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

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.