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On the pointwise existence of Cauchy $\rm{P.V.}$ integrals

Nicholas Castillo, Ovidiu Costin, Kriti Sehgal

Abstract

The Riemann-Hilbert (RH) approach, whose origins can be traced back to Riemann's PhD thesis, is well known to be far-reaching. It provides a general framework for expressing solutions of integrable problems such as ODEs or PDEs. Its generalization concerning monodromy groups of Fuchsian systems is one of Hilbert's 23 problems. In this paper we extend a basic result in scalar RH theory, the Sokhotski-Plemelj formula. Classically, this formula is derived under assumptions of Hölder continuity, although it also holds true under weaker, Dini, continuity conditions. Dini continuity is still too restrictive. We prove the Sokhotski-Plemelj formula under weaker assumptions, namely continuity at a point and an $L^1$ condition. Furthermore, we provide sufficient conditions for the existence of the Cauchy $\rm{P.V.}$ integral which are in a precise sense also necessary: weakening them runs into obstructions of a mathematical foundations nature.

On the pointwise existence of Cauchy $\rm{P.V.}$ integrals

Abstract

The Riemann-Hilbert (RH) approach, whose origins can be traced back to Riemann's PhD thesis, is well known to be far-reaching. It provides a general framework for expressing solutions of integrable problems such as ODEs or PDEs. Its generalization concerning monodromy groups of Fuchsian systems is one of Hilbert's 23 problems. In this paper we extend a basic result in scalar RH theory, the Sokhotski-Plemelj formula. Classically, this formula is derived under assumptions of Hölder continuity, although it also holds true under weaker, Dini, continuity conditions. Dini continuity is still too restrictive. We prove the Sokhotski-Plemelj formula under weaker assumptions, namely continuity at a point and an condition. Furthermore, we provide sufficient conditions for the existence of the Cauchy integral which are in a precise sense also necessary: weakening them runs into obstructions of a mathematical foundations nature.
Paper Structure (12 sections, 5 theorems, 73 equations)

This paper contains 12 sections, 5 theorems, 73 equations.

Table of Contents

  1. Abstract
  2. Preliminaries
  3. Setting of Riemann-Hilbert
  4. Results: Sufficient Conditions
  5. Results: Necessary Conditions
  6. Necessary conditions for ${\rm{P.V.}}$ integral to be defined:
  7. Necessary condition for the Sokhotski-Plemelj formulae to be valid:
  8. Acknowledgements
  9. Supplementary Material
  10. Preliminaries
  11. Main Results
  12. Proof of Lemmas

Key Result

Proposition 3

Assume that $g_o(t)/t\in L^1\left([a-\tau,b-\tau]\right)$. Then, the Cauchy principal value integral exists.

Theorems & Definitions (25)

  • Definition 1: Cauchy principal value integral
  • Definition 2
  • Proposition 3
  • proof
  • Theorem 4: Sokhotski-Plemelj theorem
  • proof
  • Remark 5
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 15 more