In the recent work G. David, J. Feneuil, and the first author have launched a program devoted to an analogue of harmonic measure for lower-dimensional sets. A relevant class of partial differential equations, analogous to the class of elliptic PDEs in the classical context, is given by linear degenerate equations with the degeneracy suitably depending on the distance to the boundary. The present paper continues this line of research and focuses on the criteria of quantitative absolute continuity of the newly defined harmonic measure with respect to the Hausdorff measure, ω ∈ A∞(σ), in terms of solvability of boundary value problems. The authors establish, in particular, square function estimates and solvability of the Dirichlet problem in BMO for domains with lower-dimensional boundaries under the underlying assumption ω ∈ A∞(σ). More generally, it is proved that in all domains with Ahlfors regular boundaries the BMO solvability of the Dirichlet problem is necessary and sufficient for the absolute continuity of the harmonic measure.
Bibliographical noteFunding Information:
Mayboroda was partially supported by the NSF INSPIRE Award DMS-1344235, NSF CAREER Award DMS-1220089, DMS-1839077 and the Simons Foundation grant 563916, SM. Zhao was partially supported by NSF grant numbers DMS-1361823, DMS-1500098 and DMS-1664867. This material is based upon work supported by the National Science Foundation under grant number DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester. MSC2010: 35J25, 42B37, 31B35. Keywords: harmonic measure, A1 Muckenhoupt weight, BMO solvability, degenerate elliptic operator.
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- A Muckenhoupt weight
- BMO solvability
- Degenerate elliptic operator
- Harmonic measure