Polarity transfer from intrinsic Hausdorff measure in balayage spaces: extending the criteria for polar sets beyond hypotheses (*) and (iv')

Hansen and Netuka (Potential Analysis 51, 2019) introduced an intrinsic Hausdorff measure $m_G$ built from the balls of a Green kernel $G$ and proved that every set of finite $m_G$ -measure is $G$ -semipolar; in balayage spaces — the axiomatic framework covering harmonic spaces of elliptic and parabolic PDEs and the potential theory of diffusions and jump processes — $G$ -semipolar Borel sets are exactly the semipolar sets. This solved affirmatively a 1985 problem of Taylor and Watson asking whether every set of finite anisotropic Hausdorff measure is semipolar for the heat equation. The parallel statement for polar sets, that every $G$ -polar Borel set is polar and hence every $m_G$ -null set is polar, was proven only under two additional hypotheses: (*), harmonicity of the balayage of the constant function 1 off compact sets, which provably fails in some balayage spaces, and (iv'), representability of bounded potentials with compact superharmonic support by $G$ . The open problem considered here is the completion of this dictionary between $G$ -analytic smallness and potential-theoretic smallness: decide whether the polarity transfer holds under the base hypotheses on $G$ alone, and decide the key open geometric instance for heat kernels with two-sided bounds on metric measure spaces, where currently only semipolarity is known.

Background

For the Laplace equation on $\mathbb{R}^n$ it is classical that small sets in the sense of Hausdorff measure are exceptional for potential theory: a set of finite $(n-2)$ -dimensional Hausdorff measure is polar, where a set is polar if the associated elliptic process never hits it, equivalently if the reduced (swept down) function of the constant 1 over the set vanishes. For the heat equation $\partial_t u = \Delta u$ on $\mathbb{R}^{n+1}$ , Taylor and Watson (1985) introduced an anisotropic (parabolic) Hausdorff measure $m_P$ built from parabolic cylinders whose radii scale like $\rho$ in space and $\rho^2$ in time; they proved that $m_P(A)=0$ implies $A$ is polar, exhibited nonpolar subsets of horizontal hyperplanes with $0 < m_P(A) < \infty$ (such sets are semipolar), and asked whether every set with $m_P(A) < \infty$ is semipolar.

The modern setting for such questions is axiomatic potential theory. A harmonic space (in the sense of Bauer and Constantinescu–Cornea) is a locally compact space with countable base equipped with a sheaf of 'harmonic' functions generalizing the solutions of an elliptic or parabolic PDE; a balayage space (Bliedtner–Hansen, 1986) is a more flexible axiomatic structure, given by a convex cone $\mathcal{W}$ of numerical functions, that covers harmonic spaces of second-order PDEs as well as the excessive cones of strong Markov processes such as diffusions and symmetric jump processes. A positive numerical function is superharmonic if it belongs to the cone locally; a potential is a positive superharmonic function whose greatest harmonic minorant is $0$ . For a set $A$ and a positive superharmonic $u$ , the reduced function $R^A_u$ is the pointwise infimum of all positive superharmonic functions dominating $u$ on $A$ , and its greatest lower semicontinuous minorant $\widehat{R}^A_u$ is the balayage of $u$ on $A$ . A set is polar if $\widehat{R}^A_1 = 0$ (the associated process almost surely never hits it), totally thin if for every point $x$ some superharmonic $u \ge 0$ has $\widehat{R}^A_u(x) < u(x)$ , and semipolar if it is a countable union of totally thin sets (the associated process hits it at most countably many times). Every polar set is semipolar; for the Laplace equation on $\mathbb{R}^n$ and for Riesz potentials every semipolar set is polar, but for jump processes semipolar sets need not be polar.

Hansen and Netuka (2019) proved the following. Given a 'Green function' $G : X \times X \to [0,\infty]$ satisfying simple regularity properties — each $G(\cdot,y)$ is a potential with superharmonic support $\{y\}$ and $\limsup_{x\to y} G(x,y)=\infty$ , $G$ is continuous off the diagonal and Borel on it, a compactness-boundedness condition, and representability of continuous real potentials with compact superharmonic support as $G\mu := \int G(\cdot,y)\,d\mu(y)$ — define $G$ -balls $B(x,\rho) := \{y : G(x,y) > 1/\rho\}$ and the intrinsic Hausdorff measure $m_G$ using the gauges $\rho$ over covers by $G$ -balls. Call a Borel set $A$ $G$ -polar if no nonzero measure supported by $A$ has bounded potential $G\mu$ , and $G$ -semipolar if no nonzero measure supported by $A$ has continuous real potential $G\mu$ (a measure-theoretic analogue, since in classical settings polar sets are exactly those supporting no measure with bounded Newtonian potential). Then: every set with $m_G(A) < \infty$ is contained in a $G$ -semipolar Borel set, and every set with $m_G(A) = 0$ is contained in a $G$ -polar Borel set. In a harmonic space or balayage space whose Green function has the properties above, $G$ -semipolar Borel sets coincide with semipolar sets, so finite $m_G$ -measure implies semipolarity — for the heat equation this answers the Taylor–Watson question affirmatively, with the intrinsic measure comparable to the parabolic Hausdorff measure: $(2n)^{-n} m_P \le m_G \le (2/n)^{n/2} m_P$ on $\mathbb{R}^{n+1}$ . Analogous equivalences between $m_G$ and ordinary Hausdorff measure $m_{\alpha-\beta}$ , and between the space-time intrinsic measure and an anisotropic measure $m_{\alpha,\beta}$ , hold for Green functions on metric measure spaces whose heat kernels admit two-sided bounds $t^{-\alpha/\beta}\Phi_j(d(x,y)t^{-1/\beta})$ (covering diffusions on manifolds and jump processes on fractals and on $d$ -sets, that is, Borel sets whose measure of balls of radius $r$ is comparable to $r^d$ ).

