Semipolar sets and intrinsic Hausdorff measure

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).