO BOX 10 11 61, D-69451 WEINHEIM, GERMANY}</style></pub-location><volume><style face="normal" font="default" size="100%">283</style></volume><pages><style face="normal" font="default" size="100%">85-93</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We prove sufficient conditions for the boundedness of the maximal operator on variable Lebesgue spaces with weights \(\varphi_{t,\gamma}(\tau)=|(\tau-t)^\gamma|\), where \(\gamma\) is a complex number, over arbitrary Carleson curves. If the curve has different spirality indices at the point $t$ and \(\gamma\) is not real, then \(\varphi_{t,\gamma}\) is an oscillating weight lying beyond the class of radial oscillating weights considered recently by V. Kokilashvili, N. Samko, and S. Samko.&lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu</style></author><author><style face="normal" font="default" size="100%">Mascarenhas, Helena</style></author><author><style face="normal" font="default" size="100%">Santos, Pedro A.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Finite section method for a Banach algebra of convolution type operators on Lp(R) with symbols generated by PC and SO</style></title><secondary-title><style face="normal" font="default" size="100%">Integral Equations and Operator Theory</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">algebraization</style></keyword><keyword><style  face="normal" font="default" size="100%">essentialization</style></keyword><keyword><style  face="normal" font="default" size="100%">Fourier multiplier</style></keyword><keyword><style  face="normal" font="default" size="100%">homogenization</style></keyword><keyword><style  face="normal" font="default" size="100%">local principle</style></keyword><keyword><style  face="normal" font="default" size="100%">slowly oscillating function</style></keyword><keyword><style  face="normal" font="default" size="100%">two projections theorem}</style></keyword><keyword><style  face="normal" font="default" size="100%">{Finite section method</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2010</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{AUG}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.springerlink.com/content/d7436748p1767u22/</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{4}</style></number><publisher><style face="normal" font="default" size="100%">{BIRKHAUSER VERLAG AG}</style></publisher><pub-location><style face="normal" font="default" size="100%">{VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND}</style></pub-location><volume><style face="normal" font="default" size="100%">67</style></volume><pages><style face="normal" font="default" size="100%">559-600</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We prove necessary and sufficient conditions for the applicability of the finite section method to an arbitrary operator in the Banach algebra generated by the operators of multiplication by piecewise continuous functions and the convolution operators with symbols in the algebra generated by piecewise continuous and slowly oscillating Fourier multipliers on \(L^p(\mathbb{R})\), \(1 &amp;lt; p &amp;lt; \infty\). &lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">4</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Ball, JA</style></author><author><style face="normal" font="default" size="100%">Bolotnikov, V</style></author><author><style face="normal" font="default" size="100%">Helton, JW</style></author><author><style face="normal" font="default" size="100%">Rodman, L</style></author><author><style face="normal" font="default" size="100%">Spitkovsky, IM</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Singular integral operators on variable Lebesgue spaces over arbitrary Carleson curves</style></title><secondary-title><style face="normal" font="default" size="100%">Topics in Operator Theory: Operators, Matrices and Analytic Functions, Vol. 1. Operator Theory: Advances and Applications, 202</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2010</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.springerlink.com/content/wk281h0005423485/</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">321-336</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces \(L^p(\Gamma)\) over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht Böttcher and Yuri Karlovich realized that these circular arcs metamorphose to so-called logarithmic leaves with a median separating point when Lyapunov curves metamorphose to arbitrary Carleson curves. We show that this result remains valid in a more general setting of variable Lebesgue spaces \(L^{p(\cdot)}(\Gamma)\) where \(p:\Gamma\to(1,\infty)\) satisfies the Dini-Lipschitz condition. One of the main ingredients of the proof is a new condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with weights related to oscillations of Carleson curves.&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Grobler, JJ</style></author><author><style face="normal" font="default" size="100%">Labuschagne, LE</style></author><author><style face="normal" font="default" size="100%">Möller, M</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Singular integral operators on variable Lebesgue spaces with radial oscillating weights</style></title><secondary-title><style face="normal" font="default" size="100%">Operator Algebras, Operator Theory and Applications.Operator Theory Advances and Applications, 195 </style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2010</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.springerlink.com/content/hxh8002466k66236/</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">185-212</style></pages><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We prove a Fredholm criterion for operators in the Banach algebra of singular integral operators with matrix piecewise continuous coefficients acting on a variable Lebesgue space with a radial oscillating weight over a logarithmic Carleson curve. The local spectra of these operators are massive and have a shape of spiralic horns depending on the value of the variable exponent, the spirality indices of the curve, and the Matuszewska-Orlicz indices of the weight at each point. These results extend (partially) the results of A. Böttcher, Yu. Karlovich, and V. Rabinovich for standard Lebesgue spaces to the case of variable Lebesgue spaces.&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Spitkovsky, Ilya M.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Connectedness of spectra of Toeplitz operators on Hardy spaces with Muckenhoupt weights over Carleson curves</style></title><secondary-title><style face="normal" font="default" size="100%">Integral Equations and Operator Theory</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Carleson curve</style></keyword><keyword><style  face="normal" font="default" size="100%">essential spectrum</style></keyword><keyword><style  face="normal" font="default" size="100%">Hardy space</style></keyword><keyword><style  face="normal" font="default" size="100%">index</style></keyword><keyword><style  face="normal" font="default" size="100%">Muckenhoupt weight</style></keyword><keyword><style  face="normal" font="default" size="100%">Pettis integral}</style></keyword><keyword><style  face="normal" font="default" size="100%">spectrum</style></keyword><keyword><style  face="normal" font="default" size="100%">{Toeplitz operator</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2009</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{SEP}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.springerlink.com/content/h278jp7j82518877/</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{1}</style></number><publisher><style face="normal" font="default" size="100%">{BIRKHAUSER VERLAG AG}</style></publisher><pub-location><style face="normal" font="default" size="100%">{VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND}</style></pub-location><volume><style face="normal" font="default" size="100%">65</style></volume><pages><style face="normal" font="default" size="100%">83-114</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Harold Widom proved in 1966 that the spectrum of a