Nonlinear parabolic thin sets and parabolic Wolff’s inequality

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