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Singular integral operators on Nakano spaces with weights having finite sets of discontinuities

Alexei Yu. Karlovich — 2011

Banach Center Publications

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.

Density of analytic polynomials in abstract Hardy spaces

Alexei Yu. Karlovich — 2017

Commentationes Mathematicae

Let X be a separable Banach function space on the unit circle 𝕋 and let H [ X ] be the abstract Hardy space built upon X . We show that the set of analytic polynomials is dense in H [ X ] if the HardyLittlewood maximal operator is bounded on the associate space X ' . This result is specified to the case of variable Lebesgue spaces.

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