Toeplitz operator \(T(a)\) acting on the Hardy space \(H^p(\mathbb{T})\) over the unit circle \(\mathbb{T}\) is a connected subset of the complex plane for every bounded measurable symbol \(a\) and \(1 &amp;lt; p &amp;lt; \infty\). In 1972, Ronald Douglas established the connectedness of the essential spectrum of \(T(a)\) on \(H^2(\mathbb{T})\). We show that, as was suspected, these results remain valid in the setting of Hardy spaces \(H^p(\Gamma,w)\), \( 1 &amp;lt; p &amp;lt; \infty \), with general Muckenhoupt weights \(w\) over arbitrary Carleson curves \(\Gamma\). &lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">1</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Adamyan, V</style></author><author><style face="normal" font="default" size="100%">Berezansky, Y</style></author><author><style face="normal" font="default" size="100%">Gohberg, I</style></author><author><style face="normal" font="default" size="100%">Gorbachuk, M</style></author><author><style face="normal" font="default" size="100%">Gorbachuk, V</style></author><author><style face="normal" font="default" size="100%">Kochubei, A</style></author><author><style face="normal" font="default" size="100%">Langer, H</style></author><author><style face="normal" font="default" size="100%">Popov, G</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Asymptotics of Toeplitz matrices with symbols in some generalized Krein algebras</style></title><secondary-title><style face="normal" font="default" size="100%">Modern Analysis and Applications: Mark Krein Centenary Conference, Vol. 1. Operator Theory Advances and Applications, 190</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2009</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-3-7643-9919-1_21</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">341-359</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(\alpha,\beta\in(0,1)\) and&lt;br /&gt;
\[&lt;br /&gt;
K^{\alpha,\beta}:=\left\{a\in L^\infty(\mathbb{T}):\&lt;br /&gt;
\sum_{k=1}^\infty |\widehat{a}(-k)|^2 k^{2\alpha}&amp;lt;\infty,\&lt;br /&gt;
\sum_{k=1}^\infty |\widehat{a}(k)|^2 k^{2\beta}&amp;lt;\infty&lt;br /&gt;
\right\}.&lt;br /&gt;
\]&lt;br /&gt;
Mark Krein proved in 1966 that \(K^{1/2,1/2}\) forms a Banach algebra. He also observed that this algebra is important in the asymptotic theory of finite Toeplitz matrices. Ten years later,  Harold Widom extended&lt;br /&gt;
earlier results of Gabor Szegö for scalar symbols and established the asymptotic trace formula&lt;br /&gt;
\[&lt;br /&gt;
\operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1)&lt;br /&gt;
\quad\text{as}\ n\to\infty&lt;br /&gt;
\]&lt;br /&gt;
for finite Toeplitz matrices \(T_n(a)\) with matrix symbols \(a\in K^{1/2,1/2}_{N\times N}\). We show that if \(\alpha+\beta\ge 1\) and \(a\in K^{\alpha,\beta}_{N\times N}\), then the Szegö-Widom asymptotic trace formula holds with \(o(1)\) replaced by \(o(n^{1-\alpha-\beta})\).&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Remark on the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights</style></title><secondary-title><style face="normal" font="default" size="100%">Journal of Function Spaces and Applications</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Carleson curve</style></keyword><keyword><style  face="normal" font="default" size="100%">Matuszewska-Orlicz indices</style></keyword><keyword><style  face="normal" font="default" size="100%">radial oscillating weight</style></keyword><keyword><style  face="normal" font="default" size="100%">submultiplicative function}</style></keyword><keyword><style  face="normal" font="default" size="100%">variable exponent</style></keyword><keyword><style  face="normal" font="default" size="100%">{Variable Lebesgue space</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2009</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.hindawi.com/journals/jfsa/2009/438146/abs/</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{3}</style></number><publisher><style face="normal" font="default" size="100%">{SCIENTIFIC HORIZON}</style></publisher><pub-location><style face="normal" font="default" size="100%">{27, KANCHANCHANGA 90, I P EXTENSION, DELHI, 110-092, INDIA}</style></pub-location><volume><style face="normal" font="default" size="100%">7</style></volume><pages><style face="normal" font="default" size="100%">301-311</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result.&lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type><notes><style face="normal" font="default" size="100%">&lt;p&gt;n/a&lt;/p&gt;
</style></notes></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Joseph A. Ball</style></author><author><style face="normal" font="default" size="100%">Yuli Eidelman</style></author><author><style face="normal" font="default" size="100%">J. William Helton</style></author><author><style face="normal" font="default" size="100%">Vadim Olshevsky</style></author><author><style face="normal" font="default" size="100%">James Rovnyak</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Higher order asymptotic formulas for traces of Toeplitz matrices with symbols in Hölder-Zygmund spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Recent Advances in Matrix and Operator Theory. Operator Theory: Advances and Applications, 179</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2008</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-3-7643-8539-2_11</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Bikhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">185-196</style></pages><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We prove a higher order asymptotic formula for traces of finite block Toeplitz matrices with symbols belonging to Hölder-Zygmund spaces. The remainder in this formula goes to zero very rapidly for very smooth symbols. This formula refines previous asymptotic trace formulas by Szegő and Widom and complement higher order asymptotic formulas for determinants of finite block Toeplitz matrices due to Böttcher and Silbermann.&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Bastos, MA</style></author><author><style face="normal" font="default" size="100%">Gohberg, I</style></author><author><style face="normal" font="default" size="100%">Lebre, AB</style></author><author><style face="normal" font="default" size="100%">Speck, FO</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Higher-order asymptotic formulas for Toeplitz matrices with symbols in generalized Hölder spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Operator Algebra, Operator Theory and Applications. Operator Theory Advances and Applications, 181</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2008</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-3-7643-8684-9_10</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">207-228</style></pages><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We prove higher-order asymptotic formulas for determinants and traces of finite block Toeplitz matrices generated by matrix functions belonging to generalized Hölder spaces with characteristic functions from the Bari-Stechkin class. We follow the approach of Böttcher and Silbermann and generalize their results for symbols in standard Hölder spaces.&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Maligranda, L</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Kato, M.</style></author><author><style face="normal" font="default" size="100%">Maligranda, L</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">On the interpolation constant for subadditive operators in Orlicz spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Proceedings of the International Symposium on Banach and Function Spaces II (ISBFS 2006), Kyushu Institute of Technology, Kitakyushu, Japan, 14-17 September 2006.