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

2014/08/19 by Alexei Yu. Karlovich, Karlovich, Alexei Yu. · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1408.4400

Abstract

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

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