Uniform rectifiability and $\varepsilon $-approximability of harmonic functions in $L^p$

Hofmann, Steve; Tapiola, Olli

doi:10.5802/aif.3359

Uniform rectifiability and

ε

-approximability of harmonic functions in

L^{p}

[Rectifiabilité uniforme et

ε

-approximation dans

L^{p}

]

Hofmann, Steve ¹ ; Tapiola, Olli ²

¹ Department of Mathematics, University of Missouri, Columbia, MO 65211 (USA)
² Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä (Finland)

Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1595-1638.

Résumé
Abstract

Soit $E$ un ensemble uniformément rectifiable de codimension $1$ dans un espace euclidien de dimension $n + 1$ et soit $Ω$ son complémentaire. Nous montrons que toute fonction harmonique est $ε$ -approchable dans $L^{p} (Ω)$ pour tout $p$ fini strictement plus grand que $1$ . Cela montre, compte tenu de résultats précédents par différents auteurs, que ponctuellement, les propriétés d’ $ε$ -approximation de type $L^{\infty}$ et $L^{p}$ de fonctions harmoniques sont équivalentes et elles caractérisent la rectifiabilité uniforme des ensembles réguliers au sens d’Ahlfors–David de codimension $1$ . Nos résultats et techniques sont des généralisations de travaux récents de T. Hytönen, A. Rosén et du premier auteur, J. M. Martell et S. Mayboroda.

Suppose that $E \subset ℝ^{n + 1}$ is a uniformly rectifiable set of codimension $1$ . We show that every harmonic function is $ε$ -approximable in $L^{p} (Ω)$ for every $p \in (1, \infty)$ , where $Ω : = ℝ^{n + 1} ∖ E$ . Together with results of many authors this shows that pointwise, $L^{\infty}$ 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.

Reçu le : 2017-10-24
Révisé le : 2018-11-28
Accepté le : 2019-03-12
Publié le : 2021-04-15

DOI : 10.5802/aif.3359

Classification : 42B37, 31B05, 42B20, 28A75
Keywords: $\varepsilon $-approximability, uniform rectifiability, Carleson measures, harmonic functions.
Mot clés : $\varepsilon $-approximation, rectifiabilité uniforme, mesures de Carleson, fonction harmonique.

Affiliations des auteurs :

Hofmann, Steve ¹ ; Tapiola, Olli ²

¹ Department of Mathematics, University of Missouri, Columbia, MO 65211 (USA)
² Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä (Finland)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Uniform rectifiability and $\varepsilon $-approximability of harmonic functions in $L^p$},
     journal = {Annales de l'Institut Fourier},
     pages = {1595--1638},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
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     doi = {10.5802/aif.3359},
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Hofmann, Steve; Tapiola, Olli. Uniform rectifiability and $\varepsilon $-approximability of harmonic functions in $L^p$. Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1595-1638. doi : 10.5802/aif.3359. https://aif.centre-mersenne.org/articles/10.5802/aif.3359/

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