Publications

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Journal Article
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 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. "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. "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.

Fernandes, Cláudio A., Alexei Yu. Karlovich, and Yuri I. Karlovich. "Banach algebras of Fourier multipliers equivalent at infinity to nice Fourier multipliers." Banach Journal of Mathematical Analysis (In Press).
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
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 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.

Karlovich, Alexei Yu. "Density of analytic polynomials in abstract Hardy spaces." Commentationes Mathematicae. 57.2 (2017): 131-141.Website
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 \(p=2\).

Karlovich, Alexei Yu, Helena Mascarenhas, and Pedro A. Santos. "Finite section method for a Banach algebra of convolution type operators on Lp(R) with symbols generated by PC and SO." Integral Equations and Operator Theory. 67.4 (2010): 559-600. AbstractWebsite

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 \(L^p(\mathbb{R})\), \(1 < p < \infty\).

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.