</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2008</style></year></dates><publisher><style face="normal" font="default" size="100%">Yokohama Publishers</style></publisher><pub-location><style face="normal" font="default" size="100%">Yokohama</style></pub-location><pages><style face="normal" font="default" size="100%">85-101</style></pages></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Michael A. Dritschel</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Algebras of singular integral operators with piecewise continuous coefficients on weighted Nakano spaces</style></title><secondary-title><style face="normal" font="default" size="100%">The Extended Field of Operator Theory. Operator Theory: Advances and Applications, 171</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2007</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-3-7643-7980-3_9</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">171-188</style></pages><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We find Fredholm criteria and a formula for the index of an arbitrary operator in the Banach algebra of singular integral operators with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over either Lyapunov curves or Radon curves without cusps. These results ``localize'' the Gohberg-Krupnik Fredhohn theory with respect to the variable exponent.&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Asymptotics of block Toeplitz determinants generated by factorable matrix functions with equal partial indices</style></title><secondary-title><style face="normal" font="default" size="100%">Mathematische Nachrichten</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">determinant</style></keyword><keyword><style  face="normal" font="default" size="100%">nonvanishing partial indices}</style></keyword><keyword><style  face="normal" font="default" size="100%">strong Szego-Widom limit theorem</style></keyword><keyword><style  face="normal" font="default" size="100%">weighted Wiener algebra</style></keyword><keyword><style  face="normal" font="default" size="100%">Wiener-Hopf factorization</style></keyword><keyword><style  face="normal" font="default" size="100%">{block Toeplitz matrix</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2007</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://onlinelibrary.wiley.com/doi/10.1002/mana.200510540/abstract</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{9-10}</style></number><publisher><style face="normal" font="default" size="100%">{WILEY-V C H VERLAG GMBH}</style></publisher><pub-location><style face="normal" font="default" size="100%">{PO BOX 10 11 61, D-69451 WEINHEIM, GERMANY}</style></pub-location><volume><style face="normal" font="default" size="100%">280</style></volume><pages><style face="normal" font="default" size="100%">1118-1127</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We prove asymptotic formulas for block Toeplitz matrices with symbols admitting right and left Wiener-Hopf factorizations such that all partial indices are equal to some integer number. We consider symbols and Wiener-Hopf factorizations in Wiener algebras with weights satisfying natural submultiplicativity, monotonicity, and regularity conditions. Our results complement known formulas for Holder continuous symbols due to Bottcher and Silbermann. &lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Asymptotics of determinants and traces of Toeplitz matrices with symbols in weighted Wiener algebras</style></title><secondary-title><style face="normal" font="default" size="100%">Zeitschrift für Analysis und ihre Anwendungen</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">canonical Wiener-Hopf factorization</style></keyword><keyword><style  face="normal" font="default" size="100%">determinant</style></keyword><keyword><style  face="normal" font="default" size="100%">strong Szego-Widom limit theorem}</style></keyword><keyword><style  face="normal" font="default" size="100%">trace</style></keyword><keyword><style  face="normal" font="default" size="100%">weighted Wiener algebra</style></keyword><keyword><style  face="normal" font="default" size="100%">{block Toeplitz matrix</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2007</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.ems-ph.org/journals/show_abstract.php?issn=0232-2064&amp;vol=26&amp;iss=1&amp;rank=3</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{1}</style></number><publisher><style face="normal" font="default" size="100%">{HELDERMANN VERLAG}</style></publisher><pub-location><style face="normal" font="default" size="100%">{LANGER GRABEN 17, 32657 LEMGO, GERMANY}</style></pub-location><volume><style face="normal" font="default" size="100%">26</style></volume><pages><style face="normal" font="default" size="100%">43-56</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We prove asymptotic formulas for determinants and traces of finite block Toeplitz matrices with symbols belonging to Wiener algebras with weights satisfying natural submultiplicativity, monotonicity, and regularity conditions. The remainders in these formulas depend on the weights and go rapidly to zero for very smooth symbols. These formulas refine or extend some previous results by Szegö, Widom, Bottcher, and Silbermann.&lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">1</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Jarosz, K</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Asymptotics of Toeplitz determinants generated by functions with Fourier coefficients in weighted Orlicz sequence classes</style></title><secondary-title><style face="normal" font="default" size="100%">Function Spaces. Contemporary Mathematics, 435</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2007</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.ams.org/books/conm/435/8380/conm435-8380.pdf</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">American Mathematical Society</style></publisher><pub-location><style face="normal" font="default" size="100%">Providence, RI</style></pub-location><pages><style face="normal" font="default" size="100%">229-243</style></pages><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We prove asymptotic formulas for Toeplitz determinants generated by functions with sequences of Fourier coefficients belonging to weighted Orlicz sequence classes. We concentrate our attention on the case of nonvanishing generating functions with nonzero Cauchy index.&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Böttcher, Albrecht</style></author><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Bernd Silbermann</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Generalized Krein algebras and asymptotics of Toeplitz determinants</style></title><secondary-title><style face="normal" font="default" size="100%">Methods of Functional Analysis and Topology</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2007</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://mfat.imath.kiev.ua/</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">13</style></volume><pages><style face="normal" font="default" size="100%">236-261</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We give a survey on generalized Krein algebras \(K_{p,q}^{\alpha,\beta}\) and their applications to Toeplitz determinants. Our methods originated in a paper by Mark Krein of 1966, where he showed that \(K_{2,2}^{1/2,1/2}\) is a Banach algebra. Subsequently, Widom proved the strong Szeg\H{o} limit theorem for block Toeplitz determinants with symbols in \((K_{2,2}^{1/2,1/2})_{N\times N}\) and later two of the authors studied symbols in the generalized Krein algebras \((K_{p,q}^{\alpha,\beta})_{N\times N}\), where \(\lambda:=1/p+1/q=\alpha+\beta\) and \(\lambda=1\). We here extend these results to \(0&amp;lt;\lambda&amp;lt;1\). The entire paper is based on fundamental work by Mark Krein, ranging from operator ideals through Toeplitz operators up to Wiener-Hopf factorization.&lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">2</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm</style></title><secondary-title><style face="normal" font="default" size="100%">Operators and Matrices</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Carleson curve</style></keyword><keyword><style  face="normal" font="default" size="100%">generalized Lebesgue space with variable exponent</style></keyword><keyword><style  face="normal" font="default" size="100%">Khvedelidze weight</style></keyword><keyword><style  face="normal" font="default" size="100%">singular integral operator}</style></keyword><keyword><style  face="normal" font="default" size="100%">{Semi-Fredholm operator</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2007</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://oam.ele-math.com/01-25/Semi-Fredholm-singular-integral-operators-with-piecewise-continuous-coefficients-on-weighted-variable-Lebesgue-spaces-are-Fredholm</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{3}</style></number><publisher><style face="normal" font="default" size="100%">{ELEMENT}</style></publisher><pub-location><style face="normal" font="default" size="100%">{R AUSTRIJE 11, 10000 ZAGREB, CROATIA}</style></pub-location><volume><style face="normal" font="default" size="100%">1</style></volume><pages><style face="normal" font="default" size="100%">427-444</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Suppose \(\Gamma\) is a Carleson Jordan curve with logarithmic whirl points, \(\varrho\) is a Khvedelidze weight, \(p:\Gamma\to(1,\infty)\) is a continuous function satisfying \(|p(\tau)-p(t)|\le -\mathrm{const}/\log|\tau-t|\) for \(|\tau-t|\le 1/2\), and \(L^{p(\cdot)}(\Gamma,\varrho)\) is a weighted generalized Lebesgue space with variable exponent. We prove that all semi-Fredholm operators in the algebra of singular integral operators with \(N\times N\) matrix piecewise continuous coefficients are Fredholm on \(L_N^{p(\cdot)}(\Gamma,\varrho)\).&lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">3</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, AY</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Higher order asymptotics of Toeplitz determinants with symbols in weighted Wiener algebras</style></title><secondary-title><style face="normal" font="default" size="100%">Journal of Mathematical Analysis and Applications</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">canonical Wiener-Hopf factorization</style></keyword><keyword><style  face="normal" font="default" size="100%">p-regularized operator determinant}</style></keyword><keyword><style  face="normal" font="default" size="100%">Schatten-von Neumann class</style></keyword><keyword><style  face="normal" font="default" size="100%">strong Szego-Widom limit theorem</style></keyword><keyword><style  face="normal" font="default" size="100%">weighted Wiener algebra</style></keyword><keyword><style  face="normal" font="default" size="100%">{Toeplitz determinant</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2006</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{AUG 15}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.sciencedirect.com/science/article/pii/S0022247X05007651</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{2}</style></number><publisher><style face="normal" font="default" size="100%">{ACADEMIC PRESS INC ELSEVIER SCIENCE}</style></publisher><pub-location><style face="normal" font="default" size="100%">{525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA}</style></pub-location><volume><style face="normal" font="default" size="100%">320</style></volume><pages><style face="normal" font="default" size="100%">944-963</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We extend a result of Bottcher and Silbermann on higher order asymptotics of determinants of block Toeplitz matrices with symbols in Wiener algebras with power weights to the case of Wiener algebras with general weights satisfying natural submultiplicativity, monotonicity, and regularity conditions. &lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">2</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Algebras of singular integral operators on Nakano spaces with Khvedelidze weights over Carleson curves with logarithmic whirl points</style></title><secondary-title><style face="normal" font="default" size="100%">Izvestiya Vysshih Uchebnyh Zavedeniy. Severo-Kavkazskiy Region. Estestvennye Nauki. Special Issue &quot;Pseudodifferential equations and some problems of mathematical physics&quot;</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2005</style></year></dates><urls><related-urls><url><style face="normal" font="default" size="100%">https://docentes.fct.unl.pt/sites/default/files/oyk/files/22_2005_simonenko-70.pdf</style></url></related-urls></urls><publisher><style face="normal" font="default" size="100%">Rostov University Press</style></publisher><pub-location><style face="normal" font="default" size="100%">Rostov-on-Don</style></pub-location><pages><style face="normal" font="default" size="100%">135-142</style></pages><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We establish a Fredholm criterion for an arbitrary operator in the Banach algebra of singular integral operators&lt;br /&gt;
with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over Carleson curves with logarithmic whirl points.&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Lerner, Andrei K.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Commutators of singular integrals on generalized Lp spaces with variable exponent</style></title><secondary-title><style face="normal" font="default" size="100%">Publicacions Matematiques</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">BMO</style></keyword><keyword><style  face="normal" font="default" size="100%">Calderon-Zygmund singular integral</style></keyword><keyword><style  face="normal" font="default" size="100%">generalized L-P space with variable exponent</style></keyword><keyword><style  face="normal" font="default" size="100%">local sharp maximal function}</style></keyword><keyword><style  face="normal" font="default" size="100%">{commutator</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2005</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://dx.doi.org/10.5565/PUBLMAT_49105_05</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{1}</style></number><publisher><style face="normal" font="default" size="100%">{UNIV AUTONOMA BARCELONA}</style></publisher><pub-location><style face="normal" font="default" size="100%">{DEPT MATHEMATICS, 08193 BELLATERRA, SPAIN}</style></pub-location><volume><style face="normal" font="default" size="100%">49</style></volume><pages><style face="normal" font="default" size="100%">111-125</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized Lp spaces with variable exponent.&lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">1</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Santos, Pedro A.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On asymptoties of Toeplitz determinants with symbols of nonstandard smoothness</style></title><secondary-title><style face="normal" font="default" size="100%">Journal of Fourier Aanalysis and Applications</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">N-function</style></keyword><keyword><style  face="normal" font="default" size="100%">Szego's strong limit theorem</style></keyword><keyword><style  face="normal" font="default" size="100%">weighted Orlicz sequence class}</style></keyword><keyword><style  face="normal" font="default" size="100%">Wiener algebra</style></keyword><keyword><style  face="normal" font="default" size="100%">{Toeplitz determinant</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2005</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/article/10.1007/s00041-004-3071-0</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{1}</style></number><publisher><style face="normal" font="default" size="100%">{BIRKHAUSER BOSTON INC}</style></publisher><pub-location><style face="normal" font="default" size="100%">{675 MASSACHUSETTS AVE, CAMBRIDGE, MA 02139 USA}</style></pub-location><volume><style face="normal" font="default" size="100%">11</style></volume><pages><style face="normal" font="default" size="100%">43-72</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We prove Szegö's strong limit theorem for Toeplitz determinants with a symbol having a nonstandard smoothness. We assume that the symbol belongs to the Wiener algebra and, moreover, the sequences of Fourier coefficients of the symbol with negative and nonnegative indices belong to weighted Orlicz classes generated by complementary \(N\)-functions both