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...

We obtain upper and lower estimates for the Green function for a second order noncoercive differential operator on a homogeneous manifold of negative curvature.

Let L be a positive Rockland operator of homogeneous degree d on a graded homogeneous group G and let $p_t$ 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 $|p_1\left(x\right)|\le Cexp(-c\tau {\left(x\right)}^{d/(d-1)})$. Moreover, if G is not stratified, more precise estimates of $p_1$ at infinity are given.

Dans cet article, en utilisant les algèbres de Jordan euclidiennes, nous étudions l’espace de Hardy $H^2\left(\Xi \right)$ d’un espace symétrique de type Cayley $ℳ=G/H$. Nous montrons que le noyau de Cauchy-Szegö de $H^2\left(\Xi \right)$ s’exprime comme somme d’une série faisant intervenir la fonction $c$ de Harish-Chandra de l’espace symétrique riemannien $D=G/K$, la fonction $c$ de l’espace symétrique $c$-dual $𝒩$ de $ℳ$ et les fonctions sphériques de l’espace symétrique ordonné $𝒩$. Nous établissons, dans le cas où la dimension de l’algèbre de Jordan associée...