The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains

Suppose that Omega subset of Rn+1, n >= 1, is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in Omega. We show that the corresponding elliptic measure omega(L) is quantitatively absolutely continuous with re...

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Autores: Bortz, S., Poggi, B., Tapiola, O., Tolsa, X.
Formato: artículo
Estado:Versión publicada
Fecha de publicación:2024
País:España
Recursos:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2072/537859
Acesso em linha:http://hdl.handle.net/2072/537859
Access Level:acceso abierto
Palavra-chave:Elliptic measure
The A∞ property
Carleson measure
ε-Approximability
Varopoulos extension
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spelling The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform DomainsBortz, S.Poggi, B.Tapiola, O.Tolsa, X.Elliptic measureThe A∞ propertyCarleson measureε-ApproximabilityVaropoulos extensionSuppose that Omega subset of Rn+1, n >= 1, is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in Omega. We show that the corresponding elliptic measure omega(L) is quantitatively absolutely continuous with respect to surface measure of partial derivative Omega in the sense that omega(L) is an element of A(infinity)(sigma) if and only if any bounded solution u to Lu = 0 in Omega is epsilon-approximable for any epsilon is an element of (0, 1). By epsilon-approximability of u we mean that there exists a function Phi = Phi(epsilon) such that parallel to u - Phi parallel to(L infinity(Omega)) <= epsilon parallel to u parallel to(L infinity(Omega)) and themeasure (mu) over tilde (Phi) with d (mu) over tilde = vertical bar del Phi(Y)vertical bar dY is a Carleson measure with L-infinity control over the Carleson norm. As a consequence of this approximability result, we show that boundary BMO functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy L-1-type Carleson measure estimates with BMO control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.Open Access Funding provided by Universitat Autonoma de Barcelona. S.B. was supported by the Simons foundation grant Travel support for Mathematicians (Grant Number 959861). B.P., O.T. and X.T. were supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement 101018680). X.T. is also partially supported by MICINN (Spain) under the Grant PID2020-114167GB-I00, the Maria de Maeztu Program for units of excellence (Spain) (CEX2020-001084-M), and 2021-SGR-00071 (Catalonia).Springer2024info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion53 p.application/pdfhttp://hdl.handle.net/2072/537859RECERCAT (Dipòsit de la Recerca de Catalunya)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésJournal Of Geometric AnalysisL'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: https://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2072/5378592026-05-29T05:05:01Z
dc.title.none.fl_str_mv The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains
title The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains
spellingShingle The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains
Bortz, S.
Elliptic measure
The A∞ property
Carleson measure
ε-Approximability
Varopoulos extension
title_short The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains
title_full The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains
title_fullStr The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains
title_full_unstemmed The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains
title_sort The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains
dc.creator.none.fl_str_mv Bortz, S.
Poggi, B.
Tapiola, O.
Tolsa, X.
author Bortz, S.
author_facet Bortz, S.
Poggi, B.
Tapiola, O.
Tolsa, X.
author_role author
author2 Poggi, B.
Tapiola, O.
Tolsa, X.
author2_role author
author
author
dc.subject.none.fl_str_mv Elliptic measure
The A∞ property
Carleson measure
ε-Approximability
Varopoulos extension
topic Elliptic measure
The A∞ property
Carleson measure
ε-Approximability
Varopoulos extension
description Suppose that Omega subset of Rn+1, n >= 1, is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in Omega. We show that the corresponding elliptic measure omega(L) is quantitatively absolutely continuous with respect to surface measure of partial derivative Omega in the sense that omega(L) is an element of A(infinity)(sigma) if and only if any bounded solution u to Lu = 0 in Omega is epsilon-approximable for any epsilon is an element of (0, 1). By epsilon-approximability of u we mean that there exists a function Phi = Phi(epsilon) such that parallel to u - Phi parallel to(L infinity(Omega)) <= epsilon parallel to u parallel to(L infinity(Omega)) and themeasure (mu) over tilde (Phi) with d (mu) over tilde = vertical bar del Phi(Y)vertical bar dY is a Carleson measure with L-infinity control over the Carleson norm. As a consequence of this approximability result, we show that boundary BMO functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy L-1-type Carleson measure estimates with BMO control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.
publishDate 2024
dc.date.none.fl_str_mv 2024
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/2072/537859
url http://hdl.handle.net/2072/537859
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Journal Of Geometric Analysis
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 53 p.
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv RECERCAT (Dipòsit de la Recerca de Catalunya)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,811543

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