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On the Local Regularity of the Hilbert Transform

Yifei Pan, Jianfei Wang, Yu Yan

TL;DR

The paper investigates the local regularity of the Hilbert transform $Hf$ for $f \in L^p(\mathbb{R})$, proving that local $C^{k,\alpha}$ regularity of $f$ near a point $x_0$ yields $Hf$ in $C^{k,\alpha_0}$ near $x_0$ for any $0<\alpha_0<\alpha$. The approach employs a local analysis via the finite Hilbert transform and a non-singular operator $T$ on a finite interval, establishing Hölder-type control that transfers regularity. Key contributions include a precise local regularity result with arbitrarily small Hölder loss, a sharp Hölder bound for $T$ with logarithmic factors, and the observation that local real analyticity is preserved by $H$; the results also relate derivatives of $Hf$ to derivatives of $f$ under suitable smoothness. Overall, the work clarifies how localized smoothness propagates through the Hilbert transform and situates these local results alongside classical global results like Privalov's theorems, while highlighting questions about the sharpness of exponent transfer.

Abstract

In this paper the local regularity of the Hilbert transform is considered, and local smoothness and real analyticity results are obtained.

On the Local Regularity of the Hilbert Transform

TL;DR

The paper investigates the local regularity of the Hilbert transform for , proving that local regularity of near a point yields in near for any . The approach employs a local analysis via the finite Hilbert transform and a non-singular operator on a finite interval, establishing Hölder-type control that transfers regularity. Key contributions include a precise local regularity result with arbitrarily small Hölder loss, a sharp Hölder bound for with logarithmic factors, and the observation that local real analyticity is preserved by ; the results also relate derivatives of to derivatives of under suitable smoothness. Overall, the work clarifies how localized smoothness propagates through the Hilbert transform and situates these local results alongside classical global results like Privalov's theorems, while highlighting questions about the sharpness of exponent transfer.

Abstract

In this paper the local regularity of the Hilbert transform is considered, and local smoothness and real analyticity results are obtained.
Paper Structure (2 sections, 5 theorems, 38 equations)

This paper contains 2 sections, 5 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. The Operator $T$ and its Hölder Continuity

Key Result

Theorem 1.1

Suppose $f \in L^p(\mathbb{R}) (1 \leq p < \infty)$. If $f$ is $C^{k, \alpha} (k \geq 0, 0<\alpha<1)$ in a neighborhood of a point $x_0 \in \mathbb{R}$, then its Hilbert transform $Hf$ is $C^{k, \alpha_0}$ in a neighborhood of $x_0$ for any $0<\alpha_0 < \alpha$.

Theorems & Definitions (6)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3