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Acta Mathematica Sinica, English Series

, Volume 29, Issue 3, pp 449–460 | Cite as

Weighted inequalities for fractional type operators with some homogeneous kernels

  • María Silvina RiverosEmail author
  • Marta Urciuolo
Article

Abstract

In this paper, we study integral operators of the form
$$T_\alpha f(x) = \int_{\mathbb{R}^n } {\left| {x - A_1 y} \right|^{ - \alpha _1 } \cdots \left| {x - A_m y} \right|^{ - \alpha _m } f(y)dy,}$$
, where A i are certain invertible matrices, α i > 0, 1 ≤ im, α 1 + … + α m = nα, 0 ≤ α < n. For \(\tfrac{1} {q} = \tfrac{1} {p} - \tfrac{\alpha } {n}\), we obtain the L p (ℝ n , w p ) − L q (ℝ n ,w q ) boundedness for weights w in A(p, q) satisfying that there exists c > 0 such that w(A i x) ≤ cw(x), a.e. x ∈ ℝ n , 1 ≤ im. Moreover, we obtain the appropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.

Keywords

Fractional operators Calderón-Zygmund operators BMO Muckenhoupt weights 

MR(2010) Subject Classification

42B20 42B25 

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Supplementary material

10114_2013_1639_MOESM1_ESM.tex (37 kb)
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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.FaMAF-UNC, CIEM-CONICET, Ciudad UniversitariaCórdobaArgentina

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