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  Volume 2, Issue 3, Article 37
$L^p-$Improving Properties for Measures on $\mathbb{R}^{4}$ Supported on Homogeneous Surfaces in Some Non Elliptic Cases

    Authors: E. Ferreyra, T. Godoy, M. Urciuolo,  
    Keywords: Singular Measures, $L^p$-Improving, Convolution Operators.  
    Date Received: 08/01/01  
    Date Accepted: 05/06/01  
    Subject Codes:


    Editors: Lubos Pick,  

In this paper we study convolution operators $ T_{mu}$ with measures $ mu$ in $ mathbb{R}^{4}$ of the form $ muleft( Eright) =int_{B}chi_{E}left(x,varphileft( xright) right) dx,$ where $ B$ is the unit ball of $ mathbb{R}^{2}$, and $ varphi$ is a homogeneous polynomial function. If $ inf_{hin S^{1}}leftvert detleft( d_{x}^{2}varphileft( h,.right) right)rightvert $ vanishes only on a finite union of lines, we prove that $ T_{mu}$ is bounded from $ L^{p}$ into $ L^{q}$ if $ left( frac{1}{p},frac{1}{q}right) $ belongs to certain explicitly described trapezoidal region.

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