satisfying the \(\Delta_2^0\)-condition and by weight sequences satisfying some regularity, and compatibility conditions. &lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">1</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Norms of Toeplitz and Hankel operators on Hardy type subspaces of rearrangement-invariant spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Integral Equations and Operator Theory</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Boyd indices</style></keyword><keyword><style  face="normal" font="default" size="100%">Hankel operator</style></keyword><keyword><style  face="normal" font="default" size="100%">Lozanovskii factorization}</style></keyword><keyword><style  face="normal" font="default" size="100%">rearrangement-invariant space</style></keyword><keyword><style  face="normal" font="default" size="100%">{Toeplitz operator</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2004</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{MAY}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/article/10.1007/s00020-002-1190-z</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{1}</style></number><publisher><style face="normal" font="default" size="100%">{BIRKHAUSER VERLAG AG}</style></publisher><pub-location><style face="normal" font="default" size="100%">{VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND}</style></pub-location><volume><style face="normal" font="default" size="100%">49</style></volume><pages><style face="normal" font="default" size="100%">43-64</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;We prove analogues of the Brown-Halmos and Nehari theorems on the norms of Toeplitz and Hankel operators, respectively, acting on subspaces of Hardy type of reflexive rearrangement-invariant spaces with nontrivial Boyd indices.&lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Kadets, V.</style></author><author><style face="normal" font="default" size="100%">Zelazko, W.</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Singular integral operators with flip and unbounded coefficients on rearrangement-invariant spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Functional Analysis and its Applications. Proceedings of the international conference, dedicated to the 110th anniversary of Stefan Banach, Lviv National University, Lviv, Ukraine, May 28--31, 2002</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2004</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.sciencedirect.com/science/article/pii/S0304020804801610</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Elsevier</style></publisher><pub-location><style face="normal" font="default" size="100%">Amsterdam</style></pub-location><pages><style face="normal" font="default" size="100%">123-131</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We prove Fredholm criteria for singular integral operators of the form&lt;br /&gt;
\[&lt;br /&gt;
P_++M_bP_-+M_uUP_-,&lt;br /&gt;
\]&lt;br /&gt;
where \(P_\pm\) are the Riesz projections, \(U\) is the flip operator, and \(M_b,M_u\) are operators of multiplication by functions \(b,u\), respectively, on a reflexive rearrangement-invariant space with nontrivial Boyd indices over the unit circle. We assume a priori that \(M_b\) is bounded, but \(M_u\) may be unbounded. The function \(u\) belongs to a class of, in general, unbounded functions that relates to the Douglas algebra \(H^\infty+C\). &lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Gohberg, Israel</style></author><author><style face="normal" font="default" size="100%">Wendland, Wolfgang</style></author><author><style face="normal" font="default" size="100%">Ferreira dos Santos, António</style></author><author><style face="normal" font="default" size="100%">Speck, Frank-Ollme</style></author><author><style face="normal" font="default" size="100%">Teixeira, Francisco Sepúlveda</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Some algebras of functions with Fourier coefficients in weighted Orlicz sequence spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Operator Theoretical Methods and Applications to Mathematical Physics. The Erhard Meister Memorial Volume. Operator Theory: Advances and Applications, 147</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2004</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-3-0348-7926-2_31</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">287-296</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;In this paper, the author proves that the set of all integrable functions whose sequences of negative (resp. nonnegative) Fourier coefficients belong to \(\ell^1\cap\ell^\Phi_{\varphi,w}\) (resp. to \(\ell^1\cap\ell^\Psi_{\psi,\varrho}\)), where \(\ell^\Phi_{\varphi,w}\) and \(\ell^\Psi_{\psi,\varrho}\) are two-weighted Orlicz sequence spaces, forms an algebra under pointwise multiplication whenever the weight sequences&lt;br /&gt;
\[&lt;br /&gt;
\varphi=\{\varphi_n\},\quad&lt;br /&gt;
\psi=\{\psi_n\},\quad&lt;br /&gt;
w=\{w_n\},\quad&lt;br /&gt;
\varrho=\{\varrho_n\}&lt;br /&gt;
\]&lt;br /&gt;
increase and satisfy the \(\Delta_2\)-condition.&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Samko, Stefan</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino</style></author><author><style face="normal" font="default" size="100%">Ferreira dos Santos, António</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Compactness of commutators arising in the Fredholm theory of singular integral operators with shifts</style></title><secondary-title><style face="normal" font="default" size="100%">Factorization, Singular Operators and Related Problems</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Boyd indices</style></keyword><keyword><style  face="normal" font="default" size="100%">commutator</style></keyword><keyword><style  face="normal" font="default" size="100%">compact operator</style></keyword><keyword><style  face="normal" font="default" size="100%">interpolation of compactness}</style></keyword><keyword><style  face="normal" font="default" size="100%">rearrangement-invariant space</style></keyword><keyword><style  face="normal" font="default" size="100%">shift operator</style></keyword><keyword><style  face="normal" font="default" size="100%">{Cauchy singular integral operator</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2003</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-94-017-0227-0_9</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Kluwer Academic Publishers</style></publisher><pub-location><style face="normal" font="default" size="100%">Dordrecht</style></pub-location><pages><style face="normal" font="default" size="100%">111-129</style></pages><isbn><style face="normal" font="default" size="100%">{1-4020-1407-4}</style></isbn><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;The paper is devoted to the compactness of the commutators \(aS_\Gamma - S_\Gamma aI\) and \(W_\alpha S_\Gamma - S_\Gamma W_\alpha\), where \(S_\Gamma\) is the Cauchy singular integral operator, \(a\) is a bounded measurable function, \(W_\alpha\) is the shift operator given by \(W_\alpha f = f\circ\alpha\), and \(\alpha\) is a bi-Lipschitz homeomorphism (shift). The cases of the unit circle and the unit interval are considered. We prove that these commutators are compact on rearrangement-invariant spaces with nontrivial Boyd indices if and only if the function a or, respectively, the derivative of the shift a has vanishing mean oscillation.&lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Proceedings Paper}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Journal of Integral Equations and Applications</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2003</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://projecteuclid.org/euclid.jiea/1181074970</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">15</style></volume><pages><style face="normal" font="default" size="100%">263-320</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We prove necessary conditions for the Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of the curve, and of the weight. As an example, we consider weighted Nakano spaces \(L^{p(\cdot)}_w\)  (weighted Lebesgue spaces with variable exponent).  