Harnack inequality for non-divergence form operators on stratified groups

We prove lower bounds for the fundamental solutions of the non-divergence form operators

   and

where the 's are Hörmander vector fields generating a stratified group and is a positive-definite matrix with Hölder continuous entries. We then prove an invariant Harnack inequality for such operators. As a byproduct we also study some relevant properties of the Green functions on bounded domains.

     

Fundamental solutions and Liouville type theorems for nonlinear integral operators

In this article we study basic properties for a class of nonlinear integral operators related to their fundamental solutions. Our goal is to establish Liouville type theorems: non-existence theorems for positive entire solutions for Iu?0 and for Iu+up?0, p>1.We prove the existence of fundamental solutions and use them, via comparison principle, to prove the theorems for entire solutions. The non-local nature of the operators poses various difficulties in the use of comparison techniques, since usual values of the functions at the boundary of the domain are replaced here by values in the complement of the domain. In particular, we are not able to prove the Hadamard Three Spheres Theorem, but we still obtain some of its consequences that are sufficient for the arguments.     

On a variational problem for radial solutions to extremal elliptic equations

On the ball |x| ≤ 1 of R ^m , m ≥ 2, a radial variational problem, related to a priori estimates for solutions to extremal elliptic equations with fixed ellipticity constant α is investigated. Such a problem has been studied and solved [see Manselli Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] in L ^p spaces, with p ≤ m. In this paper, we assume p > m and we prove the existence of a positive number α ₀ = α ₀(p,m) such that if there exists a smooth function maximizing the problem, whose representation is explicitly determined as in Manselli [Ann. Mat. Pura Appl. (IV), t. LXXXIX:31–54, 1971] This fact is no longer true if 0 < α < α ₀.      

Maximal functions for multipliers on stratified groups

In this paper we obtain a refined $L^p$ bound for maximal functions of the multiplier operators on stratified groups and maximal functions of the multi‐parameter multipliers on product spaces of stratified groups. As an application we find a refined $L^p$ bound for maximal functions of joint spectral multipliers on Heisenberg group.     

The coarea formula for real‐valued Lipschitz maps on stratified groups

We establish a coarea formula for real‐valued Lipschitz maps on stratified groups when the domain is endowed with a homogeneous distance and level sets are measured by the Q – 1 dimensional spherical Hausdorff measure. The number Q is the Hausdorff dimension of the group with respect to its Carnot–Carathéodory distance. We construct a Lipschitz function on the Heisenberg group which is not approximately differentiable on a set of positive measure, provided that the Euclidean notion of differentiability is adopted. The coarea formula for stratified groups also applies to this function, where the Euclidean one clearly fails. This phenomenon shows that the coarea formula holds for the natural class of Lipschitz functions which arises from the geometry of the group and that this class may be strictly larger than the usual one. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fundamental solutions of homogeneous elliptic differential operators

We compute fundamental solutions of homogeneous elliptic differential operators, with constant coefficients, on Rn by mean of analytic continuation of distributions. The result obtained is valid in any dimension, for any degree and can be extended to pseudodifferential operators of the same type.     

An uncertainty principle for convolution operators on discrete groups

Consider a discrete group and a bounded self-adjoint convolution operator on ; let be the spectrum of . The spectral theorem gives a unitary isomorphism between and a direct sum , where , and is a regular Borel measure supported on . Through this isomorphism corresponds to multiplication by the identity function on each summand. We prove that a nonzero function and its transform cannot be simultaneously concentrated on sets , such that and the cardinality of are both small. This can be regarded as an extension to this context of Heisenberg's classical uncertainty principle.

     

A Survey on Explicit Representation Formulae for Fundamental Solutions of Linear Partial Differential Operators

In the present article, we aim at treating the existence of fundamental solutions of linear partial differential operators with constant coefficients from the viewpoint of setting up explicit formulae yielding fundamental solutions.     

Fundamental solutions for a class of non-elliptic homogeneous differential operators

We compute temperate fundamental solutions of homogeneous differential operators with real-principal type symbols. Via analytic continuation of meromorphic distributions, fundamental solutions for these non-elliptic operators can be constructed in terms of radial averages and invariant distributions on the unit sphere.     

