Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries
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Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries Jonas Azzam1 Received: 17 December 2018 / Accepted: 4 August 2019 / © The Author(s) 2019
Abstract We show that given a domain ⊆ Rd+1 with uniformly non-flat Ahlfors s-regular boundary with s ≥ d, the dimension of its harmonic measure is strictly less than s. Keywords Harmonic measure · Dimension · Ahlfors regular sets Mathematics Subject Classification (2010) 31A15 · 28A75 · 28A78 · 31B05 · 35J25
1 Introduction The purpose of this note is to study the dimension of harmonic measure ω for a connected domain ⊆ Rd+1 . Naturally, dim ω ≤ dim ∂, but the inequality can be strict. For example, it is a classical result of Jones and Wolff [22] that if ⊆ C, then the dimension is always at most 1, even if dim ∂ > 1 (which improves on an earlier result of Makarov for simply connected planar domains [25]). In higher dimensions, the analogous property is no longer true: there are domains called Wolff snowflakes in Rd+1 whose harmonic measure can be strictly larger or strictly less than d; the d = 2 case is due to Wolff [34], and the general case is Lewis, Verchota and Vogel in [24] (note that even though the dimension can be above d, a result of Bourgain says that the dimension harmonic measure for any domain in Rd+1 can’t get too close to d + 1 [14], and it is an open problem to determine what the supremal dimension can be). While these are all very non-trivial results, these Wolff snowflakes actually have some nice geometry. In particular, they are two-sided uniform domains. We say a domain is C-uniform if for all x, y ∈ there is a curve γ ⊆ so that
H 1 (γ ) ≤ C|x − y| and
dist(z, c ) ≥ C −1 min{(x, z), (y, z)} where (a, b) denotes the length of the subarc of γ between a and b. A domain is two-sided c uniform if both and are C-uniform domains for some C. Jonas Azzam
[email protected] 1
School of Mathematics, University of Edinburgh, JCMB, Kings Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland
J. Azzam
Since two-sided uniform domains have boundaries with dimension at least d and a Wolff snowflake can have dimension less than d, this example also shows that a dimension drop for harmonic measure can occur for a domain that is quite nice in terms of its connectivity and boundary properties. Hence, it is an interesting problem to identify some general criteria for when a whole class of domains satisfy dim ω < dim ∂. A dimension drop for harmonic measure occurs for some domains whose boundaries have some self-similar structure. This phenomenon was first observed by Carleson [15] for complements of planar Cantor sets whose boundaries have dimension at least 1 (rather, he showed for a particular class of Cantor sets C, dim ω < 1). Later, Jones and Wolff showed the same result but for uniformly perfect sets satisfying a certain uniform disconnectedness property (see [22] or [18, Section X.I.2]). Makarov and Volberg showed dim ω < dim ∂ when ∂ belongs to a more general class of Cantor sets (with dim ∂ possibly below 1) [28] a
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