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

TL;DR

This paper extends the Fredholm criterion for singular integral operators from Lebesgue spaces to Nakano spaces with weights having finite discontinuities on Carleson curves, broadening the applicability of the theory.

Contribution

It generalizes the Fredholm criterion for singular integral operators to Nakano spaces with specific weights on arbitrary Carleson curves, a significant extension of prior results.

Findings

Extended Fredholm criterion to Nakano spaces with weights

Applicable to operators on arbitrary Carleson curves

Broadened the class of weights with finite discontinuities

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.