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Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators


Authors: Steve Hofmann, Carlos Kenig, Svitlana Mayboroda and Jill Pipher
Journal: J. Amer. Math. Soc. 28 (2015), 483-529
MSC (2010): Primary 42B99, 42B25, 35J25, 42B20
DOI: https://doi.org/10.1090/S0894-0347-2014-00805-5
Published electronically: May 21, 2014
MathSciNet review: 3300700
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Abstract: We consider divergence form elliptic operators $ L= {-}\textup {div}\, A(x) \nabla $, defined in the half space $ \mathbb{R}^{n+1}_+$, $ n\geq 2$, where the coefficient matrix $ A(x)$ is bounded, measurable, uniformly elliptic, $ t$-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation $ Lu=0$, and we then combine these estimates with the method of ``$ \epsilon $-approximability'' to show that $ L$-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class $ A_\infty $ with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in $ L^p$, for some $ p<\infty $). Previously, these results had been known only in the case $ n=1$.


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Additional Information

Steve Hofmann
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: hofmanns@missouri.edu

Carlos Kenig
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois, 60637
Email: cek@math.chicago.edu

Svitlana Mayboroda
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455
Email: svitlana@math.umn.edu

Jill Pipher
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
Email: jpipher@math.brown.edu

DOI: https://doi.org/10.1090/S0894-0347-2014-00805-5
Keywords: Divergence form elliptic equations, Dirichlet problem, harmonic measure, square function, non-tangential maximal function, $\epsilon$-approximability, $A_\infty$ Muckenhoupt weights, layer potentials.
Received by editor(s): February 10, 2012
Received by editor(s) in revised form: February 11, 2014
Published electronically: May 21, 2014
Additional Notes: Each of the authors was supported by the NSF
This work has been possible thanks to the support and hospitality of the University of Chicago, the University of Minnesota, the University of Missouri, Brown University, and the BIRS Centre in Banff (Canada). The authors would like to express their gratitude to these institutions.
Article copyright: © Copyright 2014 American Mathematical Society