A boundary estimate for non-negative solutions to Kolmogorov operators in non-divergence form

AbstractWe consider non-negative solutions to a class of second-order degenerate Kolmogorov operators of the form $$ \fancyscript{L}=\sum_{i,j=1}^m a_{i,j}(z)\partial_{x_i x_j}+\sum_{i=1}^m a_i(z)\partial_{x_i} +\sum_{i,j=1}^N b_{i,j}x_i\partial_{x_j}-\partial_t, $$where z = (x, t) belongs to an open set $${\Omega \subset \mathbb{R}^N\times\mathbb{R}}$$, and 1 ≤ m ≤ N. Let $${\widetilde z \in \Omega}$$, let K be a compact subset of $${\overline \Omega}$$, and let $${\Sigma \subset \partial \Omega}$$ be such that $${K \cap \partial \Omega \subset \Sigma}$$. We give sufficient geometric conditions for the validity of the following Carleson type estimate. There exists a positive constant CK, depending only on $${\Omega, \Sigma, K, \widetilde{z}}$$ and on $${\fancyscript{L}}$$, such that$$ \sup_K u \leq C_K \, u(\widetilde{z}), $$for every non-negative solution u of $${\fancyscript{L} u = 0}$$ in Ω such that $${u_{\mid \Sigma} = 0}$$.