On montre d’abord que la topologie fine est connexe et localement connexe, dans le cas d’un espace harmonique $\Omega$ satisfaisant au groupe d’axiomes $\left(A_1\right)$ de Brelot (y compris l’axiome de domination). Un autre résultat principal (qu’on n’établit complètement ici que pour le cas classique d’un espace de Green) affirme que, pour toute mesure positive $\mu$ sur $\Omega$, soit à support compact, et pour toute base $B\subset \Omega$ telle que $\mu \left(B\right)=0$, la mesure balayée ${\mu }^B$ a pour support fin la frontière fine de la réunion de toutes les composantes...

Motivated by the recent development in the theory of jump processes, we investigate its conservation property. We will show that a jump process is conservative under certain conditions for the volume-growth of the underlying space and the jump rate of the process. We will also present examples of jump processes which satisfy these conditions.

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 $L^p$.

Let $D_j$ be a bounded hyperconvex domain in ${ℂ}^{n_j}$ and set $D=D₁\times \cdots \times D_s$, j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

$M_1$ è un particolare operatore di minimizzazione per forme di Dirichlet definite su un sottoinsieme finito di un frattale $K$ che è, in un certo senso, una sorta di frontiera di $K$. Viene talvolta chiamato mappa di rinormalizzazione ed è stato usato per definire su $K$ un analogo del funzionale $u\mapsto \int {\left|\text{grad}u\right|}^2$ e un moto Browniano. In questo lavoro si provano alcuni risultati sull'unicità dell'autoforma (rispetto a $M_1$ ), e sulla convergenza dell'iterata di $M_1$ rinormalizzata. Questi risultati sono collegati con l'unicità...

Let X be a convex domain in ℂⁿ and let E be a convex subset of X. The relative extremal function $u_{E,X}$ for E in X is the supremum of the class of plurisubharmonic functions v ≤ 0 on X with v ≤ -1 on E. We show that if E is either open or compact, then the sublevel sets of $u_{E,X}$ are convex. The proof uses the theory of envelopes of disc functionals and a new result on Blaschke products.

Let $R\left(\lambda \right)$ be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that ${\int }_1^T{\left|R\left(t\right)\right|}^{2}dt=c{T}^{\frac{5}{2}}+O_{\delta }\left(T^{\frac{9}{4}+\delta }\right)$, where $c$ is a specific nonzero constant and $\delta$ is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that $R\left(t\right)=O_{\delta }\left(t^{\frac{3}{4}+\delta }\right)$.The idea of the proof is to use the Poisson summation formula to write the error term in a form which can be estimated by the method of the stationary phase. The similar result will be also proven in the $2n+1$-dimensional case.