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Journal Article
Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "Fredholmness and index of simplest singular integral operators with two slowly oscillating shifts." Operators and Matrices. 8.4 (2014): 935-955. AbstractWebsite

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.

Karlovich, Alexei Yu. "Fredholmness and index of simplest weighted singular integral operators with two slowly oscillating shifts." Banach Journal of Mathematical Analysis. 9.3 (2015): 24-42. AbstractWebsite

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 \(0<1/p+\Re\gamma<1\). We prove that the operator \(A_{ij}\) is a Fredholm operator on \(L^p(\mathbb{R}_+)\) and has zero index if \[ 0<\frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\inf_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma), \quad \frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\sup_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma)<1, \] 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 \(\gamma=0\).

Karlovich, Alexei Yu. "Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces." Journal of Integral Equations and Applications. 15.3 (2003): 263-320. AbstractWebsite

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 \(|p(\tau)-p(t)|\le A/(-\log|\tau-t|)\).

Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "The generalized Cauchy index of some semi-almost periodic functions." Boletín de la Sociedad Matemática Mexicana. 22.2 (2016): 473-485. AbstractWebsite

We compute the generalized Cauchy index of some semi-almost periodic functions, which are important
in the study of the Fredholm index of singular integral operators with shifts and slowly oscillating data.

Böttcher, Albrecht, Alexei Yu. Karlovich, and Bernd Silbermann. "Generalized Krein algebras and asymptotics of Toeplitz determinants." Methods of Functional Analysis and Topology. 13.2 (2007): 236-261. AbstractWebsite

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 \(0<\lambda<1\). The entire paper is based on fundamental work by Mark Krein, ranging from operator ideals through Toeplitz operators up to Wiener-Hopf factorization.

Karlovich, Alexei Yu. "Hardy-Littlewood maximal operator on reflexive variable Lebesgue spaces over spaces of homogeneous type." Studia Mathematica. 254.2 (2020): 149-178.
Karlovich, Alexei Yu. "Hardy-Littlewood maximal operator on the associate space of a Banach function space." Real Analysis Exchange. 44.1 (2019): 119-140.Website
Karlovich, AY. "Higher order asymptotics of Toeplitz determinants with symbols in weighted Wiener algebras." Journal of Mathematical Analysis and Applications. 320.2 (2006): 944-963. AbstractWebsite

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.

Karlovich, Alexei Yu. "The index of singular integral operators in reflexive Orlicz spaces." Mathematical Notes. 64.3 (1998): 330-341. AbstractWebsite

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.

Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "The index of weighted singular integral operators with shifts and slowly oscillating data." Journal of Mathematical Analysis and Applications. 450 (2017): 606-630.Website
Fernandes, Cláudio, Oleksiy Karlovych, and Samuel Medalha. "Invertibility of Fourier convolution operators with PC symbols on variable Lebesgue spaces with Khvedelidze weights." Journal of Mathematical Sciences. 266.3 (2022): 419-434.Website
Fernandes, Cláudio A., Alexei Yu. Karlovich, and Márcio Valente. "Invertibility of Fourier convolution operators with piecewise continuous symbols on Banach function spaces." Transactions of A. Razmadze Mathematical Institute. 175.1 (2021): 49-61.Website
Karlovich, Alexei, and Eugene Shargorodsky. "A lower estimate for weak-type Fourier multipliers." Complex Variables and Elliptic Equations. 67.3 (2022): 642-660.Website
Karlovich, Alexei Yu. "Maximal operators on variable Lebesgue spaces with weights related to oscillations of Carleson curves." Mathematische Nachrichten. 283 (2010): 85-93. AbstractWebsite

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.

Canarias, Tiago, Alexei Karlovich, and Eugene Shargorodsky. "Multiplication Is an open bilinear mapping in the Banach algebra of functions of bounded Wiener $p$-variation." Real Analysis Exchange. 46.1 (2021): 121-148.Website
Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "Necessary conditions for Fredholmness of singular integral operators with shifts and slowly oscillating data." Integral Equations and Operator Theory. 71.1 (2011): 29-53. AbstractWebsite

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 \( 1 < p < \infty\), 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.

Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "Necessary Fredholm conditions for weighted singular integral operators with shifts and slowly oscillating data." Journal of Integral Equations and Applications. 29.3 (2017): 365-399.
Fernandes, Cláudio A., Alexei Yu. Karlovich, and Yuri I. Karlovich. "Noncompactness of Fourier convolution operators on Banach function spaces." Annals of Functional Analysis. 10.4 (2019): 553-561.
Karlovich, Alexei Yu. "Noncompactness of Toeplitz operators between abstract Hardy spaces." Advances in Operator Theory. 6 (2021): 29.Website
Karlovich, Alexei Yu. "Norms of Toeplitz and Hankel operators on Hardy type subspaces of rearrangement-invariant spaces." Integral Equations and Operator Theory. 49 (2004): 43-64. AbstractWebsite

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.

Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "On a weighted singular integral operator with shifts and slowly oscillating data." Complex Analysis and Operator Theory. 10.6 (2016): 1101-1131. AbstractWebsite

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 \[ (S_2 f)(t):=\frac{1}{\pi i}\int\limits_0^\infty \left(\frac{t}{\tau}\right)^{1/2-1/p}\frac{f(\tau)}{\tau-t}\,d\tau, \quad t\in\mathbb{R}_+, \]
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 \[ \limsup_{t\to s}|c(t)|<1,\quad \limsup_{t\to s}|d(t)|<1, \quad s\in\{0,\infty\}, \] 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.

Karlovich, Alexei, and Eugene Shargorodsky. "On an analogue of a theorem by Astala and Tylli." Archiv der Mathematik. 118 (2022): 73-77.Website
Karlovich, Alexei Yu., and Pedro A. Santos. "On asymptoties of Toeplitz determinants with symbols of nonstandard smoothness." Journal of Fourier Aanalysis and Applications. 11.1 (2005): 43-72. AbstractWebsite

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.

Diening, Lars, Oleksiy Karlovych, and Eugene Shargorodsky. "On interpolation of reflexive variable Lebesgue spaces on which the Hardy-Littlewood maximal operator is bounded." Georgian Mathematical Journal . 29.3 (2022): 347-352.Website
Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "On regularization of Mellin PDO's with slowly oscillating symbols of limited smoothness." Communications in Mathematical Analysis. 17.2 (2014): 189-208. AbstractWebsite

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" 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" 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 \(C^\infty(\mathbb{R}_+\times\mathbb{R})\).