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Inversion and extension of the finite Hilbert transform on \(\mathbf {(-1,1)}\)

  • Guillermo P. CurberaEmail author
  • Susumu Okada
  • Werner J. Ricker
Article
  • 54 Downloads

Abstract

The principle of optimizing inequalities, or their equivalent operator theoretic formulation, is well established in analysis. For an operator, this corresponds to extending its action to larger domains, hopefully to the largest possible such domain (i.e., its optimal domain). Some classical operators are already optimally defined (e.g., the Hilbert transform in \(L^p(\mathbb {R})\), \(1<p<\infty \)), and others are not (e.g., the Hausdorff–Young inequality in \(L^p(\mathbb {T})\), \(1<p<2\), or the Sobolev inequality in various spaces). In this paper, a detailed investigation is undertaken of the finite Hilbert transform T acting on rearrangement invariant spaces X on \((-1,1)\), an operator whose singular kernel is neither positive nor does it possess any monotonicity properties. For a large class of such spaces X, it is shown that T is already optimally defined on X (this is known for \(L^p(-1,1)\) for all \(1<p<\infty \), except \(p=2\)). The case \(p=2\) is significantly different because the range of T is a proper dense subspace of \(L^2(-1,1)\). Nevertheless, by a completely different approach, it is established that T is also optimally defined on \(L^2(-1,1)\). Our methods are also used to show that the solution of the airfoil equation, which is well known for the spaces \(L^p(-1,1)\) whenever \(p\not =2\) (due to certain properties of T), can also be extended to the class of r.i. spaces X considered in this paper.

Keywords

Finite Hilbert transform Rearrangement invariant space Airfoil equation Fredholm operator 

Mathematics Subject Classification

Primary 44A15 46E30 Secondary 47A53 47B34 

Notes

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Facultad de Matemáticas and IMUSUniversidad de SevillaSevillaSpain
  2. 2.School of Mathematics and PhysicsUniversity of TasmaniaHobartAustralia
  3. 3.Math.-Geogr. FakultätKatholische Universität Eichstätt-IngolstadtEichstättGermany

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