Convolution operators with anisotropically homogeneous measures on ${\Bbb R}^{2n}$ with $n$-dimensional support
Volume 93 / 2002
Abstract
Let $\alpha _i,\beta _i>0,$ $1\leq i\leq n,$ and for $t>0$ and $x=( x_1,\ldots ,x_n) \in{\mathbb R}^n,$ let $t\mathbin{\bullet} x=( t^{\alpha _1}x_1,\ldots ,t^{\alpha _n}x_n)$, $t\mathbin{\circ} x=( t^{\beta _1}x_1,\ldots ,t^{\beta _n}x_n) $ and $\| x\| =\sum_{i=1}^n| x_i| ^{1/\alpha _i}$. Let $\varphi _1,\ldots,\varphi _n$ be real functions in $C^\infty ({\mathbb R}^n-\{ 0\}) $ such that $\varphi =( \varphi _1,\ldots ,\varphi _n)$ satisfies $\varphi ( t\mathbin{\bullet} x) =t\mathbin{\circ} \varphi( x)$. Let $\gamma >0$ and let $\mu $ be the Borel measure on ${\mathbb R}^{2n}$ given by $$ \mu(E)=\int_{{\mathbb R}^n}\chi _E( x,\varphi ( x)) \| x\| ^{\gamma -\alpha}\,dx, $$ where $\alpha =\sum_{i=1}^n\alpha _i$ and $dx$ denotes the Lebesgue measure on ${\mathbb R}^n$. Let $T_\mu f=\mu *f$ and let $\| T_\mu \| _{p,q}$ be the operator norm of $T_\mu $ from $L^p({\mathbb R}^{2n}) $ into $L^q({\mathbb R}^{2n})$, where the $L^p$ spaces are taken with respect to the Lebesgue measure. The type set $E_\mu $ is defined by $$ E_\mu =\{ ( 1/p, 1/q) :\| T_\mu \| _{p,q}<\infty,\,1\leq p,q\leq \infty \} . $$ In the case $\alpha _i\neq \beta _k$ for $1\leq i,k\leq n$ we characterize the type set under certain additional hypotheses on $\varphi.$
