1,720,988 research outputs found
A Wiener test à la Landis for evolutive Hörmander operators
Full text linkIn 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
Problemi di simmetria per palle della gauge nel gruppo di Heisenberg
Full text linkIn this note we focus on possible characterizations of gauge-symmetric functions in the Heisenberg group. We discuss a family of inverse problems in potential theory relating solid and surface weighted mean-value formulas, and we show a partial solution to such problems. To this aim, we review a uniqueness result for gauge balls obtained with V. Martino in [23] by means of overdetermined problems of Serrin-type. The class of competitor sets we consider enjoys partial symmetries of toric and cylindrical type.In questa nota vengono discusse possibili caratterizzazioni di funzioni gauge-simmetriche nel gruppo di Heisenberg. Viene mostrata una soluzione parziale ad una famiglia di problemi inversi legati ad opportune formule di media solida e superficiale pesate per funzioni armoniche rispetto al subLaplaciano di Heisenberg. A questo scopo, viene presentato un risultato di unicità ottenuto in [23] con V. Martino per problemi sovradeterminati di tipo Serrin in questo contesto. La classe di insiemi considerata gode di proprietà di simmetria parziale di tipo torico e cilindrico
Some global Sobolev inequalities related to Kolmogorov-type operators
Full text linkIn this note we review a recent result in [17] in collaboration with N. Garofalo, where we establish global versions of Hardy-Littlewood-Sobolev inequalities attached to hypoelliptic equations of Kolmogorov type. The relevant Sobolev spaces are defined through the fractional powers of the operator under consideration. We outline the main steps of the semigroup approach we adopt.Viene qui presentato un recente risultato ottenuto in [17] in collaborazione con N. Garofalo, in cui si dimostrano disuguaglianze globali di tipo Hardy-Littlewood-Sobolev relative ad una classe di operatori ipoellittici di tipo Kolmogorov. Nell'approccio adottato gli spazi di Sobolev sono definiti attraverso le potenze frazionarie dell'operatore in questione
A Class of Nonlocal Hypoelliptic Operators and their Extensions
Full text linkIn this paper we study nonlocal equations driven by the fractional powers of hypoelliptic operators in the form Ku = Au -partial_t u = tr(Q nabla^2 u) + - partial_t u, introduced by Hormander in his 1967 hypoellipticity paper. We show that the nonlocal operators (-K)^s , (-A)^s can be realized as the Dirichlet-to-Neumann map of doubly-degenerate extension problems. We solve such problems in L^infty, in L^p for 1 = 0. In forthcoming works we use such calculus to establish some new Sobolev and isoperimetric inequalities
On the Minkowski Formula for Hypersurfaces in Complex Space Forms
Full text linkIn this paper, we discuss various Minkowski-type formulas for real hypersurfaces in complex space forms. In particular, we investigate the formulas suggested by the natural splitting of the tangent space. In this direction, our main result concerns a new kind of 2nd Minkowski formula
Nonlocal isoperimetric inequalities for Kolmogorov-Fokker-Planck operators
No full textIn this paper we establish optimal isoperimetric inequalities for a nonlocal perimeter adapted to the fractional powers of a class of Kolmogorov-Fokker-Planck operators which are of interest in physics. These operators are very degenerate and do not possess a variational structure. The prototypical example was introduced by Kolmogorov in his 1938 paper on Brownian motion and the theory of gases. Our work has been influenced by ideas of M. Ledoux in the local case
Overdetermined problems and constant mean curvature surfaces in cones
Full text linkWe consider a partially overdetermined problem in a sector-like domain Ω in a cone Σ in RN, N ≥ 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
Hardy–Littlewood–Sobolev inequalities for a class of non-symmetric and non-doubling hypoelliptic semigroups
Full text linkIn 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
Feeling the heat in a group of Heisenberg type
No full textIn this paper we use the heat equation in a group of Heisenberg type G to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators Ls and Ls,
Harnack inequality for a class of Kolmogorov–Fokker–Planck equations in non-divergence form
Full text linkWe 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.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
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