Czechoslovak Mathematical Journal, Vol. 55, No. 2, pp. 423-432, 2005

# Weighted inequalities for integral operators with some homogeneous kernels

## Maria Silvina Riveros, Marta Urciuolo

Maria Silvina Riveros, Marta Urciuolo, FaMAF Universidad Nacional de Cordoba, Ciem-CONICET, Ciudad Universitaria 5000 Cordoba, e-mails: sriveros@mate.uncor.edu, urciuolo@mate.uncor.edu

Abstract: In this paper we study integral operators of the form
Tf(x)=\int| x-a_1y|^{-\alpha_1}\dots| x-a_my|^{-\alpha_m}f(y)\dd y,
$\alpha_1+\dots+\alpha_m=n$. We obtain the $L^p(w)$ boundedness for them, and a weighted $(1,1)$ inequality for weights $w$ in $A_p$ satisfying that there exists $c\geq1$ such that $w( a_ix) \leq cw( x)$ for a.e. $x\in\bb R^n$, $1\leq i\leq m$. Moreover, we prove $\| Tf\| _{\BMO}\leq c\| f\| _\infty$ for a wide family of functions $f\in L^\infty( \bb R^n)$.

Keywords: weights, integral operators

Classification (MSC 2000): 42B25, 42A50, 42B20

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