The polar half of the dictionary, however, is conditional. To prove that every $G$ -polar Borel set is polar — and hence that $m_G(A)=0$ implies $A$ is contained in a polar Borel set — Hansen and Netuka need two extra hypotheses: (), that for every compact $K$ the balayage $\widehat{R}^K_1$ is harmonic on $X \setminus K$ , and (iv'), that every bounded potential with compact superharmonic support is of the form $G\mu$ . Hypothesis () holds in harmonic spaces and in balayage spaces where every semipolar set is polar, but the authors note it fails in general, citing a balayage space and compact $K$ with $\widehat{R}^K_1$ not harmonic off $K$ in Bliedtner–Hansen (Potential Theory, 1986, V.9.1). Hypothesis (iv') follows from the base representability property when $G(x,\cdot)$ is lower semicontinuous at $x$ and continuous at finely isolated non-isolated points (here 'finely isolated' refers to the fine topology, the coarsest topology making all superharmonic functions continuous), by Hansen–Netuka, 'Representation of potentials' (2014, Theorem 4.1). Notably, in their metric-measure-space application for jump processes ( $\beta < \alpha$ ), the authors verify only the base properties, and so obtain only that sets of finite $m_{\alpha-\beta}$ -measure are semipolar — no polarity criterion is available there. Nothing in the paper addresses whether the hypotheses (*) and (iv') are genuinely necessary.

Problem Statement

Let $(X,\mathcal{W})$ be a balayage space with the constant function 1 superharmonic, and let $G$ be a Green function for $(X,\mathcal{W})$ having the base regularity properties (i')–(iv) of Hansen–Netuka (2019): each $G(\cdot,y)$ is a potential with superharmonic support $\{y\}$ and $\limsup_{x\to y}G(x,y)=\infty$ ; $G$ is continuous off the diagonal and Borel measurable on it; $G$ is bounded on $(X\setminus L)\times K$ for suitable compact $L$ depending on the compact $K$ ; and every continuous real potential with compact superharmonic support is representable as $G\mu$ . Recall that a Borel set $A$ is $G$ -polar if there is no nonzero measure $\mu$ with $\mu(X\setminus A)=0$ and $G\mu$ bounded, and that $A$ is polar if $\widehat{R}^A_1=0$ .

The problem, in decreasing order of primality, is:

(Polarity transfer) Decide whether every $G$ -polar Borel subset of $X$ is polar under the base properties (i')–(iv) alone, without the hypotheses (*) (harmonicity of $\widehat{R}^K_1$ on $X \setminus K$ for every compact $K$ ) and (iv') (representability of every bounded potential with compact superharmonic support as $G\mu$ ). An affirmative answer would in particular imply, via the kernel-side Corollary 3.6 of the source, that every set $A$ with $m_G(A)=0$ is contained in a polar Borel set; the two formulations are not equivalent, since a $G$ -polar set need not be $m_G$ -null (for the Newtonian kernel $G(x,y)=|x-y|^{-1}$ on $\mathbb{R}^3$ the $G$ -balls are Euclidean balls, so $m_G$ is comparable to one-dimensional Hausdorff measure; a line segment is polar, hence $G$ -polar by the classical fact that polar sets support no measure with bounded Newtonian potential, yet has positive $m_G$ -measure).
(Concrete geometric instance) In the metric-measure-space setting of Hansen–Netuka (2019, Section 7.2) — a heat semigroup on $(X,d,\mu)$ with two-sided heat kernel bounds $t^{-\alpha/\beta}\Phi_j(d(x,y)t^{-1/\beta})$ and $\beta<\alpha$ , where it is currently known only that sets of finite ordinary Hausdorff measure $m_{\alpha-\beta}(A)$ are semipolar — decide whether every set $A$ with $m_{\alpha-\beta}(A)=0$ is contained in a polar Borel set.

Items 1–2 share one objective: completing the dictionary between $G$ -analytic smallness (intrinsic Hausdorff measure and potential-boundedness conditions) and potential-theoretic smallness (semipolarity and polarity) to the same level of generality at which the semipolarity half is already established. Item 1 is the primary question and item 2 is its principal open geometric instance; the problem is settled when the polarity transfer is decided, affirmatively or negatively.

References

W. Hansen and I. Netuka, Semipolar sets and intrinsic Hausdorff measure, Potential Analysis 51 (2019), 49–69. DOI: 10.1007/s11118-018-9702-x; arXiv:1711.08918. https://doi.org/10.1007/s11118-018-9702-x
S. J. Taylor and N. A. Watson, A Hausdorff measure classification of polar sets for the heat equation, Mathematical Proceedings of the Cambridge Philosophical Society 97 (1985), 325–344. DOI: 10.1017/S0305004100062873. https://doi.org/10.1017/S0305004100062873
J. Bliedtner and W. Hansen, Potential Theory — An Analytic and Probabilistic Approach to Balayage, Universitext, Springer-Verlag, Berlin, 1986. DOI: 10.1007/978-3-642-71131-2. https://doi.org/10.1007/978-3-642-71131-2
W. Hansen and I. Netuka, Representation of potentials, Revue Roumaine de Mathématiques Pures et Appliquées 59 (2014), No. 1, 93–104. zbMATH: 1389.31018. https://zbmath.org/?q=an%3A1389.31018