BMO Solvability and Absolute Continuity of Caloric Measure
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BMO Solvability and Absolute Continuity of Caloric Measure Alyssa Genschaw1
· Steve Hofmann1
Received: 16 May 2019 / Accepted: 31 January 2020 / © Springer Nature B.V. 2020
Abstract We show that BMO-solvability implies scale invariant quantitative absolute continuity (specifically, the weak-A∞ property) of caloric measure with respect to surface measure, for an open set ⊂ Rn+1 , assuming as a background hypothesis only that the essential boundary of satisfies an appropriate parabolic version of Ahlfors-David regularity, entailing some backwards in time thickness. Since the weak-A∞ property of the caloric measure is equivalent to Lp solvability of the initial-Dirichlet problem, we may then deduce that BMO-solvability implies Lp solvability for some finite p. Keywords BMO · Dirichlet problem · Caloric measure · Parabolic measure · Divergence form parabolic equations · Weak-A∞ · Ahlfors-David regularity Mathematics Subject Classification (2010) 42B99 · 42B25 · 35J25 · 42B20
1 Introduction In the setting of divergence form elliptic PDE, it is well known that solvability of the Dirichlet problem with Lp data is equivalent to scale-invariant absolute continuity of elliptic-harmonic measure (specifically that elliptic-harmonic measure belongs to the Muckenhoupt weight class A∞ with respect to surface measure on the boundary). To be more precise, in a Lipschitz or even chord-arc domain, one obtains that the Dirichlet problem is solvable with data in Lp () for some 1 < p < ∞, if and only if elliptic-harmonic measure ω with some fixed pole is absolutely continuous with respect to surface measure σ on the boundary, and the Poisson kernel dω/dσ satisfies a reverse H¨older condition with exponent p = p/(p − 1); see the monograph of Kenig [19], and the references cited there. In fact, The authors were supported by NSF grant number DMS-1664047. Alyssa Genschaw
[email protected] Steve Hofmann [email protected] 1
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
A. Genschaw, S. Hofmann
the equivalence between Lp solvability and quantitative absolute continuity holds much more generally, for any open set with an Ahlfors-David regular boundary (see [14] for a proof, although the result is somewhat folkloric); in this generality, the A∞ /reverse-H¨older property is (necessarily) replaced by its weak version, which does not entail doubling. These results have endpoint versions, as well: in [5], Dindos, Kenig and Pipher showed that in a Lipschitz domain (or even a chord-arc domain) elliptic-harmonic measure satisfies an A∞ condition with respect to surface measure, if and only if a natural Carleson measure/BMO estimate holds for solutions of the Dirichlet problem with continuous data. The results of [5] have been extended to the setting of a 1-sided Chord-arc domain by Z. Zhao [28]. In the above works, the proofs relied substantially on quantitative connectivity of the domain, in the form of the Harnack Chain condition. More recently, the second named author and P. Le [14] proved an
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