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The finite Hilbert transform on $(-1,1)$

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

Abstract

We present a detailed survey of recent developments in the study of the finite Hilbert transform and its corresponding inversion problem in rearrangement invariant spaces on $(-1,1)$.

The finite Hilbert transform on $(-1,1)$

Abstract

We present a detailed survey of recent developments in the study of the finite Hilbert transform and its corresponding inversion problem in rearrangement invariant spaces on .
Paper Structure (9 sections, 38 theorems, 75 equations)

This paper contains 9 sections, 38 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The airfoil equation: $L^p$-theory
  4. The finite Hilbert transform in rearrangement invariant spaces
  5. Inversion of the finite Hilbert transform and the airfoil equation
  6. Extension of the finite Hilbert transform
  7. The finite Hilbert transform acting on $L\textnormal{log} L$
  8. The fine spectrum of the finite Hilbert transform
  9. Integral representation of the finite Hilbert transform

Key Result

Theorem 3.1

Let $1 < p < \infty$ and $\rho$ be the weight function where $\gamma, \delta \in (-1/p, 1/p')$. Then the function $\rho T(f/\rho)$ belongs to $L^p$ for every $f \in L^p$ and the resulting linear operator is continuous from $L^p$ into $L^p$.

Theorems & Definitions (38)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Lemma 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 5.1
  • Theorem 5.2
  • Corollary 5.3
  • ...and 28 more