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BMO Solvability and Absolute Continuity of Harmonic Measure

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Abstract

We show that for a uniformly elliptic divergence form operator L, defined in an open set \(\Omega \) with Ahlfors–David regular boundary, BMO solvability implies scale-invariant quantitative absolute continuity (the weak-\(A_\infty \) property) of elliptic-harmonic measure with respect to surface measure on \(\partial \Omega \). We do not impose any connectivity hypothesis, qualitative, or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of \(\Omega \). In this generality, our results are new even for the Laplacian. Moreover, we obtain a partial converse, assuming in addition that \(\Omega \) satisfies an interior Corkscrew condition, in the special case that L is the Laplacian.

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Notes

  1. It might be more accurate to refer to this property as “VMO-solvability,” but BMO solvability seems to be the established terminology in the literature. Under less austere circumstances, e.g., in a Lipschitz or (more generally) a Chord-arc domain, it can be seen that the two notions are ultimately equivalent; see [6] for a discussion of this point, although for the reader’s convenience, we shall show below (see Remark 4.5) that in fact the equivalence holds for a domain with an ADR boundary, in the presence of “\(S<N\) estimates” in \(L^p\) (thus, in particular, in the special case of the Laplacian, in a domain satisfying an interior Corkscrew condition). In the more general setting of our Theorem 1.1 this matter is not settled.

  2. See also [11] and [20] for more general versions of the result of [12].

  3. We shall say precisely what this means in the sequel; see Proposition 4.1, and Remark 4.2.

  4. i.e., either \(\Omega \) is a bounded domain, or both \(\Omega \) and \(\partial \Omega \) are unbounded. We discuss a variant of our results, valid in the case that \(\partial \Omega \) is bounded but \(\Omega \) is unbounded, in Sect. 5.

  5. We remark that in fact, the \(L^\infty \) version of (1.3) actually characterizes uniform rectifiability: a converse to the result of [13] has recently been obtained in [9].

  6. The BMO estimate for f in (3.6) follows from the fact that \(\mathcal {M}(1_A)^{1/2}\) is an A\(_1\) weight with A\(_1\) constant depending only on dimension, and that the log of an A\(_1\) weight w belongs to BMO, with BMO norm depending only on the A\(_1\) constant of w; see, e.g., [8, Ch. 2, Theorems 3.3 and 3.4].

  7. Recall that our ambient dimension is \(n+1\), with \(n\ge 2\), so that the fundamental solution of L decays to 0 at infinity.

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Acknowledgements

We are grateful to Simon Bortz for a suggestion which has simplified one of our arguments in Sect. 4. We also thank the referee for a careful reading of the manuscript, for suggesting numerous improvements to the exposition, and especially for pointing out an error in the original proof of Lemma 3.3. Finally, we thank Chema Martell for suggesting the approach that we have used in Sect. 5 to treat the case of an unbounded domain with bounded boundary. The authors were supported by NSF Grant Number DMS-1361701. Steve Hofmann was also supported by NSF Grant Number DMS-1664047

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  1. Phi Le

    Present address: Department of Mathematics, Syracuse University, Syracuse, NY, 13244-1150, USA

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  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Steve Hofmann & Phi Le

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Hofmann, S., Le, P. BMO Solvability and Absolute Continuity of Harmonic Measure. J Geom Anal 28, 3278–3299 (2018). https://doi.org/10.1007/s12220-017-9959-0

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