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Hölder regularity of the $\bar\partial-$equation on the polydisc

Yu Jun Loo, Alexander Tumanov

TL;DR

The paper solves the $\bar\partial$-equation on the polydisc $\mathbb{D}^n$ with Hölder regularity preserved: for any $k\ge0$ and $0<\alpha<1$, there exists a bounded linear operator $H$ mapping $Z^{k+\alpha}_{(0,1)}(\mathbb{D}^n)$ to $C^{k+\alpha}(\mathbb{D}^n)$ such that $\bar\partial u=g$ has solution $u=H[g]$ and $KH=0$, where $K$ is the Cauchy torus integral. The construction uses Henkin's weighted formula and is shown to coincide with the Nijenhuis–Woolf operator $T$, yielding a canonical solution operator ($KH=KT=0$). The base case $k=0$ is established via sectorial Hölder estimates for a reduced integral operator, and the general case follows by induction on $k$ using differentiation properties of the singular integrals. The result demonstrates optimal Hölder regularity preservation for product domains and suggests extensions to broader product-domain settings, highlighting the interplay between Henkin’s and NW formalisms.

Abstract

In this note, we show the existence of a solution operator to the $\bar\partial-$equation in the polydisc that preserves Hölder regularity. This solution operator is constructed using Henkin's formula. It is a well-known fact that solution operators to the $\bar\partial-$equation on product domains do not improve Hölder regularity. Hence, this solution operator is optimal in that regard.

Hölder regularity of the $\bar\partial-$equation on the polydisc

TL;DR

The paper solves the -equation on the polydisc with Hölder regularity preserved: for any and , there exists a bounded linear operator mapping to such that has solution and , where is the Cauchy torus integral. The construction uses Henkin's weighted formula and is shown to coincide with the Nijenhuis–Woolf operator , yielding a canonical solution operator (). The base case is established via sectorial Hölder estimates for a reduced integral operator, and the general case follows by induction on using differentiation properties of the singular integrals. The result demonstrates optimal Hölder regularity preservation for product domains and suggests extensions to broader product-domain settings, highlighting the interplay between Henkin’s and NW formalisms.

Abstract

In this note, we show the existence of a solution operator to the equation in the polydisc that preserves Hölder regularity. This solution operator is constructed using Henkin's formula. It is a well-known fact that solution operators to the equation on product domains do not improve Hölder regularity. Hence, this solution operator is optimal in that regard.
Paper Structure (5 sections, 8 theorems, 28 equations)

This paper contains 5 sections, 8 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Proof of Theorem \ref{['k = 0 main result']}
  4. Proof of Theorem \ref{['main theorem']}
  5. Conclusion and extensions

Key Result

Theorem 1

For any integer $k \geq 0$, and $0<\alpha < 1$, Let $Z^{k+\alpha}_{(0,1)}(\mathbb{D}^n) \subseteq C^{k+\alpha}_{(0,1)}(\mathbb{D}^n)$ denote the subspace of $\bar{\partial}-$closed, Hölder $k+ \alpha$, $(0,1)-$forms on the polydisc. Then for all $g \in Z^{k+\alpha}_{(0,1)}(\mathbb{D}^n)$, the equati Moreover, $H$ is canonical in the sense that for any $f \in C^{k+\alpha}_{(0,1)}(\mathbb{D}^n)$, H

Theorems & Definitions (20)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Proposition 1
  • ...and 10 more