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A
Fernandes, Cláudio A., Alexei Yu. Karlovich, and Yuri. I. Karlovich. "Algebra of convolution type operators with continuous data on Banach function spaces." Banach Center Publications. 119 (2019): 157-171.Website
Karlovich, Alexei Yu. "Algebras of continuous Fourier multipliers on variable Lebesgue spaces." Mediterranean Journal of Mathematics. 17.102 (2020): 19 pages.Website
Karlovich, Alexei Yu. "Algebras of singular integral operators on Nakano spaces with Khvedelidze weights over Carleson curves with logarithmic whirl points." Izvestiya Vysshih Uchebnyh Zavedeniy. Severo-Kavkazskiy Region. Estestvennye Nauki. Special Issue "Pseudodifferential equations and some problems of mathematical physics". Rostov-on-Don: Rostov University Press, 2005. 135-142. Abstract22_2005_simonenko-70.pdf

We establish a Fredholm criterion for 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 Carleson curves with logarithmic whirl points.

Karlovich, Alexei Yu. "Algebras of singular integral operators on rearrangement-invariant spaces and Nikolski ideals." The New York Journal of Mathematics. 8 (2002): 215-234. AbstractWebsite

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.

Karlovich, Alexei Yu. "Algebras of singular integral operators with PC coefficients in rearrangement-invariant spaces with Muckenhoupt weights." Journal of Operator Theory. 47 (2002): 303-323. AbstractWebsite

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.

Karlovich, Alexei Yu. "Algebras of singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces." Mathematische Nachrichten. 179 (1996): 187-222. AbstractWebsite

We consider singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces \(L_M(\Gamma)\), which are generalizations of the Lebesgue spaces \(L_p(\Gamma)\), \(1 < p < \infty\). 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 \(L_M^n(\Gamma)\).

Karlovich, Alexei Yu. "Algebras of singular integral operators with piecewise continuous coefficients on weighted Nakano spaces." The Extended Field of Operator Theory. Operator Theory: Advances and Applications, 171. Ed. Michael A. Dritschel. Basel: Birkhäuser, 2007. 171-188. Abstract

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.

Karlovich, Alexei Yu. "Asymptotics of block Toeplitz determinants generated by factorable matrix functions with equal partial indices." Mathematische Nachrichten. 280 (2007): 1118-1127. AbstractWebsite

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.

Karlovich, Alexei Yu. "Asymptotics of determinants and traces of Toeplitz matrices with symbols in weighted Wiener algebras." Zeitschrift für Analysis und ihre Anwendungen. 26.1 (2007): 43-56. AbstractWebsite

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.

Karlovich, Alexei Yu. "Asymptotics of Toeplitz determinants generated by functions with Fourier coefficients in weighted Orlicz sequence classes." Function Spaces. Contemporary Mathematics, 435. Ed. K. Jarosz. Providence, RI: American Mathematical Society, 2007. 229-243. Abstract

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.

Karlovich, Alexei Yu. "Asymptotics of Toeplitz matrices with symbols in some generalized Krein algebras." Modern Analysis and Applications: Mark Krein Centenary Conference, Vol. 1. Operator Theory Advances and Applications, 190. Eds. V. Adamyan, Y. Berezansky, I. Gohberg, M. Gorbachuk, V. Gorbachuk, A. Kochubei, H. Langer, and G. Popov. Basel: Birkhäuser, 2009. 341-359. Abstract

Let \(\alpha,\beta\in(0,1)\) and
\[
K^{\alpha,\beta}:=\left\{a\in L^\infty(\mathbb{T}):\
\sum_{k=1}^\infty |\widehat{a}(-k)|^2 k^{2\alpha}<\infty,\
\sum_{k=1}^\infty |\widehat{a}(k)|^2 k^{2\beta}<\infty
\right\}.
\]
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
earlier results of Gabor Szegö for scalar symbols and established the asymptotic trace formula
\[
\operatorname{trace}f(T_n(a))=(n+1)G_f(a)+E_f(a)+o(1)
\quad\text{as}\ n\to\infty
\]
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 \(o(n^{1-\alpha-\beta})\).

