A Carleson-type estimate in Lipschitz type domains for non-negative solutions to Kolmogorov operators (Q2844280)

The authors consider nonnegative solutions to a class of second order degenerate differential operators of Kolmogorov type of the form NEWLINE\[NEWLINE L = \sum_{i,j=1}^m a_{i,j}(z)\partial_{x_i x_j} + \sum_{i=1}^m a_{i}(z)\partial_{x_i} + \sum_{i,j=1}^N b_{i,j}x_i\partial_{x_j} - \partial_t NEWLINE\]NEWLINE where \(z = (x,t) \in\mathbb R^{N+1}\) and \(1\leq m\leq N\).NEWLINENEWLINEBy assuming that the domain is Lipschitz (more precisely the authors use a suitable notion of Lipschitz continuity defined in terms of dilations) they prove a scale-invariant Carleson type estimate.NEWLINENEWLINEThis result generalizes classical results proved for second order uniformly parabolic equations. The boundary regularity for the Kolmogorov operator is a very recent research topic, and this result is an important step to establish a regularity theory for the free boundaries occurring in the obstacle problem for Kolmogorov operators.