Geodesic
Weighted inequalities for integral operators with some homogeneous kernels
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 2, pp. 423-432
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In this paper we study integral operators of the form \[ Tf(x)=\int | x-a_1y|^{-\alpha _1}\dots | x-a_my|^{-\alpha _m}f(y)\mathrm{d}y, \] $\alpha _1+\dots +\alpha _m=n$. 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 $c\ge 1$ such that $w( a_ix) \le cw( x)$ for a.e. $x\in \mathbb R^n$, $1\le i\le m$. Moreover, we prove $\Vert Tf\Vert _{{\mathrm BMO}}\le c\Vert f\Vert _\infty $ for a wide family of functions $f\in L^\infty ( \mathbb R^n)$.
In this paper we study integral operators of the form \[ Tf(x)=\int | x-a_1y|^{-\alpha _1}\dots | x-a_my|^{-\alpha _m}f(y)\mathrm{d}y, \] $\alpha _1+\dots +\alpha _m=n$. 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 $c\ge 1$ such that $w( a_ix) \le cw( x)$ for a.e. $x\in \mathbb R^n$, $1\le i\le m$. Moreover, we prove $\Vert Tf\Vert _{{\mathrm BMO}}\le c\Vert f\Vert _\infty $ for a wide family of functions $f\in L^\infty ( \mathbb R^n)$.
Classification : 42A50, 42B20, 42B25
Keywords: weights; integral operators 
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     title = {Weighted inequalities for integral operators with some homogeneous kernels},
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Riveros, María Silvina; Urciuolo, Marta. Weighted inequalities for integral operators with some homogeneous kernels. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 2, pp. 423-432. http://geodesic.mathdoc.fr/item/CMJ_2005_55_2_a11/

[1] R. Coifmann and C. Fefferman: Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51 (1974), 241–250. | DOI | MR

[2] J. Duoandikoetxea: Análisis de Fourier. Ediciones de la Universidad Autónoma de Madrid, Editorial Siglo  XXI, 1990.

[3] T. Godoy and M. Urciuolo: About the $L^p$  boundedness of some integral operators. Revista de la UMA 38 (1993), 192–195. | MR

[4] T.  Godoy and M.  Urciuolo: On certain integral operators of fractional type. Acta Math. Hungar. 82 (1999), 99–105. | DOI | MR

[5] F. John and L. Nirenberg: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14 (1961), 415–426. | DOI | MR

[6] B. Muckenhoupt: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972), 207–226. | DOI | MR | Zbl

[7] F.  Ricci and P.  Sjögren: Two parameter maximal functions in the Heisenberg group. Math.  Z. 199 (1988), 565–575. | MR

[8] A. de la Torre and J. L. Torrea: One-sided discrete square function. Studia Math. 156 (2003), 243–260. | DOI | MR