We prove the equivalence of various capacitary strong type estimates. Some of them appear in the characterization of the measures $\mu$ that are admissible data for the existence of solutions to semilinear elliptic problems with power growth. Other estimates are known to characterize the measures $\mu$ for which the Sobolev space $W^{2,p}$ can be imbedded into $L^p\left(\mu \right)$. The motivation comes from the semilinear problems: simpler descriptions of admissible data are given. The proof surprisingly involves the theory of singular...

We show the equivalence of some different definitions of p-superharmonic functions given in the literature. We also provide several other characterizations of p-superharmonicity. This is done in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. There are many examples of such spaces. A new one given here is the union of a line (with the one-dimensional Lebesgue measure) and a triangle (with a two-dimensional weighted Lebesgue measure). Our results also...

Étant donnés $p\in [1,+\infty [$ et un arbre $T$ dont chaque sommet est de valence au moins $3$, on étudie la constante de Sobolev d’exposant $p$ de $T$, c’est-à-dire la plus petite constante ${\sigma }_p$ telle que pour tout $u\in {\ell }_p\left(T^0\right)$ on ait ${\parallel u\parallel }_p^p\le {\sigma }_p{\parallel \mathrm{d}u\parallel }_p^p$. Notre motivation vient de la recherche de graphes finis avec des petites constantes de Poincaré d’exposant $p$, en vue d’obtenir des exemples de groupes qui ont la propriété de point fixe sur les espaces $L^p$.