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On harmonic approximation of Lipschitz functions on compacts in $\mathbb{R}^d$

Nikolai A. Shirokov, Andrei V. Vasin

TL;DR

The authors prove a quantitative Jackson–Bernstein type theorem for harmonic approximation of Lipschitz-type functions on porous compacts in $\mathbb{R}^d$. They combine Whitney extension, smooth regularization, Vitushkin localization, and Taylor expansions of the fundamental solution to construct harmonic approximants in $K_{\delta}$ with rate $\omega(\delta)$ and controlled gradients, under a uniform gradient bound. The main contribution is a general characterization of $\mathrm{Lip}_{\omega}(K)$ for porous sets (including Ahlfors–David regular ones) without Dini-type restrictions, extending prior chord-arc and bi-Lipschitz image results to higher dimensions. The approach provides a constructive, local-to-global framework for harmonic approximation via a carefully designed partition of unity and localized harmonic sums, with explicit dependence on the modulus of continuity and porosity constants.

Abstract

Given a porous compact $K \subset \mathbb{R}^d$ and a continuity modulus $ω$, we prove a quantitative Jackson-Bernstein type theorem on harmonic approximation. That is, a function $f$ belongs to the class $\mathrm{Lip}_ω(K)$ if and only if $f$ can be approximated uniformly on $K$ with a rate of $ω(δ)$ by a function that is harmonic in the $δ$-neighborhood of $K$, provided the uniform estimate $ω(δ)/δ$ on the gradient holds.

On harmonic approximation of Lipschitz functions on compacts in $\mathbb{R}^d$

TL;DR

The authors prove a quantitative Jackson–Bernstein type theorem for harmonic approximation of Lipschitz-type functions on porous compacts in . They combine Whitney extension, smooth regularization, Vitushkin localization, and Taylor expansions of the fundamental solution to construct harmonic approximants in with rate and controlled gradients, under a uniform gradient bound. The main contribution is a general characterization of for porous sets (including Ahlfors–David regular ones) without Dini-type restrictions, extending prior chord-arc and bi-Lipschitz image results to higher dimensions. The approach provides a constructive, local-to-global framework for harmonic approximation via a carefully designed partition of unity and localized harmonic sums, with explicit dependence on the modulus of continuity and porosity constants.

Abstract

Given a porous compact and a continuity modulus , we prove a quantitative Jackson-Bernstein type theorem on harmonic approximation. That is, a function belongs to the class if and only if can be approximated uniformly on with a rate of by a function that is harmonic in the -neighborhood of , provided the uniform estimate on the gradient holds.

Paper Structure

This paper contains 15 sections, 5 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main result
  4. Organization and notations
  5. Preliminaries
  6. Whitney extension
  7. Regularization
  8. Reduction of proof
  9. Porous sets
  10. Proof of Proposition \ref{['pr1']}
  11. Partition of unity
  12. Vitushkin type localization
  13. Taylor expansion of localization
  14. Approximation function construction
  15. Proof of Proposition \ref{['pr1']}

Key Result

Theorem 1.1

Let $K \subset \mathbb{R}^d$, $d>2$, be a porous compact, and let $\omega (t)$ be a continuity modulus. Given a continuous function $f$ on $K$, then $f \in \mathrm{Lip}_{\omega}(K)$ if and only if for any $\delta >0$ there is a harmonic function $\mathcal{G}_{\delta}$ in a $\delta$- neighborhood $K_ and where the constants $C_1$ and $C_2$ are independent of $\delta$ and $f$.

Theorems & Definitions (13)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1.1
  • Remark 2
  • Proposition 2.1
  • Remark 3
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • ...and 3 more