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Homeomorphic Extensions in Bi-Orlicz-Sobolev Spaces

Yizhe Zhu

TL;DR

This work addresses when boundary self-homeomorphisms of the unit circle admit bi–Orlicz–Sobolev extensions to the disk, formulated via $\Phi$- and $\Psi$-Douglas conditions. It develops a constructive dyadic extension in the upper half-plane and establishes sharp energy estimates showing $h\in W^{1,\Phi}$ and $h^{-1}\in W^{1,\Psi}$. It proves the equivalence between the boundary Douglas-type conditions and the existence of the extension, and derives discrete reformulations and corollaries, including a growth criterion $\int_1^{\infty} \frac{\Phi(t)}{t^3} dt<\infty$ ensuring both $h$ and $h^{-1}$ lie in $W^{1,\Phi}$. The results generalize the classical Sobolev ($p$-Douglas) theory to the Orlicz framework, providing flexible control over regularity and inverse regularity for circle-to-disk extensions.

Abstract

We provide a complete characterization of those self-homeomorphisms of the unit circle that admit homeomorphic extensions to the unit disk belonging to bi--Orlicz--Sobolev spaces. Our results generalize classical criteria from the Sobolev setting to the more flexible Orlicz framework.

Homeomorphic Extensions in Bi-Orlicz-Sobolev Spaces

TL;DR

This work addresses when boundary self-homeomorphisms of the unit circle admit bi–Orlicz–Sobolev extensions to the disk, formulated via - and -Douglas conditions. It develops a constructive dyadic extension in the upper half-plane and establishes sharp energy estimates showing and . It proves the equivalence between the boundary Douglas-type conditions and the existence of the extension, and derives discrete reformulations and corollaries, including a growth criterion ensuring both and lie in . The results generalize the classical Sobolev (-Douglas) theory to the Orlicz framework, providing flexible control over regularity and inverse regularity for circle-to-disk extensions.

Abstract

We provide a complete characterization of those self-homeomorphisms of the unit circle that admit homeomorphic extensions to the unit disk belonging to bi--Orlicz--Sobolev spaces. Our results generalize classical criteria from the Sobolev setting to the more flexible Orlicz framework.
Paper Structure (5 sections, 6 theorems, 48 equations, 2 figures)

This paper contains 5 sections, 6 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. N-functions
  4. Alternative to $\Phi$-Douglas condition
  5. Proof of Theorem \ref{['thm:rkcp>2']} and Corollaries

Key Result

Theorem 1.1

Let $\varphi \colon \partial \mathbb{D} \xrightarrow{\rm onto} \partial \mathbb{D}$ be a homeomorphism. Suppose $\Phi, \Psi \colon [0, \infty) \to [0, \infty)$ are $N$-functions that satisfy the doubling condition and ${\rm (aInc)}_p$ on $t\ge t_0$ for some $p>1$. Then $\varphi$ satisfies the $\Phi$

Figures (2)

  • Figure 1: Construction of $h$
  • Figure 2:

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 2 more