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Estimates of singular integrals and multilinear commutators in weighted Morrey spaces
Journal of Inequalities and Applications volume 2012, Article number: 302 (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.
Let be a variable kernel. The singular integral operator is defined by
and its multilinear commutator with the BMO function
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.
Theorem 1.1 Let , . If , then there exists a constant such that
When , for any and ellipsoid ℰ, there exists a constant such that
If is a constant kernel and a metric ρ is Euclidean one, this result is just Theorem 3.3 in .
Theorem 1.2 Let , . If , , , then there exists a constant such that
where . When , for any and ellipsoid ℰ, then there exists a constant such that
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.
2 Some notations and lemmas
In this section, we introduce some basic definitions and lemmas needed for the proof of the main results.
Let be real numbers, and . Following Fabes and Riviève , there exists a function ρ such that defines a distance between any two points . Thus endowed with the metric ρ results in a homogeneous metric space [1, 3]. The balls with respect to centered at the origin and of radius r are the ellipsoids
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.
Definition 2.1 The function is called a variable kernel with mixed homogeneity if
for every fixed x, the function is a constant kernel satisfying
for any , ,
for every multiindex β, independent of x.
Definition 2.2 Let , and w be a weight function. Then a weighted Morrey space is defined by
the supremum is taken over all ellipsoid ℰ in .
Definition 2.3 For the function and any ellipsoid ℰ, b is called a BMO function if
where . The quantity is a norm in the BMO modulo constant function under which BMO results in a Banach space (see ).
Definition 2.4 Let . For any locally integrable function w and ellipsoid ℰ, if
holds, then w belongs to the Muckenhoupt class . We denote .
When , if there exists such that
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.
Suppose , there exists a constant such that
for any ellipsoid ℰ.
Lemma B 
Suppose , then the norm of is equivalent to the norm of , where
Lemma C 
Let the ellipsoid centered at with side length of r. For any positive integer i, denotes the ellipsoid centered at with side length of , we have the inequality
Lemma D 
Suppose , and , if is the classical Calderón-Zygmund operator with a constant kernel, then the operator is bounded on .
If , and , then there exists a constant such that
for all and any ellipsoid ℰ.
Definition 2.6 Let . The Orlicz maximal operator is given by
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 .
Lemma E Let and . Suppose , , then there exists a constant such that
Although the commutators with a BMO function are not of weak type , we have the following inequality.
Lemma F Let . There exists a constant such that
By the above inequality, we have the following result.
Lemma G Let . There exists a constant such that, for all ,
Finally, we need the spherical harmonics and their properties (see more detail in [1, 15, 16]). Recall that any homogeneous polynomial of degree m that satisfies is called an n-dimensional solid harmonic of degree m. Its restriction to the unit sphere will be called an n-dimensional spherical harmonic of degree m. Denote by the space of all n-dimensional spherical harmonics of degree m. In general, it results in a finite-dimensional linear space with such that , and
Furthermore, let be an orthonormal basis of . Then is a complete orthonormal system in and
If, for instance, , then is the Fourier series expansion of with respect to ( substitutes ) and
for any integer l. In particular, the expansion of ϕ into spherical harmonics converges uniformly to ϕ. For more detail, we can see .
3 Proof of the theorems
In this section, we shall use the complete orthonormal system in and some lemmas as above to finish the theorems.
Proof of Theorem 1.1 In order to ensure the existence of the operator (1.1) in , , we restrict our consideration to the function , for which the norm of is finite. For the sake of convenience, we still denote these spaces by . Let and . In view of the properties of the kernel K with respect to the second variable and the complete of in , we get
Replacing the kernel with its series expansion, (1.1) can be written as
From the properties of (2.1)-(2.3), the series expansion , where the integer l is preliminarily chosen greater than . Along with the for a.a. , by the Fubini dominated convergence theorem, we have
where . Instead of the operators , we shall study the existence and boundedness in of the operators with a kernel . Observe that is a constant kernel and satisfies
From Lemma D, it follows
for . Consequently, by the above inequality and (2.1)-(2.3), we show
where the integer l is preliminary chosen greater that . For , by Lemma D, we have
for any and ellipsoid ℰ. Therefore, one gets
thus we complete the proof of Theorem 1.1. □
Next we begin with the second theorem, for which further discussion is needed.
Proof of Theorem 1.2 As above, we use the series expansion of a kernel , the operator is divided into
Instead of the operator , we only consider the existence and boundedness in of the operators .
Let . For any ellipsoid ℰ, we only need to obtain the inequality
In fact, by the series expansion of a kernel , we have
where the integer l is chosen greater than . Next, fix the above ellipsoid and decompose , where , denotes the characteristic function of , then we have
By using Lemma E, we get
For the term II, without loss of generality, we can assume . Thus, the operator can be divided into four parts,
where , . For the term , observing that and , we have . Thus, it yields
since , and by the definition of a weighted Morrey space, we get
The third inequality is obtained by Lemma A.
For , note that , . By Hölder’s inequality and , we get
Indeed, by Lemma B we know is equivalent to , . Let , , . For any ellipsoid ℰ, by using Lemma B and Lemma C, we show
Thus, since , it yields
The last inequality is obtained by Lemma A and the D’Alembert judge method of positive series.
For , by the inequality since , by Lemma B, we have
By Hölder’s inequality, Lemma B and Lemma C, we get
indeed for , by using Lemma A, we have that
Thus, we conclude
In the same way, we shall get the result of
Which together with (3.1)-(3.7), for , the proof of Theorem 1.2 is finished.
Now, we are in a position to consider the case . In general, the singularity of the commutator is stronger than the singular integral, and the endpoint case of the commutator is not even obtained. Thus, the result for the case of the multilinear commutator is interesting. We split f as above by , which yields
for any ellipsoid ℰ, and integer . For the term III, we use Lemma G. It follows that
For the last term IV, without loss of generality, we still suppose . By homogeneity, it is enough to assume , and hence we only need to prove
In fact, by Lemma F, we get
where . We use the Fefferman-Stein maximal inequality
for any functions f and . This yields
For , since , it follows that
To estimate the term , we first consider the form
for any , and . By simple geometric observation, we have
Therefore, we obtain
Since satisfies the doubling condition and Lemma A, we estimate the term as follows:
The last inequality is similar to (3.4). Noting that , from (3.8)-(3.11), we conclude
Thus, the proof of Theorem 1.2 is completed. □
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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).
The authors declare that they have no competing interests.
XFY conceived of the study and drafted the manuscript. XSZ participated in the discussion. All authors read and approved the final manuscript.