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

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 $X^{\mathrm{\prime }}({\mathbb{R}}^n)$ and a maximally modulated Calder'on-Zygmund singular integral operator $T^{\mathrm{\Phi }}$ is of weak type $(r,r)$ for all $r\in (1,\mathrm{\infty })$, then $T^{\mathrm{\Phi }}$ 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 $p:\mathbb{R}\to (1,\mathrm{\infty })$. Applications of the above result to the boundedness and compactness of pseudodifferential operators with $L^{\mathrm{\infty }}(\mathbb{R},V(\mathbb{R}))$-symbols on variable Lebesgue spaces $L^{p(\cdot )}(\mathbb{R})$ are considered. Here the Banach algebra $L^{\mathrm{\infty }}(\mathbb{R},V(\mathbb{R}))$ 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.