## Publications

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2016
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., 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
2015
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. "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. "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. "Maximally modulated singular integral operators and their applications to pseudodifferential operators on Banach function spaces." Function Spaces in Analysis. Contemporary Mathematics, 645. Ed. Krzysztof Jarosz. Providence, Rhode Island: American Mathematical Society, 2015. 165-178. Abstract

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.

Karlovich, Alexei Yu. "The Stechkin inequality for Fourier multipliers on variable Lebesgue spaces." Mathematical Inequalities and Applications. 18.4 (2015): 1473-1481. Abstract

We prove the Stechkin inequality for Fourier multipliers on variable Lebesgue spaces under some natural assumptions on variable exponents.

2014
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 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., 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., 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 $$C^\infty(\mathbb{R}_+\times\mathbb{R})$$.

2013
Karlovich, Alexei Yu., and Ilya M. Spitkovsky. "Pseudodifferential operators on variable Lebesgue spaces." Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Operator Theory: Advances and Applications, 228. Eds. Yuri I. Karlovich, Luigi Rodino, Bernd Silbermann, and Ilya M. Spitkovsky. Basel: Birkhäuser, 2013. 173-183. Abstract

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 $$0<\rho\le 1$$, $$0\le\delta<1$$, 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 $\lim_{R\to\infty}\inf_{|x|+|\xi|\ge R}|a(x,\xi)|>0$ 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 $$\mathcal{M}^*(\mathbb{R}^n)$$.

2011
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., 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, 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, Yuri I. Karlovich, and Amarino B. Lebre. "Sufficient conditions for Fredholmness of singular integral operators with shifts and slowly oscillating data." Integral Equations and Operator Theory. 70.4 (2011): 451-483. AbstractWebsite

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 $$1 < p < \infty$$, 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 $$\infty$$.

Karlovich, Alexei Yu. "Singular integral operators on Nakano spaces with weights having finite sets of discontinuities." Function spaces IX. Proceedings of the 9th international conference, Kraków, Poland, July 6–11, 2009. Banach Center Publications, 92. Eds. Henryk Hudzik, Grzegorz Lewicki, Julian Musielak, Marian Nowak, and Leszek Skrzypczak. Warszawa: Polish Academy of Sciences, Institute of Mathematics, 2011. 143-166. Abstract

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.

2010
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.

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. "Singular integral operators on variable Lebesgue spaces over arbitrary Carleson curves." Topics in Operator Theory: Operators, Matrices and Analytic Functions, Vol. 1. Operator Theory: Advances and Applications, 202. Eds. JA Ball, V. Bolotnikov, JW Helton, L. Rodman, and IM Spitkovsky. Basel: Birkhäuser, 2010. 321-336. Abstract

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.

Karlovich, Alexei Yu. "Singular integral operators on variable Lebesgue spaces with radial oscillating weights." Operator Algebras, Operator Theory and Applications.Operator Theory Advances and Applications, 195 . Eds. JJ Grobler, LE Labuschagne, and M. Möller. Basel: Birkhäuser, 2010. 185-212. Abstract

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.

2009
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. "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 $$o(n^{1-\alpha-\beta})$$.

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.

2008
Karlovich, Alexei Yu. "Higher order asymptotic formulas for traces of Toeplitz matrices with symbols in Hölder-Zygmund spaces." Recent Advances in Matrix and Operator Theory. Operator Theory: Advances and Applications, 179. Eds. Joseph A. Ball, Yuli Eidelman, William J. Helton, Vadim Olshevsky, and James Rovnyak. Basel: Bikhäuser, 2008. 185-196. Abstract

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.