Nonlinear parabolic thin sets and parabolic Wolff inequalities

TL;DR

This paper develops a parabolic analogue of Wolff's inequality using intrinsic scaling and dyadic rectangles, leading to new characterizations of thinness and capacity properties for heat operators and degenerate operators.

Contribution

It introduces a parabolic version of Wolff's inequality and applies it to characterize thinness and capacity for heat and degenerate operators in parabolic settings.

Findings

Established a parabolic Wolff inequality with intrinsic scaling.

Characterized parabolic thinness and proved Kellogg and Choquet properties.

Showed sets of irregular boundary points are negligible with respect to specific capacities.

Abstract

We prove a parabolic analogue of Wolff's inequality adapted to the intrinsic scaling $\delta_c(x,t)=(cx,c^2t)$ and formulated in terms of time-backward parabolic dyadic rectangles. As a consequence, we obtain equivalent characterizations of parabolic $(\alpha,q)$-thinness in this geometric setting and establish the associated Kellogg and Choquet properties. We further use the notion of $(\alpha,2)$-thinness defined in terms of fractional heat balls and prove that the sets of irregular boundary points $z_0\in\partial\Omega$ for the heat operator $\partial_t-\Delta$ and for the degenerate operator $\mathscr{L}a=\partial_t(|y|^a\cdot)-\operatorname{div}(|y|^a\nabla\cdot)$ in $\Omega\subset\mathbb{R}^{d+1}$ are negligible with respect to the thermal capacity $\mathrm{cap}^{\mathcal T}$ and the parabolic Bessel capacity $C_{\alpha,2}$, respectively.