Publications

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2022
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
2011
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. "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, 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 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.

2008
Karlovich, Alexei Yu., and L. Maligranda. "On the interpolation constant for subadditive operators in Orlicz spaces." Proceedings of the International Symposium on Banach and Function Spaces II (ISBFS 2006), Kyushu Institute of Technology, Kitakyushu, Japan, 14-17 September 2006. Eds. M. Kato, and L. Maligranda. Yokohama: Yokohama Publishers, 2008. 85-101.
2001
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 \(C=1\).