Moreover, our necessary conditions become also sufficient for weighted Nakano spaces over nice curves whenever \(w\) is a Khvedelidze weight, and the variable exponent \(p(t)\) satisfies the estimate \(|p(\tau)-p(t)|\le A/(-\log|\tau-t|)\). &lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">3</style></issue></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author><author><style face="normal" font="default" size="100%">Lebre, Amarino B.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Böttcher, Albrecht</style></author><author><style face="normal" font="default" size="100%">Marinus A. Kaashoek</style></author><author><style face="normal" font="default" size="100%">Amarino Brites Lebre</style></author><author><style face="normal" font="default" size="100%">Ferreira dos Santos, António</style></author><author><style face="normal" font="default" size="100%">Frank-Olme Speck</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">Invertibility of functional operators with slowly oscillating non-Carleman shifts</style></title><secondary-title><style face="normal" font="default" size="100%">Singular Integral Operators, Factorization and Applications. Operator Theory: Advances and Applications, 142</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2003</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-3-0348-8007-7_9</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">147-174</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We prove criteria for the invertibility of the binomial functional operator&lt;br /&gt;
\[&lt;br /&gt;
A=aI-bW_\alpha&lt;br /&gt;
\]&lt;br /&gt;
in the Lebesgue spaces \(L^p(0,1)\), \( 1 &amp;lt; p &amp;lt; \infty\), where \(a\) and \(b\) are continuous functions on \((0,1)\), \(I\) is the identity operator, \(W_\alpha\) is the shift operator, \(W_\alpha f=f\circ\alpha\), generated by a non-Carleman shift \(\alpha:[0,1]\to[0,1]\) which has only two fixed points \(0\) and \(1\). We suppose that \(\log\alpha'\) is bounded and continuous on \((0,1)\) and that \(a,b,\alpha'\) slowly oscillate at \(0\) and \(1\). The main difficulty connected with slow oscillation is overcome by using the method of limit operators.&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Algebras of singular integral operators with PC coefficients in rearrangement-invariant spaces with Muckenhoupt weights</style></title><secondary-title><style face="normal" font="default" size="100%">Journal of Operator Theory</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Fredholmness</style></keyword><keyword><style  face="normal" font="default" size="100%">Muckenhoupt weight}</style></keyword><keyword><style  face="normal" font="default" size="100%">rearrangement-invariant space</style></keyword><keyword><style  face="normal" font="default" size="100%">{singular integral operator</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2002</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{SPR}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.theta.ro/jot/archive/2002-047-002/2002-047-002-004.html</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{2}</style></number><publisher><style face="normal" font="default" size="100%">{THETA FOUNDATION}</style></publisher><pub-location><style face="normal" font="default" size="100%">{C/O INST MATHEMATICS, PO BOX 1-764, BUCHAREST 70700, ROMANIA}</style></pub-location><volume><style face="normal" font="default" size="100%">47</style></volume><pages><style face="normal" font="default" size="100%">303-323</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;In this paper we extend results on Fredholmness of singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces \(X(\Gamma)\) with nontrivial Boyd indices \(\alpha_X,\beta_X\) [K98] to the weighted case. Suppose a weight \(w\) belongs to the Muckenhoupt classes \(A_{\frac{1}{\alpha_X}}(\Gamma)\) and \(A_{\frac{1}{\beta_X}}(\Gamma)\). We prove that these conditions guarantee the boundedness of the Cauchy singular integral operator \(S\) in the weighted rearrangement-invariant space \(X(\Gamma,w)\). Under a ``disintegration condition'' we construct a symbol calculus for the Banach algebra generated by singular integral operators with matrix-valued piecewise continuous coefficients and get a formula for the index of an arbitrary operator from this algebra. We give nontrivial examples of spaces, for which this ``disintegration condition'' is satisfied. One of such spaces is a Lebesgue space with a general Muckenhoupt weight over an arbitrary Carleson curve.&lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">One-sided invertibility of binomial functional operators with a shift on rearrangement-invariant spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Integral Equations and Operator Theory</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2002</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{FEB}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/article/10.1007/BF01275516</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{2}</style></number><publisher><style face="normal" font="default" size="100%">{BIRKHAUSER VERLAG AG}</style></publisher><pub-location><style face="normal" font="default" size="100%">{VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND}</style></pub-location><volume><style face="normal" font="default" size="100%">42</style></volume><pages><style face="normal" font="default" size="100%">201-228</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(\Gamma\) be an oriented Jordan smooth curve and \(\alpha\) a diffeomorphism of $\Gamma$ onto itself which has an arbitrary nonempty set of periodic points. We prove criteria for one-sided invertibility of the binomial functional operator \(A=aI-bW\) where $a$ and $b$ are continuous functions, \(I\) is the identity operator, \(W\) is the shift operator, \(Wf=f\circ\alpha\), on a reflexive rearrangement-invariant space \(X(\Gamma)\) with Boyd indices \(\alpha_X,\beta_X\) and Zippin indices \(p_X,q_X\) satisfying inequalities&lt;br /&gt;
\[&lt;br /&gt;
0&amp;lt;\alpha_X=p_X\le q_X=\beta_X&amp;lt;1.&lt;br /&gt;
\]&lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Algebras of singular integral operators on rearrangement-invariant spaces and Nikolski ideals</style></title><secondary-title><style face="normal" font="default" size="100%">The New York Journal of Mathematics</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2002</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.emis.de/journals/NYJM/j/2002/8-14.html</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">8</style></volume><pages><style face="normal" font="default" size="100%">215-234</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We construct a presymbol for the Banach algebra \(\operatorname{alg}(\Omega, S)\) generated by the Cauchy singular integral operator \(S\) and the operators of multiplication by functions in a Banach subalgebra \(\Omega\) of \(L^\infty\). This presymbol is a homomorphism \(\operatorname{alg}(\Omega,S)\to\Omega\oplus\Omega\) whose kernel coincides with the commutator ideal of \(\operatorname{alg}(\Omega,S)\). In terms of the presymbol, necessary conditions for Fredholmness of an operator in \(\operatorname{alg}(\Omega,S)\) are proved. All operators are considered on reflexive rearrangement-invariant spaces with nontrivial Boyd indices over the unit circle.&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Karlovich, Yuri I.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Invertibility in Banach algebras of functional operators with non-Carleman shifts</style></title><secondary-title><style face="normal" font="default" size="100%">Ukrains'kyj matematychnyj kongres -- 2001. Pratsi. Sektsiya 11. Funktsional'nyj analiz</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2002</style></year></dates><urls><related-urls><url><style face="normal" font="default" size="100%">https://docentes.fct.unl.pt/sites/default/files/oyk/files/13_2002_ukrainian_math_congress-kyiv-01.pdf</style></url></related-urls></urls><publisher><style face="normal" font="default" size="100%">Instytut Matematyky NAN Ukrainy</style></publisher><pub-location><style face="normal" font="default" size="100%">Kyiv</style></pub-location><pages><style face="normal" font="default" size="100%">107-124</style></pages><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We prove the inverse closedness of the Banach algebra \(\mathfrak{A}_p\) of functional operators with non-Carleman shifts, which have only two fixed points, in the Banach algebra of all bounded linear operators on \(L^p\). We suppose that \(1 \le p \le \infty\) and the generators of the algebra \(\mathfrak{A}_p\) have essentially bounded data. An invertibility criterion for functional operators in \(\mathfrak{A}_p\) is obtained in terms of&lt;br /&gt;
