Maximally modulated singular integral operators and their applications to pseudodifferential operators on Banach function spaces

Citation:
Karlovich, Alexei Yu. "Maximally modulated singular integral operators and their applications to pseudodifferential operators on Banach function spaces." Function Spaces in Analysis. Contemporary Mathematics, 645. Ed. Krzysztof Jarosz. Providence, Rhode Island: American Mathematical Society, 2015. 165-178.

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 Calderón-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.