The boundedness of the Hilbert transformation from one rearrangement invariant Banach space into another and applications

TL;DR

This paper investigates the boundedness of the Hilbert transformation between Lorentz spaces, characterizes the optimal range of a truncation operator in Schatten-Lorentz ideals, and explores applications to commutator estimates and operator Lipschitz functions.

Contribution

It extends classical results by providing new boundedness criteria and characterizations in Lorentz and Schatten-Lorentz spaces, with applications to operator theory.

Findings

Boundedness of Hilbert transform in Lorentz spaces established

Optimal range of triangular truncation in Schatten-Lorentz ideals characterized

Sharp commutator estimates derived for Schatten-Lorentz ideals

Abstract

In this paper, we study the boundedness of the Hilbert transformation in Lorentz function spaces, thereby complementing classical results of Boyd. We also characterize the optimal range of a triangular truncation operator in Schatten-Lorentz ideals. These results further entail sharp commutator estimates and applications to operator Lipschitz functions in Schatten-Lorentz ideals.