Publications

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2024
Karlovych, Oleksiy, and Alina Shalukhina. "Maximal noncompactness of singular integral operators on Lp spaces with power weights." Tbilisi Analysis and PDE Seminar. Extended Abstracts of the 2020-2023 Seminar Talks. Eds. Rolnad Duduchava, Eugene Shargorodsky, and George Tephnadze. Cham: Birkhäuser, 2024. 87-97.
Karlovych, Oleksiy, and Eugene Shargorodsky. "A remark on piecewise linear interpolation of continuous Fourier multipliers." Tbilisi Analysis and PDE Seminar. Extended Abstracts of the 2020-2023 Seminar Talks. Eds. Rolnad Duduchava, Eugene Shargorodsky, and George Tephnadze. Cham: Birkhäuser, 2024. 99-107.
2023
Diening, Lars, Oleksiy Karlovych, and Eugene Shargorodsky. "Addendum to "On interpolation of reflexive variable Lebesgue spaces on which the Hardy–Littlewood maximal operator is bounded"." Georgian Mathematical Journal . 30.2 (2023): 211-212.Website
2022
Diening, Lars, Oleksiy Karlovych, and Eugene Shargorodsky. "On interpolation of reflexive variable Lebesgue spaces on which the Hardy-Littlewood maximal operator is bounded." Georgian Mathematical Journal . 29.3 (2022): 347-352.Website
2007
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.