arXiv subjects

Giulio Tralli

Publications and source records attributed to Giulio Tralli.

26 records · Page 2Linked to original sources

arXiv2019

Hardy-Littlewood-Sobolev inequalities for a class of non-symmetric and non-doubling hypoelliptic semigroups

In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper Hörmander discussed a general class of degenerate Ornstein-Uhlenbeck operators that includes Kolmogorov's as a special case. In this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy-Littlewood-Sobolev type in the situation when the drift matrix has nonnegative trace. Our work has been influenced by ideas of E. Stein and Varopoulos in the framework of symmetric semigroups. One of our objectives is to show that such ideas can be pushed to successfully handle the present degenerate non-symmetric setting.

math.AP↗

arXiv2019

Harnack inequality for a class of Kolmogorov-Fokker-Planck equations in non-divergence form

We prove invariant Harnack inequalities for certain classes of non-divergence form equations of Kolmogorov type. The operators we consider exhibit invariance properties with respect to a homogeneous Lie group structure. The coefficient matrix is assumed either to satisfy a Cordes-Landis condition on the eigenvalues, or to admit a uniform modulus of continuity.

math.AP↗

arXiv2018

Overdetermined problems and constant mean curvature surfaces in cones

We consider a partially overdetermined problem in a sector-like domain $Ω$ in a cone $Σ$ in $\mathbb{R}^N$, $N\geq 2$, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that $Ω$ is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces $Γ$ with boundary which satisfy a "gluing" condition with respect to the cone $Σ$. We prove that if either the cone is convex or the surface is a radial graph then $Γ$ must be a spherical cap. Finally we show that, under the condition that the relative boundary of the domain or the surface intersects orthogonally the cone, no other assumptions are needed.

math.AP↗

arXiv2017

Harnack's inequality for a class of non-divergent equations in the Heisenberg group

We prove an invariant Harnack's inequality for operators in non-divergence form structured on Heisenberg vector fields when the coefficient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach from [DGL08] to obtain Harnack's inequality.

math.AP↗

arXiv2016

$C^{1,α}$ estimates for the parallel refractor

We consider the parallel refractor problem when the planar radiating source lies in a medium having higher refractive index than the medium in which the target is located. We prove local $C^{1,α}$ estimates for parallel refractors under suitable geometric assumptions on the source and target, and under local regularity hypotheses on the target set. We also discuss existence of refractors under energy conservation assumptions.

math.AP↗

arXiv2015

Wiener-type tests from a two-sided Gaussian bound

In this paper we are concerned with hypoelliptic diffusion operators $\mathcal{H}$. Our main aim is to show, with an axiomatic approach, that a Wiener-type test of $\mathcal{H}$-regularity of boundary points can be derived starting from the following basic assumptions: Gaussian bounds of the fundamental solution of $\mathcal{H}$ with respect to a distance satisfying doubling condition and segment property. As a main step towards 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.

math.AP↗

arXiv2014

Wiener criterion for X-elliptic operators

In this note we prove a Wiener criterion of regularity of boundary points for the Dirichlet problem related to $X$-elliptic operators in divergence form enjoying the doubling condition and the Poincaré inequality. As a step towards this result, we exhibit some other characterizations of regularity in terms of the capacitary potentials. Finally, we also show that a cone-type criterion holds true in our setting.

math.AP↗

arXiv2013

A certain critical density property for invariant Harnack inequalities in H-type groups

We consider second order linear degenerate-elliptic operators which are elliptic with respect to horizontal directions generating a stratified algebra of H-type. Extending a result by Gutiérrez and Tournier for the Heisenberg group, we prove a critical density estimate by assuming a condition of Cordes-Landis type. We then deduce an invariant Harnack inequality for the non-negative solutions from a result by Di Fazio, Gutiérrez, and Lanconelli.

math.AP↗