Maximal degenerate representations of SL.
Maximal functions related to subelliptic operators invariant under an action of a nilpotent Lie group
Maximal functions related to subelliptic operators invariant under an action of a solvable Lie group
On the domain S_a = {(x,e^b): x ∈ N, b ∈ ℝ, b > a} where N is a simply connected nilpotent Lie group, a certain N-left-invariant, second order, degenerate elliptic operator L is considered. N × {e^a} is the Poisson boundary for L-harmonic functions F, i.e. F is the Poisson integral F(xe^b) = ʃ_N f(xy)dμ^b_a(x), for an f in L^∞(N). The main theorem of the paper asserts that the maximal function M^a f(x) = sup{|ʃf(xy)dμ_a^b(y)| : b > a} is of weak type (1,1).
Maximal functions related to subelliptic operators with polynomially growing coefficients.
Minimality and unique ergodicity for subgroup actions
Let be an -algebraic semisimple group, an algebraic -subgroup, and a lattice in . Partially answering a question posed by Hillel Furstenberg in 1972, we prove that if the action of on is minimal, then it is uniquely ergodic. Our proof uses in an essential way Marina Ratner’s classification of probability measures on invariant under unipotent elements, and the study of “tubes” in .
Monotone functions on symmetric spaces.
Moser-Trudinger and logarithmic HLS inequalities for systems
We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere , on a bounded domain and on all of . In some cases we also address the question of existence of minimizers.
Multiplicité un pour les espaces symétriques exponentiels
Netradiční integrální geometrie
New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals [Book]
Newtonian spaces: An extension of Sobolev spaces to metric measure spaces.
This paper studies a possible definition of Sobolev spaces in abstract metric spaces, and answers in the affirmative the question whether this definition yields a Banach space. The paper also explores the relationship between this definition and the Hajlasz spaces. For specialized metric spaces the Sobolev embedding theorems are proven. Different versions of capacities are also explored, and these various definitions are compared. The main tool used in this paper is the concept of moduli of path...
Noncoercive differential operators on homogeneous manifolds of negative curvature and their Green functions
We obtain upper and lower estimates for the Green function for a second order noncoercive differential operator on a homogeneous manifold of negative curvature.
Nonlinear Schrödinger equation on real hyperbolic spaces
Nonlinear subelliptic Schrödinger equations with external magnetic field.
Note on semigroups generated by positive Rockland operators on graded homogeneous groups
Let L be a positive Rockland operator of homogeneous degree d on a graded homogeneous group G and let be the convolution kernels of the semigroup generated by L. We prove that if τ(x) is a Riemannian distance of x from the unit element, then there are constants c>0 and C such that . Moreover, if G is not stratified, more precise estimates of at infinity are given.
Noyau de Cauchy-Szegö d'un espace symétrique de type Cayley
Dans cet article, en utilisant les algèbres de Jordan euclidiennes, nous étudions l’espace de Hardy d’un espace symétrique de type Cayley . Nous montrons que le noyau de Cauchy-Szegö de s’exprime comme somme d’une série faisant intervenir la fonction de Harish-Chandra de l’espace symétrique riemannien , la fonction de l’espace symétrique -dual de et les fonctions sphériques de l’espace symétrique ordonné . Nous établissons, dans le cas où la dimension de l’algèbre de Jordan associée...
Noyau de diffusion sur les espaces homogènes compacts
Objects Dual to Subsemigroups of Groups.
On a Class of Generalized Gelfand Pairs.
On a zonal polynomial integral.