Publications

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Journal Article
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 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 (In Press).
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 (In Press).
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. "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 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.

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

Karlovich, Alexei Yu, and Ilya M. Spitkovsky. "On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces." Journal of Mathematical Analysis and Appliactions. 384.2 (2011): 706-725. AbstractWebsite

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.

Karlovich, Alexei Yu. "On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces." Integral Equations and Operator Theory. 38 (2000): 28-50. AbstractWebsite

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 \(\Gamma\):
\[
|S|\ge\cot(\pi\lambda_{\Gamma,w}/2)
\]
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 \(\beta(Q_tw)\).

Karlovich, Alexei Yu., and Lech Maligranda. "On the interpolation constant for Orlicz spaces." Proceedings of the American Mathematical Society. 129 (2001): 2727-2739. AbstractWebsite

In this paper we deal with the interpolation from Lebesgue spaces \(L^p\) and \(L^q\), into an Orlicz space \(L^\varphi\), where \( 1 \le p < q \le \infty \) 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,
\[
\|T\|_{L^\varphi\to L^\varphi}
\le C\max\Big\{
\|T\|_{L^p\to L^p},
\|T\|_{L^q\to L^q}
\Big\},
\]
where the interpolation constant \(C\) depends only on \(p\) and \(q\). We give estimates for \(C\), which imply \(C<4\). Moreover, if either \( 1 < p < q \le 2\) or \( 2 \le p < q < \infty\), then \(C< 2\). 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 \(C=1\).

Karlovich, Alexei Yu., Yuri I. Karlovich, and Amarino B. Lebre. "One-sided invertibility criteria for binomial functional operators with shift and slowly oscillating data." Mediterranean Journal of Mathematics. 13.6 (2016): 4413-4435.Website
Karlovich, Alexei Yu., and Yuri I. Karlovich. "One-sided invertibility of binomial functional operators with a shift on rearrangement-invariant spaces." Integral Equations and Operator Theory. 42 (2002): 201-228. AbstractWebsite

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
\[
0<\alpha_X=p_X\le q_X=\beta_X<1.
\]

Karlovich, Alexei Yu. "Remark on the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights." Journal of Function Spaces and Applications. 7 (2009): 301-311. AbstractWebsite

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.

Fernandes, Cláudio A., and Alexei Yu. Karlovich. "Semi-almost periodic Fourier multipliers on rearrangement-invariant spaces with suitable Muckenhoupt weights." Boletín de la Sociedad Matemática Mexicana. 26.3 (2020): 1135-1162.Website
Karlovich, Alexei Yu. "Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm." Operators and Matrices. 1.3 (2007): 427-444. AbstractWebsite

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 \(L_N^{p(\cdot)}(\Gamma,\varrho)\).

Karlovich, Alexei Yu. "Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces." Integral Equations and Operator Theory. 32 (1998): 436-481. AbstractWebsite

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.