Inégalités à poids pour l'opérateur de Hardy–Littlewood–Sobolev dans les espaces métriques mesurés à deux demi-dimensions
Tom 105 / 2006
Streszczenie
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 $1< p\leq q < \infty$.