the invertibility of a family of discrete operators on \(l^p\). An effective invertibility criterion  is established for binomial difference operators with \(l^\infty\) coefficients on the spaces \(l^p\). Using the reduction to binomial difference operators, we give effective criteria of invertibility  for binomial functional operators on the spaces \(L^p\).&lt;/p&gt;
</style></abstract></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Criteria for one-sided invertibility of a functional operator in rearrangement-invariant spaces of fundamental type</style></title><secondary-title><style face="normal" font="default" size="100%">Mathematische Nachrichten</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Boyd indices</style></keyword><keyword><style  face="normal" font="default" size="100%">one-sided invertibility</style></keyword><keyword><style  face="normal" font="default" size="100%">rearrangement-invariant space of fundamental type</style></keyword><keyword><style  face="normal" font="default" size="100%">Zippin indices}</style></keyword><keyword><style  face="normal" font="default" size="100%">{functional operator with shift</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2001</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://onlinelibrary.wiley.com/doi/10.1002/1522-2616(200109)229:1%3C91::AID-MANA91%3E3.0.CO;2-X/abstract</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">{WILEY-V C H VERLAG GMBH}</style></publisher><pub-location><style face="normal" font="default" size="100%">{PO BOX 10 11 61, D-69451 WEINHEIM, GERMANY}</style></pub-location><volume><style face="normal" font="default" size="100%">229</style></volume><pages><style face="normal" font="default" size="100%">91-118</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;Let \(\gamma\) be a simple open smooth curve and \(\alpha\) be an orientation-preserving diffeomorphism of \(\gamma\) onto itself which has only two fixed points. Criteria for one-sided invertibility of the functional operator&lt;br /&gt;
\[&lt;br /&gt;
A=aI-bW,&lt;br /&gt;
\]&lt;br /&gt;
where \(a\) and \(b\) are continuous functions, \(I\) is the identity operator, \(W\) is the shift operator: \((Wf)(t)=f[\alpha(t)]\), in a reflexive rearrangement-invariant space of fundamental type \(X(\gamma)\) with nontrivial Boyd indices, are obtained. &lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Maligranda, Lech</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On the interpolation constant for Orlicz spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Proceedings of the American Mathematical Society</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">concave function}</style></keyword><keyword><style  face="normal" font="default" size="100%">convex function</style></keyword><keyword><style  face="normal" font="default" size="100%">interpolation constant</style></keyword><keyword><style  face="normal" font="default" size="100%">interpolation of operators</style></keyword><keyword><style  face="normal" font="default" size="100%">K-functional</style></keyword><keyword><style  face="normal" font="default" size="100%">{Orlicz spaces</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2001</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.ams.org/journals/proc/2001-129-09/S0002-9939-01-06162-7/</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{9}</style></number><publisher><style face="normal" font="default" size="100%">{AMER MATHEMATICAL SOC}</style></publisher><pub-location><style face="normal" font="default" size="100%">{201 CHARLES ST, PROVIDENCE, RI 02940-2213 USA}</style></pub-location><volume><style face="normal" font="default" size="100%">129</style></volume><pages><style face="normal" font="default" size="100%">2727-2739</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;In this paper we deal with the interpolation from Lebesgue spaces \(L^p\) and \(L^q\), into an Orlicz space \(L^\varphi\), where \( 1 \le p &amp;lt; q \le \infty \) and \(\varphi^{-1}(t)=t^{1/p}\rho(t^{1/q-1/p})\) for some concave function \(\rho\), with the special attention to the interpolation constant \(C\). For a bounded linear operator \(T\) in \(L^p\) and \(L^q\), we prove modular inequalities, which allow us to get the estimate, for both, the Orlicz norm and the Luxemburg norm,&lt;br /&gt;
\[&lt;br /&gt;
\|T\|_{L^\varphi\to L^\varphi}&lt;br /&gt;
\le C\max\Big\{&lt;br /&gt;
\|T\|_{L^p\to L^p},&lt;br /&gt;
\|T\|_{L^q\to L^q}&lt;br /&gt;
\Big\},&lt;br /&gt;
\]&lt;br /&gt;
where the interpolation constant \(C\) depends only on \(p\) and \(q\). We give estimates for \(C\), which imply \(C&amp;lt;4\). Moreover, if either \( 1 &amp;lt; p &amp;lt; q \le 2\) or \( 2 \le p &amp;lt; q &amp;lt; \infty\), then \(C&amp;lt; 2\). If \(q=\infty\), then \(C\le 2^{1-1/p}\), and, in particular, for the case \(p=1\) this gives the classical Orlicz interpolation theorem with the constant \(C=1\).&lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Integral Equations and Operator Theory</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2000</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{SEP}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/article/10.1007/BF01192300</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{1}</style></number><publisher><style face="normal" font="default" size="100%">{BIRKHAUSER VERLAG AG}</style></publisher><pub-location><style face="normal" font="default" size="100%">{VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND}</style></pub-location><volume><style face="normal" font="default" size="100%">38</style></volume><pages><style face="normal" font="default" size="100%">28-50</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;In this paper we extend necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients in rearrangement-invariant spaces [K98] to the weighted case \(X(\Gamma,w)\). These conditions are formulated in terms of indices \(\alpha(Q_tw)\) and \(\beta(Q_tw)\) of a submultiplicative function \(Q_tw\), which is associated with local properties of the space, of the curve, and of the weight at the point \(t\Gamma\). Using these results we obtain a lower estimate for the essential norm \(S\) of the Cauchy singular integral operator \(S\) in reflexive weighted rearrangement-invariant spaces \(X(\Gamma, w)\) over arbitrary Carleson curves \(\Gamma\):&lt;br /&gt;
\[&lt;br /&gt;
|S|\ge\cot(\pi\lambda_{\Gamma,w}/2)&lt;br /&gt;
\]&lt;br /&gt;
