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

Nicholas Castillo; Ovidiu Costin; Kriti Sehgal

Home/Archive/arXiv

arXiv · 2311.13392

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

Nicholas Castillo·Ovidiu Costin·Kriti Sehgal

View original source ↗Read source document ↗Export metadata ↓

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 H\"older 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.

Explore related subjects

math.CV math.CA

Keep this discovery

BibTeX RIS

Nicholas Castillo, Ovidiu Costin, Kriti Sehgal. 2023-11-22. On the pointwise existence of Cauchy $\rm{P.V.}$ integrals. https://arxiv.org/abs/2311.13392

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The two-dimensional Matkowski--Sut\^o equation with holomorphic and strictly increasing generators

We study the two-dimensional Matkowski--Sut\^o equation, which asks for two quasi-arithmetic means whose sum is twice the arithmetic mean, in two settings. For holomorphic injective generators with convex images on a convex domain in the complex plane, the solutions are exactly the affine pairs and the exponential pairs with a nonzero complex exponent, up to affine changes of the generators. The admissible exponents depend on the shape of the domain and are described by a curvature criterion for its boundary. In the monotone-operator framework of T\'oth, we construct an infinite-dimensional family of non-affine shear pairs on the whole plane. Their generators are strictly increasing in the sense of monotone operators and need not be differentiable. These pairs solve the weighted equation for any number of variables. The rigidity of the one-dimensional problem, due to Dar\'oczy and P\'ales, persists under holomorphy but not under monotonicity.

math.CV↗

arXiv2026

A counterexample to an open problem of Dorff

The classical P\'olya-Schoenberg conjecture, proved by Ruscheweyh-Sheil-Small, asserts that the convolution of two normalized convex univalent functions is again convex. This property fails to carry over to planar harmonic mappings. In 2001, Dorff posed the open problem whether the self-convolution of a normalized convex harmonic mapping with bounded image must remain in the same class. We construct a normalized sense-preserving harmonic diffeomorphism that maps the unit disk onto an ellipse; its self-convolution has vanishing Jacobian at some interior point of the unit disk, which provides a negative answer to Dorff's open problem.

math.CV↗

arXiv2026

Analytic Construction of Rational Curves on Fano Manifolds

Inspired by methods for constructing entire curves in Oka geometry, we give an analytic construction of rational curves on a complex Fano manifold $X$. Yau's theorem provides a K\"ahler metric with positive Ricci curvature. Using this curvature to guide deformations of holomorphic discs, we construct maps from discs of radii tending to infinity with uniformly bounded area. A central point is to preserve the derivative normalization through the limiting process. This yields a nonconstant entire map $f:\mathbb C\rightarrow X$ of finite area. This map extends across infinity to a nonconstant holomorphic map $\mathbb P^1\to X$. Combined with algebraic arguments in characteristic zero, the construction yields proofs of the rational connectedness of Fano manifolds and of Hartshorne's conjecture on ample tangent bundles.

math.CV↗