Pseudodifferential Operators on Variable Lebesgue Spaces

TL;DR

This paper establishes boundedness and Fredholmness of pseudodifferential operators on variable Lebesgue spaces with broad classes of exponents, extending previous results to less regular exponent functions.

Contribution

It generalizes known boundedness and Fredholm criteria for pseudodifferential operators to variable Lebesgue spaces with minimal regularity assumptions on the exponent functions.

Findings

Boundedness of perators with symbols in Hörmander classes on variable Lebesgue spaces.

Fredholmness of perators under certain conditions on the symbol and variable exponents.

Extension of previous results to broader classes of variable exponents.

Abstract

Let $\mathcal{M}(\mathbb{R}^n)$ be the class of bounded away from one and infinity functions $p:\mathbb{R}^n\to[1,\infty]$ such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^{p(\cdot)}(\mathbb{R}^n)$. We show that if $a$ belongs to the H\"ormander class $S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$, $0\le\delta<1$, then the pseudodifferential operator $\Op(a)$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(\R^n)$ provided that $p\in\cM(\R^n)$. Let $\mathcal{M}^*(\mathbb{R}^n)$ be the class of variable exponents $p\in\mathcal{M}(\mathbb{R}^n)$ represented as $1/p(x)=\theta/p_0+(1-\theta)/p_1(x)$ where $p_0\in(1,\infty)$, $\theta\in(0,1)$, and $p_1\in\mathcal{M}(\mathbb{R}^n)$. We prove that if $a\in S_{1,0}^0$ slowly oscillates at infinity in the first variable, then the condition \[ \lim_{R\to\infty}\inf_{|x|+|\xi|\ge R}|a(x,\xi)|>0 \] is sufficient for the Fredholmness of $\Op(a)$ on $L^{p(\cdot)}(\R^n)$ whenever $p\in\cM^*(\R^n)$. Both theorems generalize pioneering results by Rabinovich and Samko \cite{RS08} obtained for globally log-H\"older continuous exponents $p$, constituting a proper subset of $\mathcal{M}^*(\mathbb{R}^n)$.