Nonlinear parabolic thin sets and parabolic Wolff inequalities

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.