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dc.contributor.authorHofmann, Steve
dc.contributor.authorTapiola, Olli
dc.date.accessioned2021-08-18T12:37:37Z
dc.date.available2021-08-18T12:37:37Z
dc.date.issued2021
dc.identifier.citationHofmann, S., & Tapiola, O. (2021). Uniform rectifiability implies Varopoulos extensions. <i>Advances in Mathematics</i>, <i>390</i>, Article 107961. <a href="https://doi.org/10.1016/j.aim.2021.107961" target="_blank">https://doi.org/10.1016/j.aim.2021.107961</a>
dc.identifier.otherCONVID_99281118
dc.identifier.urihttps://jyx.jyu.fi/handle/123456789/77424
dc.description.abstractWe construct extensions of Varopolous type for functions f∈BMO(E), for any uniformly rectifiable set E of codimension one. More precisely, let Ω⊂Rn+1 be an open set satisfying the corkscrew condition, with an n-dimensional uniformly rectifiable boundary ∂Ω, and let ≔σ≔Hn⌊∂Ω denote the surface measure on ∂Ω. We show that if f∈BMO(∂Ω,dσ) with compact support on ∂Ω, then there exists a smooth function V in Ω such that |∇V(Y)|dY is a Carleson measure with Carleson norm controlled by the BMO norm of f, and such that V converges in some non-tangential sense to f almost everywhere with respect to σ. Our results should be compared to recent geometric characterizations of Lp-solvability and of BMO-solvability of the Dirichlet problem, by Azzam, the first author, Martell, Mourgoglou and Tolsa and by the first author and Le, respectively. In combination, this latter pair of results shows that one can construct, for all f∈Cc(∂Ω), a harmonic extension u, with |∇u(Y)|2dist(Y,∂Ω)dY a Carleson measure with Carleson norm controlled by the BMO norm of f, only in the presence of an appropriate quantitative connectivity condition.en
dc.format.mimetypeapplication/pdf
dc.language.isoeng
dc.publisherElsevier
dc.relation.ispartofseriesAdvances in Mathematics
dc.rightsCC BY 4.0
dc.subject.otheruniform rectifiability
dc.subject.otherCarleson measure estimate
dc.subject.otherepsilon-approximability
dc.subject.otherBMO
dc.subject.othersolvability of the Dirichlet problem
dc.subject.otherharmonic measure
dc.titleUniform rectifiability implies Varopoulos extensions
dc.typearticle
dc.identifier.urnURN:NBN:fi:jyu-202108184587
dc.contributor.laitosMatematiikan ja tilastotieteen laitosfi
dc.contributor.laitosDepartment of Mathematics and Statisticsen
dc.contributor.oppiaineMatematiikkafi
dc.contributor.oppiaineMathematicsen
dc.type.urihttp://purl.org/eprint/type/JournalArticle
dc.type.coarhttp://purl.org/coar/resource_type/c_2df8fbb1
dc.description.reviewstatuspeerReviewed
dc.relation.issn0001-8708
dc.relation.volume390
dc.type.versionpublishedVersion
dc.rights.copyright© 2021 the Authors
dc.rights.accesslevelopenAccessfi
dc.subject.ysoharmoninen analyysi
dc.subject.ysoosittaisdifferentiaaliyhtälöt
dc.format.contentfulltext
jyx.subject.urihttp://www.yso.fi/onto/yso/p28124
jyx.subject.urihttp://www.yso.fi/onto/yso/p12392
dc.rights.urlhttps://creativecommons.org/licenses/by/4.0/
dc.relation.doi10.1016/j.aim.2021.107961
jyx.fundinginformationS.H. was supported by NSF grant DMS-1664047. O.T. was partially supported by Emil Aaltosen Säätiö through Foundations' Post Doc Pool grant.
dc.type.okmA1


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