In this paper, we are concerned with hypoelliptic diffusion operators H. Our main aim is to show, with an axiomatic approach, that a Wiener-type test of H-regularity of boundary points can be derived starting from the following basic assumptions: Gaussian bounds of the fundamental solution of H with respect to a distance satisfying doubling condition and segment property. As a main step toward this result, we establish some estimates at the boundary of the continuity modulus for the generalized Perron–Wiener solution to the relevant Dirichlet problem. The estimates involve Wiener-type series, with the capacities modeled on the Gaussian bounds. We finally prove boundary Hölder estimates of the solution under a suitable exterior cone condition.
Lanconelli, E., Tralli, G., Uguzzoni, F. (2017). Wiener-type tests from a two-sided Gaussian bound. ANNALI DI MATEMATICA PURA ED APPLICATA, 196(1), 217-244 [10.1007/s10231-016-0570-y].
Wiener-type tests from a two-sided Gaussian bound
LANCONELLI, ERMANNO;TRALLI, GIULIO;UGUZZONI, FRANCESCO
2017
Abstract
In this paper, we are concerned with hypoelliptic diffusion operators H. Our main aim is to show, with an axiomatic approach, that a Wiener-type test of H-regularity of boundary points can be derived starting from the following basic assumptions: Gaussian bounds of the fundamental solution of H with respect to a distance satisfying doubling condition and segment property. As a main step toward this result, we establish some estimates at the boundary of the continuity modulus for the generalized Perron–Wiener solution to the relevant Dirichlet problem. The estimates involve Wiener-type series, with the capacities modeled on the Gaussian bounds. We finally prove boundary Hölder estimates of the solution under a suitable exterior cone condition.File | Dimensione | Formato | |
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