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| Home > Articles > Published articles > The A∞ Condition, ε-Approximators, and Varopoulos Extensions in Uniform Domains |
(Universitat Autònoma de Barcelona. Departament de Matemàtiques) (Universitat Autònoma de Barcelona. Departament de Matemàtiques) (Universitat Autònoma de Barcelona. Departament de Matemàtiques)| Date: | 2024 |
| Abstract: | Suppose that ⊂ 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 . We show that the corresponding elliptic measure ωL is quantitatively absolutely continuous with respect to surface measure of ∂ in the sense that ωL ∈ A∞(σ ) if and only if any bounded solution u to Lu = 0 in is ε-approximable for any ε ∈ (0, 1). By ε-approximability of u we mean that there exists a function = ε such that u-L∞() ≤ εuL∞() and the measure μ with dμ = |∇(Y )| dY is a Carleson measure with L∞ 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 L1-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. |
| Grants: | European Commission 101018680 Agencia Estatal de Investigación PID2020-114167GB-I00 Agencia Estatal de Investigación CEX2020-001084-M Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00071 |
| Note: | Altres ajuts: acords transformatius de la UAB |
| Rights: | Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.  |
| Language: | Anglès |
| Document: | Article ; recerca ; Versió publicada |
| Subject: | Elliptic measure ; The A∞ property ; Carleson measure ; ε-Approximability ; Varopoulos extension |
| Published in: | Journal of geometric analysis, Vol. 34, Issue 7 (July 2024) , art. 218, ISSN 1559-002X |
