Weighted boundedness of multilinear operators associated to singular integral operators with non-smooth kernels

Daqing Lu

doi:10.1186/1029-242x-2014-276

Abstract

In this paper, we establish the weighted sharp maximal function inequalities for a multilinear operator associated to a singular integral operator with non-smooth kernel. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces. MSC:42B20, 42B25.

Highlights

Introduction and preliminariesAs the development of singular integral operators, their commutators and multilinear operators have been well studied
In [ – ], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on Lp(Rn) for < p < ∞
In [, ], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on Triebel-Lizorkin and Lp(Rn) ( < p < ∞) spaces is obtained

Summary

The multilinear operator related to the operator T is defined by

For < η < n, ≤ p < ∞ and the non-negative weight function w, set. A = w ∈ Lploc Rn : M(w)(x) ≤ Cw(x), a.e. Given a non-negative weight function w. Let w be a non-negative weight function on Rn and f be a locally integrable function on Rn. Set, for ≤ η < n and ≤ p < n/η, f. Theorem Let T be a singular integral operator with non-smooth kernel as given in Definition , w ∈ A , < p < ∞ and Dαb ∈ BMO(w) for all α with |α| = m. F WLq ≤ Np,q(f ) ≤ q/(q – p) /p f WLq. Lemma (see [ , ]) Let T be a singular integral operator with non-smooth kernel as given in Definition.

Qis the cube centered at x and having side length

AtQ T

Dα bf WL