Abstract
This paper studies the regularity problem for block uniformly elliptic operators in divergence form with complex bounded measurable coefficients. We consider the case where the boundary data belongs to Lebesgue spaces with weights in the Muckenhoupt classes. Our results generalize those of S. Mayboroda (and those of P. Auscher and S. Stahlhut employing the first order method) who considered the unweighted case. To obtain our main results we use the weighted Hardy space theory associated with elliptic operators recently developed by the last two named authors. One of the novel contributions of this paper is the use of an “inhomogeneous” vertical square function which is shown to be controlled by the gradient of the function to which is applied in weighted Lebesgue spaces.
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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 second author acknowledges financial support from the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S). The second and third authors were partially supported by the Spanish Ministry of Science and Innovation, MTM PID2019-107914GB-I00.
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Chen, L., Martell, J.M. & Prisuelos-Arribas, C. The Regularity Problem for Uniformly Elliptic Operators in Weighted Spaces. Potential Anal 58, 409–439 (2023). https://doi.org/10.1007/s11118-021-09945-w
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DOI: https://doi.org/10.1007/s11118-021-09945-w
Keywords
- Regulatity problem
- Uniformly elliptic operators in divergence form
- Muckenhoupt weights
- Singular non-integral operators
- Square functions
- Heat and Poisson semigroups
- A priori estimates
- Off-diagonal estimates
- Square roots of elliptic operators
- Kato’s conjecture