Maximally Modulated Singular Integral Operators and their Applications to Pseudodifferential Operators on Banach Function Spaces

TL;DR

This paper establishes conditions under which maximally modulated singular integral operators are bounded on Banach function spaces, with applications to pseudodifferential operators on variable Lebesgue spaces.

Contribution

It proves the boundedness of maximally modulated Calderón-Zygmund operators on Banach function spaces assuming weak type conditions and maximal operator boundedness.

Findings

Boundedness of maximally modulated Hilbert transform on variable Lebesgue spaces.

Extension of boundedness results to pseudodifferential operators with bounded variation symbols.

Application to compactness criteria for pseudodifferential operators.

Abstract

We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$ and a maximally modulated Calder\'on-Zygmund singular integral operator $T^\Phi$ is of weak type $(r,r)$ for all $r\in(1,\infty)$, then $T^\Phi$ extends to a bounded operator on $X(\mathbb{R}^n)$. This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R})$ under natural assumptions on the variable exponent $p:\mathbb{R}\to(1,\infty)$. Applications of the above result to the boundedness and compactness of pseudodifferential operators with $L^\infty(\mathbb{R},V(\mathbb{R}))$-symbols on variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R})$ are considered. Here the Banach algebra $L^\infty(\mathbb{R},V(\mathbb{R}))$ consists of all bounded measurable $V(\mathbb{R})$-valued functions on $\mathbb{R}$ where $V(\mathbb{R})$ is the Banach algebra of all functions of bounded total variation.