Wiener-type tests from a two-sided Gaussian bound

A Wiener test à la Landis for evolutive Hörmander operators

In this paper we prove a Wiener-type characterization of boundary regularity, in the spirit of a classical result by Landis, for a class of evolutive Hörmander operators. We actually show the validity of our criterion for a larger class of degenerate-parabolic operators with a fundamental solution satisfying suitable two-sided Gaussian bounds. Our condition is expressed in terms of a series of balayages or, (as it turns out to be) equivalently, Riesz-potentials.

Two-sided Gaussian bounds for fundamental solutions of non-divergence form parabolic operators with Hölder continuous coefficients

We establish two-sided Gaussian bounds for fundamental solutions of general non-divergence form parabolic operators with Hölder continuous coefficients. The result we obtain is essentially based on parametrix method.

关于非散度形式二阶抛物方程基本解的双侧高斯估计

Two-sided Gaussian estimates for fundamental solutions of second-order parabolic equations in non-divergence form

本文在最小正则性假设下，通过局部有界性和对偶算子的弱Harnack不等式，建立了二阶抛物方程基本解的双侧高斯界，并避免了前人工作中所需的归一化对偶解。

Gaussian lower bound for the Neumann Green function of ageneral parabolic operator

Based on the fact that the Neumann Green function can be constructed as a perturbation of the fundamental solution by a single-layer potential, we establish gaussian two-sided bounds for the Neumann Green function for a general parabolic operator. We build our analysis on classical tools coming from the construction of a fundamental solution of a general parabolic operator by means of the so-called parametrix method. At the same time we provide a simple proof for the gaussian two-sided bounds for the fundamental solution. We also indicate how our method can be adapted to get a gaussian lower b…

Gaussian Estimates: A Brief History

Two-sided Gaussian estimates for the fundamental solution of a second order linear parabolic differential equation are upper and lower bounds in terms of the fundamental solution of the classical heat conduction equation. In his seminal 1958 paper Nash stated, without proof, two-sided non-Gaussian bounds for the fundamental solution of a uniformly parabolic divergence structure equation assuming only boundedness of the coefficients. In his 1967-1968 papers Aronson derived truly Gaussian estimates for the fundamental solutions of a large class of linear parabolic equations (including the diverg…

Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Let $Ω\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, $Ω$ satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that $Ω$ satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider $L_0 u=-\mathrm{div}(A_0\nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$, two real (non-necessarily symmetric) uniformly elliptic operators in $Ω$, and w…