On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

TL;DR

This paper investigates the Fredholm properties of singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces, establishing invertibility conditions linked to asymptotic behavior at infinity.

Contribution

It proves that Fredholmness of certain operators on variable Lebesgue spaces implies invertibility of associated operators on standard Lebesgue spaces with specific exponents.

Findings

Fredholm operators imply invertibility on standard Lebesgue spaces

Invertibility depends on asymptotic limits of coefficients

Results connect variable and classical Lebesgue space theories

Abstract

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.