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

Abstract

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 formOur estimate is scale-invariant and generalizes previous results valid for second order uniformly parabolic equations to the class of operators considered.

Boundary estimates for certain degenerate and singular parabolic equations

“…In the elliptic context we mention [33] for the Laplace operator in nontangentially accessible domains, [8], and [3], [4], [25] for elliptic operators in divergence and non-divergence form, respectively, [40], [41], for the p-Laplace operator, [12], [13] for the Kolmogorov operator.…”

“…Indeed, in the critical and sub-critical range, explicit counterexamples rule out the possibility of a Harnack inequality. Only so-called Harnack-type estimates are possible, where, however, the ratio of infimum over supremum in proper space-time cylinders depends on the solution itself (for more details, see [21,Chapter 6,[11][12][13][14][15]).…”

A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term

“…In the homogeneous case, perhaps the most flexible proof of the Carleson estimate is due to [11] and has been adapted to many situations, see e.g., [1,3,4,14,17,18]. This proof relies on two basic estimates:…”

“…14), the definition of G and the condition G(1 2) = 0 to estimate(6.18)e e K+1 2−ε ≤ e R−(1+ε)s ds ≤ e e R−(1+ε) 4by possibly enlarging H min (ε). Therefore we may estimate the value of ψ at x by (6.18) and (6.15) and getψ(x) = G(3 8) − G(1 2) e R−(1+ε)s ds ≥ ce e R−7 16(1+ε) ≥ c e e R−(1+ε) 4 e −3 16−ε ≥ c e e K+1 2−ε e −3 16−ε (by (6.18)) = c e e K+1 4 e 1 16−2ε…”

Wiener-type tests from a two-sided Gaussian bound

“…In these settings, more literature is available relating to the boundary behavior, in sufficiently regular domains, of nonnegative solutions to evolution equations (see e.g. the recent papers [6,7,13,27], and the references therein).…”