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## Convolution operators with anisotropically homogeneous measures on ${\Bbb R}^{2n}$ with $n$-dimensional support

### Tom 93 / 2002

Colloquium Mathematicum 93 (2002), 285-293 MSC: Primary 42B15; Secondary 42B20, 44A35. DOI: 10.4064/cm93-2-8

#### Streszczenie

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 $\varphi.$

#### Autorzy

• E. FerreyraFaMAF, Universidad Nacional de Córdoba
CIEM
5000 Córdoba, Argentina
e-mail
• T. GodoyFaMAF, Universidad Nacional de Córdoba
CIEM
5000 Córdoba, Argentina
e-mail
• M. UrciuoloFaMAF, Universidad Nacional de Córdoba
CIEM 