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Convolution Operators with Homogeneous Singular Measures on $R^{3}$ of Polynomial Type. The Remainder Case.  
 
  Authors: M. Urciuolo,  
  Keywords: Convolution operators, Singular measures.  
  Date Received: 22/12/05  
  Date Accepted: 23/09/06  
  Subject Codes:

42B20, 26B10.

  
  Editors: Alberto Fiorenza,  
 
  Abstract:

Let $ \varphi\left( y_{1},y_{2}\right) =y_{2}^{l}P\left( y_{1},y_{2}\right) $ where $ P$ is a polynomial function of degree $ l$ such that $ P\left( 1,0\right) \neq0$. Let $ \mu_{\delta}$ be the Borel measure on $ \mathbb{R}^{3} $ defined by $ \mu_{\delta}\left( E\right) =\int_{V_{\delta}}\chi _{E}\left( x,\varphi\left( x\right) \right) dx$ where

$\displaystyle V_{\delta}=\left\{ x=\left( x_{1},x_{2}\right) \in\mathbb{R}^{2}:... ...and }\left\vert x_{1}\right\vert \leq\delta\left\vert x_{2}\right\vert \right\}$    

and let $ T_{\mu_{\delta}}$ be the convolution operator with the measure $ \mu_{\delta}.$ In this paper we explicitly describe the type set
$\displaystyle E_{\mu_{\delta}}:=\left\{ \left( \frac{1}{p},\frac{1}{q}\right) \... ...eft[ 0,1\right] :\left\Vert T_{\mu_{\delta}}\right\Vert _{p,q}<\infty\right\} ,$    

for $ \delta$ small enough. ;


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