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

Cinti, Chiara; Nyström, Kaj; Polidoro, Sergio

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 2, pp. 439-465.

We prove a Carleson type estimate, in Lipschitz type domains, for non-negative solutions to a class of second order degenerate differential operators of Kolmogorov type of the form

$$Ł=\sum _{i,j=1}^m{a}_{i,j}\left(z\right){\partial }_{x_i{x}_j}+\sum _{i=1}^m{a}_i\left(z\right){\partial }_{x_i}+\sum _{i,j=1}^N{b}_{i,j}{x}_i{\partial }_{x_j}-{\partial }_t,$$

where $z=(x,t)\in {ℝ}^{N+1}$, $1\le m\le N$. Our estimate is scale-invariant and generalizes previous results valid for second order uniformly parabolic equations to the class of operators considered.

Classification : 35K65, 35K70, 35H20, 35R03

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