where \(\lambda_{\Gamma,w} :=inf_{t\in\Gamma} min\{\alpha(Q_tw), 1 - \beta(Q_tw)\}\). In some cases we give formulas for computation of \(\alpha(Q_tw)\) and \(\beta(Q_tw)\). &lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">The index of singular integral operators in reflexive Orlicz spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Mathematical Notes</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Carleson curve</style></keyword><keyword><style  face="normal" font="default" size="100%">cusp</style></keyword><keyword><style  face="normal" font="default" size="100%">index formula</style></keyword><keyword><style  face="normal" font="default" size="100%">logarithmic spiral}</style></keyword><keyword><style  face="normal" font="default" size="100%">Singular integral operator</style></keyword><keyword><style  face="normal" font="default" size="100%">symbolic calculus</style></keyword><keyword><style  face="normal" font="default" size="100%">{reflexive Orlicz space</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">1998</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{SEP-OCT}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/article/10.1007%2FBF02314841</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{3-4}</style></number><publisher><style face="normal" font="default" size="100%">{PLENUM PUBL CORP}</style></publisher><pub-location><style face="normal" font="default" size="100%">{CONSULTANTS BUREAU, 233 SPRING ST, NEW YORK, NY 10013 USA}</style></pub-location><volume><style face="normal" font="default" size="100%">64</style></volume><pages><style face="normal" font="default" size="100%">330-341</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We consider the Banach algebra \(\mathfrak{A}\) of singular integral operators with matrix piecewise continuous coefficients in the reflexive Orlicz space \(L_M^n(\Gamma)\). We assume that \(\Gamma\) belongs to a certain wide subclass of the class of Carleson curves; this subclass includes curves with cusps, as well as curves of the logarithmic spiral type. We obtain an index formula for an arbitrary operator from the algebra \(\mathfrak{A}\) in terms of the symbol of this operator.&lt;/p&gt;
</style></abstract><issue><style face="normal" font="default" size="100%">3</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Integral Equations and Operator Theory</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">1998</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{DEC}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/article/10.1007/BF01194990</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{4}</style></number><publisher><style face="normal" font="default" size="100%">{BIRKHAUSER VERLAG AG}</style></publisher><pub-location><style face="normal" font="default" size="100%">{VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND}</style></pub-location><volume><style face="normal" font="default" size="100%">32</style></volume><pages><style face="normal" font="default" size="100%">436-481</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;The paper is devoted to some only recently uncovered phenomena emerging in the study of singular integral operators (SIO's) with piecewise continuous (PC) coefficients in reflexive rearrangement-invariant spaces over Carleson curves. We deal with several kinds of indices of submultiplicative functions which describe properties of spaces (Boyd and Zippin indices) and curves (spirality indices). We consider some ``disintegration condition{''} which combines properties of spaces and curves, the Boyd and spirality indices. We show that the essential spectrum of SIO associated with the Riemann boundary value problem with PC coefficient arises from the essential range of the coefficient by filling in certain massive connected sets (so-called logarithmic leaves) between the endpoints of jumps. These results combined with the Allan-Douglas local principle and with the two projections theorem enable us to study the Banach algebra \(\mathfrak{A}\) generated by SIO's with matrix-valued piecewise continuous coefficients. We construct a symbol calculus for this Banach algebra which provides a Fredholm criterion and gives a basis for an index formula for arbitrary SIO's from \(\mathfrak{A}\) in terms of their symbols.&lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Singular integral operators with regulated coefficients in reflexive Orlicz spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Siberian Mathematical Journal</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">1997</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{MAR-APR}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/article/10.1007/BF02674624</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{2}</style></number><publisher><style face="normal" font="default" size="100%">{PLENUM PUBL CORP}</style></publisher><pub-location><style face="normal" font="default" size="100%">{CONSULTANTS BUREAU, 233 SPRING ST, NEW YORK, NY 10013}</style></pub-location><volume><style face="normal" font="default" size="100%">38</style></volume><pages><style face="normal" font="default" size="100%">253-266</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><issue><style face="normal" font="default" size="100%">2</style></issue><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Doklady Akademii Nauk</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">1996</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{JUL}</style></date></pub-dates></dates><urls><related-urls><url><style face="normal" font="default" size="100%">https://docentes.fct.unl.pt/sites/default/files/oyk/files/03_1996_doklady.pdf</style></url></related-urls></urls><number><style face="normal" font="default" size="100%">{1}</style></number><publisher><style face="normal" font="default" size="100%">{MEZHDUNARODNAYA KNIGA}</style></publisher><pub-location><style face="normal" font="default" size="100%">{39 DIMITROVA UL., 113095 MOSCOW, RUSSIA}</style></pub-location><volume><style face="normal" font="default" size="100%">349</style></volume><pages><style face="normal" font="default" size="100%">10-12</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Algebras of singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Mathematische Nachrichten</style></secondary-title></titles><keywords><keyword><style  face="normal" font="default" size="100%">Boyd indices</style></keyword><keyword><style  face="normal" font="default" size="100%">Fredholmness}</style></keyword><keyword><style  face="normal" font="default" size="100%">reflexive Orlicz space</style></keyword><keyword><style  face="normal" font="default" size="100%">{singular integral operator</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">1996</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://onlinelibrary.wiley.com/doi/10.1002/mana.19961790112/abstract;jsessionid=12A98E1CA7FADB04EF49A6BF53EE9A4F.d03t01</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">{WILEY-V C H VERLAG GMBH}</style></publisher><pub-location><style face="normal" font="default" size="100%">{PO BOX 10 11 61, D-69451 WEINHEIM, GERMANY}</style></pub-location><volume><style face="normal" font="default" size="100%">179</style></volume><pages><style face="normal" font="default" size="100%">187-222</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;We consider singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces \(L_M(\Gamma)\), which are generalizations of the Lebesgue spaces \(L_p(\Gamma)\), \(1 &amp;lt; p &amp;lt; \infty\). We suppose that \(\Gamma\) belongs to a large class of Carleson curves, including curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. For the singular integral operator associated with the Riemann boundary value problem with a piecewise continuous coefficient \(G\), we establish a Fredholm criterion and an index formula in terms of the essential range of \(G\) complemented by spiralic horns depending on the Boyd indices of \(L_M(\Gamma)\) and contour properties. Our main result is a symbol calculus for the closed algebra of singular integral operators with piecewise continuous matrix-valued coefficients on \(L_M^n(\Gamma)\). &lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>10</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Алексей Карлович</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Об алгебре сингулярных интегральных операторов в рефлексивных пространствах Орлича на кривых Карлесона</style></title><secondary-title><style face="normal" font="default" size="100%">Краевые задачи, специальные функции и дробное исчисление</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">1996</style></year></dates><urls><related-urls><url><style face="normal" font="default" size="100%">https://docentes.fct.unl.pt/sites/default/files/oyk/files/02_1996_gahov-90_minsk-96.pdf</style></url></related-urls></urls><publisher><style face="normal" font="default" size="100%">Издательство Университетское</style></publisher><pub-location><style face="normal" font="default" size="100%">Минск</style></pub-location><pages><style face="normal" font="default" size="100%">127-132</style></pages></record></records></xml>