Measure theoretic aspects of the finite Hilbert transform

TL;DR

This paper explores the measure theoretic properties of the finite Hilbert transform on Zygmund spaces, establishing an integral representation that reveals new operator characteristics such as non-compactness and non-order-boundedness.

Contribution

It introduces an integral representation of the finite Hilbert transform via an $L^1$-valued measure, enabling the application of measure theoretic methods to derive new operator properties.

Findings

The finite Hilbert transform is not order bounded.

It is not completely continuous.

It is not weakly compact.

Abstract

The finite Hilbert transform $T$, when acting in the classical Zygmund space $\logl$ (over $(-1,1)$), was intensively studied in \cite{curbera-okada-ricker-log}. In this note an integral representation of $T$ is established via the $L^1(-1,1)$-valued measure $\mlog\colon A\mapsto T(\chi_A)$ for each Borel set $A\subseteq(-1,1)$. This integral representation, together with various non-trivial properties of $\mlog$, allow the use of measure theoretic methods (not available in \cite{curbera-okada-ricker-log}) to establish new properties of $T$. For instance, as an operator between Banach function spaces $T$ is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for $T$ plays a crucial role.