Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Murat Akman; Steve Hofmann; José María Martell; Tatiana Toro

doi:10.1515/acv-2021-0053

Published by De Gruyter August 30, 2022

Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Murat Akman , Steve Hofmann , José María Martell and Tatiana Toro

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2021-0053

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Abstract

Let Ω ⊂ ℝ n + 1 , n ≥ 2 , be a 1-sided non-tangentially accessible domain (also known as uniform domain), that is, Ω satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that Ω satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider two real-valued (non-necessarily symmetric) uniformly elliptic operators

L 0 ⁢ u = - div ⁡ ( A 0 ⁢ ∇ ⁡ u ) and L ⁢ u = - div ⁡ ( A ⁢ ∇ ⁡ u )

in Ω, and write ω L 0 and ω L for the respective associated elliptic measures. The goal of this article and its companion [M. Akman, S. Hofmann, J. M. Martell and T. Toro, Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition, preprint 2021, https://arxiv.org/abs/1901.08261v3] is to find sufficient conditions guaranteeing that ω L satisfies an A ∞ -condition or a RH q -condition with respect to ω L 0 . In this paper, we are interested in obtaining a square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work of Dahlberg, Jerison and Kenig and are fundamental for the proof of the perturbation results in the paper cited above.

Keywords: Uniformly elliptic operators; elliptic measure; Green function; capacity density condition; Ahlfors-regularity; reverse Hölder; Carleson measures; square function estimates; non-tangential maximal function estimates; dyadic analysis; sawtooth domains; perturbation

MSC 2010: 31B05; 35J08; 35J25; 42B37; 42B25; 42B99

Communicated by Kaj Nyström

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1664047

Award Identifier / Grant number: DMS-2000048

Award Identifier / Grant number: DMS-1664867

Award Identifier / Grant number: DMS-1954545

Award Identifier / Grant number: DMS-1440140

Funding source: Agencia Estatal de Investigación

Award Identifier / Grant number: CEX2019-000904-S

Award Identifier / Grant number: PID2019-107914GB-I00

Funding source: European Research Council

Award Identifier / Grant number: 615112 HAPDEGMT

Funding source: Simons Foundation

Award Identifier / Grant number: 614610

Funding statement: The second author was partially supported by NSF grants DMS-1664047 and DMS-2000048. The third author acknowledges financial support from MCIN, Agencia Estatal de Investigación, grants CEX2019-000904-S and PID2019-107914GB-I00, and that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The fourth author was partially supported by the Craig McKibben & Sarah Merner Professor in Mathematics, by NSF grants DMS-1664867 and DMS-1954545, and by the Simons Foundation Fellowship 614610. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.

Acknowledgements

The authors would like to thank the referee for a careful reading of the paper and for his/her comments and suggestions to clarify certain matters, and to make improvements in the presentation.

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Received: 2021-06-17

Revised: 2021-12-30

Accepted: 2022-06-02

Published Online: 2022-08-30

Published in Print: 2023-07-01

Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Abstract

Acknowledgements

References

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