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

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