Nonlinear parabolic thin sets and parabolic Wolff’s inequality

We prove a parabolic analogue of Wolff’s inequality adapted to the intrinsic scaling δ c(x,t)=(cx,c²t) δ c ( x , t ) = ( c x , c 2 t ) and formulated in terms of time-backward parabolic dyadic rectangles. As a consequence, we obtain equivalent characterizations of parabolic (α ,q) ( α , q ) -thinness in this geometric setting and establish the associated Kellogg and Choquet properties. We further use the notion of (α ,2) ( α , 2 ) -thinness defined in terms of fractional heat balls and prove that the sets of irregular boundary points z₀∈ ∂ Ω z 0 ∈ ∂ Ω for the heat operator ∂ ₜ-Δ ∂ t - Δ and for the degenerate operator La=∂ ₜ(|y|ᵃ· )-div(|y|ᵃ∇ · ) L a = ∂ t ( | y | a · ) - div ( | y | a ∇ · ) in Ω ⊂ Rᵈ⁺¹ Ω ⊂ R d + 1 are negligible with respect to the thermal capacity cap T cap T and the parabolic Bessel capacity Cα ,2 C α , 2 , respectively.