Inégalités à poids pour l'opérateur de Hardy–Littlewood–Sobolev dans les espaces métriques mesurés à deux demi-dimensions

David Mascré¹

¹ Département de Mathématiques Université de Cergy-Pontoise 95 302 Cergy-Pontoise Cedex, France

Colloquium Mathematicum, Tome 105 (2006) no. 1, pp. 77-104.

Résumé

On a metric measure space $(X, \varrho, \mu)$, consider the weight functions $$\eqalign{ w_{\alpha}(x)= \cases{\varrho(x,z_0)^{-\alpha_0} \mbox{if }\varrho(x,z_0)1,\cr \varrho(x,z_0)^{-\alpha_1} \mbox{if }\varrho(x,z_0)\geq1,\cr}\cr w_{\beta}(x)=\cases{\varrho(x,z_0)^{-\beta_0} \mbox{if } \varrho(x,z_0)1,\cr \varrho(x,z_0)^{-\beta_1} \mbox{if }\varrho(x,z_0)\geq1,\cr}\cr} $$ where $z_0$ is a given point of $X$, and let $\kappa_a:X\times X \rightarrow {{\mathbb R}}_+$ be an operator kernel satisfying $$ \kappa_a(x,y) \leq \cases{c\varrho(x,y)^{a-d} \mbox{for all }x,y \in X\mbox{ such that }\varrho(x,y)1,\cr c\varrho(x,y)^{a-D} \mbox{for all }x,y \in X\mbox{ such that } \varrho(x,y)\geq 1,\cr} $$ where $0 a \min(d,D)$, and $d$ and $D$ are respectively the local and global volume growth rate of the space $X$. We determine conditions on $a, \alpha_0, \alpha_1, \beta_0, \beta_1 \in {{\mathbb R}}$ for the Hardy–Littlewood–Sobolev operator with kernel $\kappa(x,y)=w_{\beta}(x)\kappa_a(x,y)w_{\alpha}(y)$ to be bounded from $L^p(X)$ to $L^{q}(X)$ for $1 p\leq q \infty$.

Détail

DOI : 10.4064/cm105-1-9

Mots-clés : metric measure space varrho consider weight functions eqalign alpha cases varrho alpha mbox varrho varrho alpha mbox varrho geq beta cases varrho beta mbox varrho varrho beta mbox varrho geq where given point kappa times rightarrow mathbb operator kernel satisfying kappa leq cases varrho a d mbox mbox varrho varrho a d mbox mbox varrho geq where min respectively local global volume growth rate space determine conditions alpha alpha beta beta mathbb hardy littlewood sobolev operator kernel kappa beta kappa w alpha bounded leq infty