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Transference of
scale-invariant estimates from Lipschitz to nontangentially
accessible to uniformly rectifiable domains
Steve Hofmann, José María Martell and Svitlana Mayboroda |
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Vol. 17 (2024), No. 9, 3251–3334
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Abstract |
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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 -approximability, for solutions to the second-order divergence-form elliptic partial differential equations . In more general situations, notably, in an open set 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 -approximability remain valid in such , for solutions of , provided that such solutions enjoy these properties in Lipschitz subdomains of . 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. |
Keywords
Carleson measures, square functions, nontangential maximal
functions, $\varepsilon$-approximability, uniform
rectifiability, harmonic functions
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Mathematical Subject Classification
Primary: 42B25, 42B37
Secondary: 28A75, 28A78, 31B05, 42B20
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Milestones
Received: 22 May 2022
Revised: 4 May 2023
Accepted: 13 June 2023
Published: 1 November 2024
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Authors |
| Steve Hofmann | |
| Department of Mathematics University of Missouri Columbia, MO United States |
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| José María Martell | |
| Instituto de Ciencias Matemáticas
CSIC-UAM-UC3M-UCM Consejo Superior de Investigaciones Científicas Madrid Spain |
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| Svitlana Mayboroda | |
| Department of Mathematics University of Minnesota Minneapolis, MN United States |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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