Uniform rectifiability and ε-approximability of harmonic functions in L p
[Rectifiabilité uniforme et ε-approximation dans L p ]
Annales de l'Institut Fourier, Tome 70 (2020) no. 4, pp. 1595-1638.

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 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 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(1,), where Ω:= n+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.

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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.
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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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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