Semipolar sets and intrinsic Hausdorff measure

TL;DR

This paper introduces an intrinsic Hausdorff measure based on a Green function and demonstrates its equivalence to classical measures in potential theory and heat kernel contexts, solving an open problem related to the heat equation.

Contribution

It establishes a new intrinsic Hausdorff measure linked to Green functions and proves its equivalence to classical measures in various potential-theoretic and heat kernel settings.

Findings

Every set with finite intrinsic Hausdorff measure is contained in a G-semipolar set.

The intrinsic Hausdorff measure is equivalent to classical Hausdorff measures in potential theory.

The measure is equivalent to an anisotropic Hausdorff measure for the heat equation case.

Abstract

Given a "Green function" $G$ on a locally compact space $X$ with countable base, a Borel set $A$ in $X$ is called $G$-semipolar, if there is no measure $\nu\ne 0$ supported by $A$ such that $G\nu:=\int G(\cdot,y)\,d\nu(y)$ is a continuous real function on $X$. Introducing an intrinsic Hausdorff measure $m_G$ using $G$-balls $B(x,\rho):=\{y\in X\colon G(x,y)>1/\rho\}$, it is shown that every set $A$ in $X$ with $m_G(A)<\infty$ is contained in a $G$-semipolar Borel set. This is of interest, since $G$-semipolar sets are semipolar in the potential-theoretic sense (countable unions of totally thin sets, hit by a corresponding process at most countably many times) provided $G$ is really a Green function for a harmonic space or, more generally, a balayage space. For classical potential theory and Riesz potentials on $R^n$ or, more generally, for Green functions on a metric measure space $(X,d,\mu)$ (where balls are relatively compact) given by a continuous heat kernel $(x,y,t)\mapsto p_t(x,y)$ with upper and lower bounds of the form $t^{-\alpha/\beta}\Phi_j(d(x,y)t^{-1/\beta})$, $j=1,2$, the intrinsic Hausdorff measure is equivalent to an ordinary Hausdorff measure $m_{\alpha-\beta}$. It is shown that for the corresponding space-time situation on $X\times R$ (heat equation on $R^n \times R$ in the classical case of the Gauss-Weierstrass kernel) the intrinsic Hausdorff measure is equivalent to an anisotropic Hausdorff measure $m_{\alpha,\beta}$ (with $\alpha=n$ and $\beta=2$ for the heat equation). In particular, our result solves an open problem for the heat equation (which was the initial motivation for the paper).