Abstract
We give a systematic treatment of caloric measure null sets on the essential boundary $\partial_eE$ of an arbitrary open set $E$ in ${\bf R}$. We discuss two characterisations of such sets and present some basic properties. We investigate the dependence of caloric measure null sets on the open set $E$. Thus, if $D$ is an open subset of $E$ and $Z\subseteq\partial_eE\cap\partial_eD$, we show that $Z$ is caloric measure null for $D$ if it is caloric measure null for $E$. We also give conditions on $E$ and $Z$ which imply that the reverse implication is true. We know from \cite{watson2011} that any polar subset of $\partial_eD$ is caloric measure null for $D$, but the reverse implication is not generally true. In our final result we show that, for subsets of a certain component of $\partial_eD$, caloric measure null sets are necessarily polar.
Highlights
Caloric measure is sometimes called parabolic measure, sometimes harmonic measure for the heat equation
Its null sets have been studied by several authors for particular boundaries, for example in [3, 4, 5, 6, 7, 14, 15]. To my knowledge they have never been given a systematic treatment for arbitrary open sets
We recall from [10] that a subset Z of ∂eE is a caloric measure null set for E if μEp (Z) = 0 for all p ∈ E
Summary
Caloric measure is sometimes called parabolic measure, sometimes harmonic measure for the heat equation. For any point p ∈ E there is a unique nonnegative Borel measure μEp on ∂eE such that SfE(p) = ∂eE f dμEp holds for every f ∈ C(∂eE) The completion of this measure is called the caloric measure relative to E and p; it is denoted by μEp. If f is resolutive for E, SfE has the representation SfE(p) = ∂eE f dμEp for all p ∈ E. We recall from [10] that a subset Z of ∂eE is a caloric measure null set for E if μEp (Z) = 0 for all p ∈ E. Under certain conditions on E and Z, that the reverse implication is true These results are analogues of known results for harmonic measure null sets. Note that if q ∈ ∂a∗D there is an upper half-ball H∗(q, r) ⊆ D, so that q ∈/ ∂sD and q ∈ ∂eD
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