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(R) and on its associate space X(R) and a maximally modulated Calderon-Zygmund singular integral operator TΦ is of weak type (r, r) for all r ∈ (1,∞), then TΦ extends to a bounded operator on X(R). This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces Lp(·)(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 Lp(·)(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.