Carleson estimates for the singular parabolic $p$-Laplacian in time-dependent domains

Abstract

We deal with the parabolic $p$-Laplacian in the so-called singular super-critical range $\frac{2N}{N+1}<p<2$, and we prove Carleson estimates for non-negative solutions in suitable non-cylindrical domains $\Omega \subset {\mathbb{R}}^{N+1}$. The sets $\Omega$ satisfy a proper NTA condition, tailored on the parabolic $p$-Laplacian. As an intermediate step, we show that in these domains non-negative solutions which vanish at the boundary, are Hölder continuous up to the same boundary.