Weighted Morrey Estimates for Multilinear Fourier Multiplier Operators
The multilinear Fourier multipliers and their commutators with Sobolev regularity are studied. The purpose of this paper is to establish that these operators are bounded on certain product Morrey spaces . Based on the boundedness of these operators from to , we obtained that they are also bounded from to , with , , , and .
Recently some authors have taken so much interest in the text of multilinear Fourier multipliers with Sobolev regularity. To state some interesting results, we recall some necessary notations and definitions. Let ; the multilinear Fourier multiplier operator is defined by for all , where and is the Fourier transform of . It is well known that  the boundedness of from to holds if satisfying the condition for all multi-indice with and all with . Grafakos and Torres  improved the multiplier theorem of Coifman and Meyer to the indices by the multilinear Calderón-Zygmund operator theory in the case of . An important progress in this topic was given by Tomita. Let satisfy Set Tomita  proved that if for some , then is bounded from to provided that and . Grafakos and Si in  obtained that maps from to , if satisfies (5) and . Miyachi and Tomita  considered the problem to find minimal smoothness condition for multilinear Fourier multiplier. Let where . Miyachi and Tomita  proved that if for each , then is bounded from to provided that , , and with . Moreover, they also gave minimal smoothness condition for which is bounded from to .
Let , , and . Fujita and Tomita  proved the following inequality: if and , where and in what follows . Li et al.  obtained the endpoint cases. Hu and Lin  also obtained this result from another approach. Replacing by , Bui and Duong  and Li and Sun  proved that if , then (8) also holds. Jiao  gave a generalization of the above inequality with the class , which generalizes the class introduced by Lerner et al. . Fujita and Tomita showed a counterexample to answer the question whether the inequality (8) holds under the conditions and .
We still recall the weighted Morrey spaces which were introduced by Komori and Shirai . A weight is a nonnegative, locally integrable function on . Let ; a weight function is said to belong to the class , if there is a constant such that for any cube , and belongs to the class , if there is a constant such that, for any cube , We denote .
Definition 1 (See ). Let , let , and let be a weight function on . The weighted Morrey space is defined by where
Our main results can be stated as follows.
Theorem 2. Let be a multiplier satisfying for and let be the operator defined by (1) and . Set . If and the weight for and such that , then where .
Given a multilinear Fourier multiplier operator and , we define the commutators to be with
Theorem 3. Let be a multiplier satisfying for and let be the operator defined by (1) and . Set . If and the weight for and such that , then for any , where and .
Because the regularity condition is stronger than that of , we have the following corollaries.
Corollary 4. Let be a multiplier satisfying for and let be the operator defined by (1) and . Set . If and the weight for and such that , then where .
Corollary 5. Let be a multiplier satisfying for and let be the operator defined by (1) and . Set . If and the weight for and such that , then, for any , where and .
Remark 6. For and , we also extend Hörmander’s theorem  to the weighted Morrey spaces.
2. Some Notations and Lemmas
We begin with the definitions of Hardy-Littlewood maximal function, and of the sharp maximal function, For , we also define the following maximal functions and . The following classical result belongs to Fefferman and Stein .
Lemma 7. Let , and . Then there exists some constant such that
Similarly, we have the responding lemma on weighted Morrey spaces as a consequent result.
Lemma 8. Let , , and . Then there exists some constant such that
For , , , and set , we define This maximal function is the generalization of which is introduced by Lerner et al. , we refer to  for some properties of it. The following lemma is the special example of [11, Theorem 2.1].
Lemma 9. Let , , and for and . Then we have and if at least one , then where .
Lemma 10. Let , , , , and for and . Then we have
Lemma 11 (See ). Let and . Suppose that satisfies Then is bounded from to .
For and , the weighted Lebesgue space of mixed type is defined by the norm
Lemma 12 (See ). Let , , and for . Then there exists a constant such that for all with .
By the reverse Hölder inequality, we have the following lemma.
Lemma 13. Assume that , with . Let ; then there exist constants such that .
The following lemma is the key to our main results.
Lemma 14. Let “” be a multplier satisfying for and let be the operator defined by (1). If , and , where , and . Then for all with for , where .
Proof. By Lemma 13, ; then . Fix a point and a cube such that . It suffices to prove that
for some constant . We decompose with for all and . Then
where . Then we can write
Applying Kolmogorov’s inequality to , we have
since is bounded from to by Lemma 11.
Taking we claim that, for any , Let At first we consider the case ,Denote and ; it follows from Lemma 12 thatGiven that , then we have that On the other hand, a similar process follows in ; we get thatwhere . Since we have From Lemma 13, , it deduces that So It remains to consider the case that there exists a proper subset of ,