Uniform rectifiability and ε-approximability of harmonic functions in L^p

2017/10/16 by Steve Hofmann, Hofmann, Steve, Olli Tapiola +1
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.1710.05528

openalex publication_date 2017/10/16 · openalex created_date 2017/11/10 · openalex updated_date 2026/07/28

Abstract

Suppose that E ⊂ ℝⁿ⁺¹ is a uniformly rectifiable set of codimension 1. We show that every harmonic function is ε-approximable in L^p(Ω) for every p ∈ (1,∞), where Ω:= ℝⁿ⁺¹ ∖ E. Together with results of many authors this shows that pointwise, L^∞ and L^p type ε-approximability properties of harmonic functions are all equivalent and they characterize uniform rectifiability for codimension 1 Ahlfors-David regular sets. Our results and techniques are generalizations of recent works of T. Hytönen and A. Rosén and the first author, J. M. Martell and S. Mayboroda.

Citations

• Hp spaces of several variables

Related

• ε-Approximability of Harmonic Functions in L^p Implies Uniform Rectifiability
• Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in L^p implies uniform rectifiability
• Harmonic measure and approximation of uniformly rectifiable sets
• Proximinality and uniformly approximable sets in L^p
• Interactions between Quantitative Rectifiability, Singular Integrals, and Boundary Value Problems for Harmonic Functions