B
Karlovich, Alexei Yu. "Banach algebra of the Fourier multipliers on weighted Banach function spaces." Concrete Operators. 2.1 (2015): 27-36. AbstractWebsite

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.

Karlovich, Alexei Yu. "Boundedness of pseudodifferential operators on Banach function spaces." Operator Theory, Operator Algebras and Applications. Operator Theory: Advances and Applications, 242. Eds. Maria Amélia Bastos, Amarino Lebre, Stefan Samko, and Ilya M. Spitkovsky. Basel: Birkhäuser/Springer, 2014. 185-195. Abstract

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 \(0<\rho\le 1\), \(0\le\delta<1\) or to the the Miyachi class \(S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)\) with \(0\le\delta\le\rho\le 1\), \(0\le\delta<1\), and \(\varkappa>0\). This result is applied to the case of variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R}^n)\).

Karlovich, Alexei, and Eugene Shargorodsky. "The Brown-Halmos theorem for a pair of abstract Hardy spaces." Journal of Mathematical Analysis and Applications. 472 (2019): 246-265.Website
C
Fernandes, Cláudio A., Alexei Yu. Karlovich, and Yuri I. Karlovich. "Calkin images of Fourier convolution operators with slowly oscillating symbols." Proceedings of IWOTA 2019. In Press.
Karlovich, Alexei Yu., and Ilya M. Spitkovsky. "The Cauchy singular integral operator on weighted variable Lebesgue spaces." Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, 236. Eds. Manuel Cepedello Boiso, Håkan Hedenmalm, Marinus A. Kaashoek, Alfonso Montes Rodríguez, and Sergei Treil. Basel: Birkhäuser, 2014. 275-291. Abstract

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 $$ \sup_{-\infty < a < b < \infty} \frac{1}{b-a} \|w\chi_{(a,b)}\|_{p(\cdot)} \|w^{-1}\chi_{(a,b)}\|_{p'(\cdot)}<\infty \quad (1/p(x)+1/p'(x)=1). $$

Karlovich, Alexei Yu. "Commutators of convolution type operators on some Banach function spaces." Annals of Functional Analysis. 6.4 (2015): 191-205. AbstractWebsite

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.

Karlovich, Alexei Yu., and Andrei K. Lerner. "Commutators of singular integrals on generalized Lp spaces with variable exponent." Publicacions Matematiques. 49.1 (2005): 111-125. AbstractWebsite

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.

Karlovich, Alexei Yu., and Yuri I. Karlovich. "Compactness of commutators arising in the Fredholm theory of singular integral operators with shifts." Factorization, Singular Operators and Related Problems. Eds. Stefan Samko, Amarino Lebre, and António Ferreira dos Santos. Dordrecht: Kluwer Academic Publishers, 2003. 111-129. Abstract

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.

Karlovich, Alexei Yu., and Ilya M. Spitkovsky. "Connectedness of spectra of Toeplitz operators on Hardy spaces with Muckenhoupt weights over Carleson curves." Integral Equations and Operator Theory. 65.1 (2009): 83-114. AbstractWebsite

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 \(1 < p < \infty\). 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 \(H^p(\Gamma,w)\), \( 1 < p < \infty \), with general Muckenhoupt weights \(w\) over arbitrary Carleson curves \(\Gamma\).

Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "Criteria for n(d)-normality of weighted singular integral operators with shifts and slowly oscillating data." Proceedings of the London Mathematical Society. 116.4 (2018): 997-1027 .Website
Karlovich, Alexei Yu. "Criteria for one-sided invertibility of a functional operator in rearrangement-invariant spaces of fundamental type." Mathematische Nachrichten. 229 (2001): 91-118. AbstractWebsite

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
\[
A=aI-bW,
\]
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.

D
Karlovich, Alexei Yu. "Density of analytic polynomials in abstract Hardy spaces." Commentationes Mathematicae. 57.2 (2017): 131-141.Website
E
Karlovich, Alexei Yu., Helena Mascarenhas, and Pedro A. Santos. "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)." Integral Equations and Operator Theory. 69.3 (2011): 447-449. AbstractWebsite

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 \(p=2\).