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Measure theoretic aspects of the finite Hilbert transform

Guillermo P. Curbera, Susumu Okada, Werner J. Ricker

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 $\mlog\colon A\mapsto T(χ_A)$ 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.

Measure theoretic aspects of the finite Hilbert transform

Abstract

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

Paper Structure

This paper contains 6 sections, 17 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rearrangement invariant spaces
  4. The finite Hilbert transform
  5. The vector measure associated to $T$
  6. Integral representation of $T$

Key Result

Lemma \oldthetheorem

The finite Hilbert transform $T$ maps $L\textnormal{log} L$ into $L^1$ and the resulting linear operator $T_{\textnormal{log}}\colon L\textnormal{log} L \to L^1$ is bounded.

Theorems & Definitions (36)

  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 26 more