Anisotropic estimates for sub-elliptic operators

In the 1970's,Folland and Stein studied a family of subelliptic scalar operators L_λwhich arise naturally in the(?)_b-complex.They introduced weighted Sobolev spaces as the natural spaces for this complex,and then obtained sharp estimates for(?)b in these spaces using integral kernels and approximate inverses.In the 1990's,Rumin introduced a differential complex for compact contact manifolds,showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator,and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper,we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.     

Weak solutions for equations defined by accretive operators II: relaxation limits

In the first part of this paper we define solutions for certain nonlinear equations defined by accretive operators, “dissipative solution”. This kind of solution is equivalent to the viscosity solutions for Hamilton-Jacobi equations and to the entropy solutions for conservation laws.In this paper we use dissipative solutions to obtain several relaxation limits for systems of semilinear transport equations and quasilinear conservation laws. These converge to diffusion second-order equations and in one case to a single conservation law. The relaxation limit is obtained using a version of the perturbed test function method to pass to the limit. This guarantees existence for the considered equations.     

Surjectivity of partial differential operators with good fundamental solutions

We characterize the surjectivity of a linear partial differential operator with constant coefficients on E(Ω) as well as on D^′(Ω) in terms of the existence of “good” shifted fundamental solutions. This characterization complements results of Meise, Taylor, and Vogt as well as Frerick and the present author.     

Link of groups and homogeneous Hörmander operators

We study a notion of link of Lie groups suggested by the structure of the partial differential operators of Kolmogorov type. As an application of our link procedure we construct explicit examples of stratified Lie groups, with dimension and step arbitrarily large. We also give a set of examples of hypoelliptic second-order operators which are left translation invariant and homogeneous of degree two on the previous groups.

     

A family of fundamental solutions of elliptic partial differential operators with quaternion constant coefficients

The purpose of this paper is to construct a family of fundamental solutions for elliptic partial differential operators with quaternion constant coefficients. The elements of such family are expressed by means of functions, which depend jointly real analytically on the coefficients of the operators and on the spatial variable. We show some regularity properties in the frame of Schauder spaces for the corresponding single layer potentials. Ultimately, we exploit our construction by showing a real analyticity result for perturbations of the layer potentials corresponding to complex elliptic partial differential operators of order two. Copyright © 2012 John Wiley & Sons, Ltd.     

Fundamental solutions of stochastic differential equations with drift

The paper deals with the pathwise uniqueness of solutions to one-dimensional time homogeneous stochastic differential equations with a diffusion coefficient σ satisfying the local time condition and measurable drift term b. We show that if the functions σ and b satisfy a non-degeneracy condition and fundamental solution to considered equation is unique in law, then pathwise uniqueness of solutions holds. Our result is in some sense negative, more precisely we give an example of an equation with Holder continuous diffusion coefficient and nondegenerate drift for which a fundamental solution is not unique in law and pathwise uniqueness of solutions does not hold.     

Self-similar solutions for a class of non-divergence form equations

In this paper, we study the self-similar solutions for a non-divergence form equation of the form $$u(x, t)=(t + 1)^{-\alpha}f((t + 1)^{\beta}|x|^2).$$ We first establish the existence and uniqueness of solutions f with compact supports, which implies that the self-similar solution is shrink. On the basis of this, we also establish the convergent rates of these solutions on the boundary of the supports. On the other hands, we also consider the convergent speeds of solutions, and compare which with Dirac function as t tends to infinity.     

Existence of periodic solutions for nonlinear evolution equations with pseudo monotone operators

In this paper, we study the existence of -periodic solutions for the problem

where is a -periodic, pseudo monotone mapping from a reflexive Banach space into its dual.

     

Symmetry properties for solutions of nonlocal equations involving nonlinear operators

We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional p-Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.     

On a class of pseudodifferential operators on the product of compact Lie groups

In this paper, a bisingular pseudodifferential calculus, along the lines of the one introduced by L. Rodino in his paper of 1975, is developed in the global setting of a product of compact Lie groups. The approach follows that introduced by M. Ruzhansky and V. Turunen in their book of 2010 (see also V. Fischer's paper of 2015), in that it exploits the harmonic analysis of the groups involved.