- Open Access
Estimates of singular integrals and multilinear commutators in weighted Morrey spaces
© Ye and Zhu; licensee Springer 2012
- Received: 24 August 2012
- Accepted: 5 November 2012
- Published: 17 December 2012
Suppose T is a singular integral operator whose kernel is a variable kernel with mixed homogeneity; the purpose of this paper is to study the continuity of the operator in weighted Morrey spaces , , . A special attention is paid also to the multilinear commutator of this operator with BMO function.
- mixed homogeneity
- multilinear commutators
- weighted Morrey spaces
- Orlicz maximal operator
where , , . The variable kernel depends on some parameter x and possesses ‘good’ properties with respect to the second variable ξ, which was firstly introduced by Fabes and Rieviève in . They generalized the classical Calderón and Zygmund kernel and the parabolic kernel studied by Jones in . By introducing a new metric ρ, Fabes and Rieviève studied (1.1) in , where was endowed with the topology induced by ρ and defined by ellipsoids.
By using this metric ρ, Softova in  obtained that the integral operator (1.1) and commutator were continuous in generalized Morrey spaces , , ω satisfying suitable conditions.
The weighted Morrey spaces were introduced by Komori and Shirai . Moreover, they showed some classical integral operators and corresponding commutators were bounded in weighted Morrey spaces. Recently, Wang [7–9] obtained that some other kind of integral operators (e.g., Bochner-Riesz operator, Marcinkiewicz operators etc.) and commutators were also bounded in weighted Morrey spaces. He Sha  showed that multilinear operators were bounded on weighted Morrey spaces with the symbol of . The main purpose of this paper is to discuss the continuity of the singular integral operator whose kernel is a variable kernel with mixed homogeneity and its multilinear commutator in the weighted Morrey spaces , , , where the weight function ω is weight. Furthermore, we shall give the weighted weak type estimate of theses operators in the weighted Morrey spaces , . Our main results are stated as follows.
If is a constant kernel and a metric ρ is Euclidean one, this result is just Theorem 3.3 in .
where and .
In what follows, we denote by C positive constants which are independent of the main parameters but may vary from line to line.
In this section, we introduce some basic definitions and lemmas needed for the proof of the main results.
with Lebesgue measure . It is easy to see that the unit sphere with respect to this metric coincides with the unit sphere with respect to the Euclidean one.
for every fixed x, the function is a constant kernel satisfying
- (2)for any , ,
for every multiindex β, independent of x.
the supremum is taken over all ellipsoid ℰ in .
where . The quantity is a norm in the BMO modulo constant function under which BMO results in a Banach space (see ).
holds, then w belongs to the Muckenhoupt class . We denote .
for almost every .
Remark 2.5 Given a weight function , , it also satisfies the doubling condition : for any ellipsoid ℰ, there exists a constant such that .
In fact, , we have the following inequality.
for any ellipsoid ℰ.
Lemma B 
Lemma C 
Lemma D 
Suppose , and , if is the classical Calderón-Zygmund operator with a constant kernel, then the operator is bounded on .
for all and any ellipsoid ℰ.
From the above definition, observe that . This inequality will be relevant in our work.
Aside from the properties of an weight function and a BMO function, we need some estimates of multilinear commutators. The following results were proved by Pérez and González .
Although the commutators with a BMO function are not of weak type , we have the following inequality.
By the above inequality, we have the following result.
for any integer l. In particular, the expansion of ϕ into spherical harmonics converges uniformly to ϕ. For more detail, we can see .
In this section, we shall use the complete orthonormal system in and some lemmas as above to finish the theorems.
thus we complete the proof of Theorem 1.1. □
Next we begin with the second theorem, for which further discussion is needed.
Instead of the operator , we only consider the existence and boundedness in of the operators .
The third inequality is obtained by Lemma A.
The last inequality is obtained by Lemma A and the D’Alembert judge method of positive series.
Which together with (3.1)-(3.7), for , the proof of Theorem 1.2 is finished.
Thus, the proof of Theorem 1.2 is completed. □
XFY research was partially supported by the National Natural Sciences Foundation of China (10961015; 11161021). XSZ research was supported by the National Natural Sciences Foundation of China (11161021).
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