TRANSFERENCE OF SCALE-INVARIANT ESTIMATES FROM LIPSCHITZ TO NONTANGENTIALLY ACCESSIBLE TO UNIFORMLY RECTIFIABLE DOMAINS
In relatively nice geometric settings, in particular, on Lipschitz domains, absolute continuity of elliptic measure with respect to the surface measure is equivalent to Carleson measure estimates, to square function estimates, and to ε-approximability, for solutions to the second-order divergence-fo...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/381584 |
| Acceso en línea: | http://hdl.handle.net/10261/381584 https://www.scopus.com/inward/record.uri?eid=2-s2.0-85204609400&doi=10.2140%2fapde.2024.17.3251&partnerID=40&md5=4e4277d54c39564569de596c11fb4598 |
| Access Level: | acceso abierto |
| Palabra clave: | Carleson measures Harmonic functions Nontangential maximal functions Square functions ε-approximability Uniform rectifiability |
| Sumario: | In relatively nice geometric settings, in particular, on Lipschitz domains, absolute continuity of elliptic measure with respect to the surface measure is equivalent to Carleson measure estimates, to square function estimates, and to ε-approximability, for solutions to the second-order divergence-form elliptic partial differential equations Lu = − div(A∇u) = 0. In more general situations, notably, in an open set Ω with a uniformly rectifiable boundary, absolute continuity of elliptic measure with respect to the surface measure may fail, already for the Laplacian. In the present paper, extending and clarifying our previous work (Duke Math J. 165:12 (2016), 2331–2389), we demonstrate that nonetheless, Carleson measure estimates, square function estimates, and ε-approximability remain valid in such Ω, for solutions of Lu = 0, provided that such solutions enjoy these properties in Lipschitz subdomains of Ω. Moreover, we establish a general real-variable transference principle, from Lipschitz to chord-arc domains, and from chord-arc to open sets with uniformly rectifiable boundary, that is not restricted to harmonic functions or even to solutions of elliptic equations. In particular, this allows one to deduce the first Carleson measure estimates and square function bounds for higher-order systems on open sets with uniformly rectifiable boundaries and to treat subsolutions and subharmonic functions. © 2024 MSP (Mathematical Sciences Publishers). |
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