Research Article | Open Access
Zengyan Si, "Some Weighted Estimates for Multilinear Fourier Multiplier Operators", Abstract and Applied Analysis, vol. 2013, Article ID 987205, 10 pages, 2013. https://doi.org/10.1155/2013/987205
Some Weighted Estimates for Multilinear Fourier Multiplier Operators
We first provide a weighted Fourier multiplier theorem for multilinear operators which extends Theorem 1.2 in Fujita and Tomita (2012) by using -based Sobolev spaces (). Then, by using a different method, we obtain a result parallel to Theorem 6.2 which is an improvement of Theorem 1.2 under assumption (i) in Fujita and Tomita (2012).
During the last several years, considerable attention has been paid to the study of multilinear Fourier multiplier operators. Let be the Schwartz space of all rapidly decreasing smooth functions on , for some . The multilinear Fourier multiplier operator associated with a symbol is defined by for , .
Coifman and Meyer  proved that if is a bounded function on that satisfies away from the origin for all sufficiently large multi-indices , then is bounded from the product to for all satisfying . The multiplier theorem of Coifman and Meyer was extended to indices (and larger than ) by Grafakos and Torres  and Kenig and Stein  (when ). Exploiting the idea of the proof of the Hörmander multiplier theorem in , Tomita  gave a Hörmander type theorem for multilinear Fourier multipliers with more weaker smoothness condition assumed on than (2). Grafakos and Si  gave similar results for by using -based Sobolev spaces . Grafakos et al.  proved the -boundedness of with multipliers of limited smoothness.
In order to state other known results, we first introduce some notations. The Laplacian on is , that is, the sum of the second partials of in every variable. We define the operator , where for . Let be the -based Sobolev space with norm where .
Let and let the product type Sobolev space consist of all functions such that the following norm of is finite:
where and .
Let be such that and for .
Let be the set of all Schwartz functions on , whose Fourier transform is supported in an annulus of the form , is nonvanishing in a smaller annulus (for some choice of constants ), and satisfies
The weighted estimate for is also an interesting topic in harmonic analysis. And it has attracted many authors in this area. Recently, Fujita and Tomita  established some weighted estimates of under the Hörmander condition and classical weights. For other works about the weighted estimates for , see [9, 10] and the references therein.
Theorem A (see ). Let , , and . Assume (i) and ;(ii) and , .
If satisfies , then is bounded from to .
An improvement of Theorem 1.2 is stated as follows.
Theorem B (see ). Let , , and , . Assume and for . If satisfies , then is bounded from to , where .
The first purpose of this paper is to improve Theorem A by using -based Sobolev spaces . The second purpose is to give a new proof of Theorem B. The following are the main results.
Theorem 1. For some , suppose that and satisfy, for some ,
If , , and the weights satisfy one of the following two conditions: (i) and ,(ii), , and ,then there is a number satisfying , such that the -linear operator , associated with the multiplier , is bounded from to , whenever for all , and is given by .
Theorem 2. Let and let and . If satisfies , then is bounded from to , whenever , , and with .
2. The Proof of Theorem 1
In this section we discuss the proof of Theorem 1. We begin with some definitions for maximal operators. Throughout the paper, denotes the Hardy-Littlewood maximal operator defined by where moves over all cubes containing . For , is the maximal function defined by In addition, is the sharp maximal function of Fefferman and Stein: where denotes the average of over and a variant of is given by
We prepare some lemmas which will be used later.
Lemma 3 (see ). Let and . Then (1) ; (2) there exists such that .
Lemma 4 (see ). Let , and . Then there exist positive finite constants such that for all sequences of locally integrable functions on .
Lemma 5. Let be the Littlewood-Paley operator given by , , where is a Schwartz function whose Fourier transform is supported in the annulus , for some , and satisfies , for some constant . Let and . Then there is a constant , such that for functions one has
Proof. The proof follows from similar steps in Lemma 4 of  and combines the method used in Remark 2.6 of . Let be a Schwartz function with integral one. Then, If , the weighted Hardy space coincides with the weighted Triebel-Lizorkin space for . Hence, if , we have The proof is complete.
Now we give the proof of Theorem 1.
Proof. Since the proof follows from similar steps in Theorem 1 in , we just give the different parts. For each , we let be the set of points in such that and we introduce nonnegative smooth functions on that are supported in such that
for all , with the understanding that the variable with the hat is missing. These functions introduce a partition of unity of subordinate to a conical neighborhood of the region .
Each region can be written as the union of sets: with . We need to work with a finer partition of unity, subordinate to each . To achieve this, for each , we introduce smooth functions on supported in such that for all in the support of with .
We now have obtained the following partition of unity of : where the dots indicate the variables of each function.
We now introduce a nonnegative smooth bump supported in the interval and equal to 1 in the interval and we decompose into a finite number of multipliers: where We will prove the required assertion for each piece of this decomposition, that is, for the multipliers and for each pair in the previous sum. In view of the symmetry of the decomposition, it suffices to consider the case of a fixed pair in the previous sum. To simplify notation, we fix the pair ; thus, for the rest of the proof we fix and and we prove boundedness for the -linear operators whose symbols are and . These correspond to the -linear operators and , respectively.
We first prove Theorem 1 under assumption (i). Since , we can take such that and . We first consider , where are fixed Schwartz functions. We fix a Schwartz radial function whose Fourier transform is supported in the annulus and satisfies Associated with we define the Littlewood-Paley operator , where for . We also define an operator by setting where is a smooth function whose Fourier transform is equal to 1 on the ball and vanishes outside the double of this ball. As in [6, page 143], by using Lemma 5 we get
We will use the following estimate for (see [6, page 145]):
We now square the previous expression, we sum over , and we take square roots. Since , the hypothesis implies , and thus each term is bounded on . We obtain where the last step holds due to Lemma 4 with and the weighted Littlewood-Paley theorem.
Next we deal with . Following [6, page 146], we write for some other Littlewood-Paley operator which is given on the Fourier transform by multiplication with a bump , where is equal to one on the annulus and vanishes on a larger annulus. Also, is given by convolution with , where is a smooth function whose Fourier transform is equal to 1 on the ball and vanishes outside the double of this ball.
Summing over and taking norms yield where the last step holds due to the Cauchy-Schwarz inequality and we omitted the prime from the term with for the matter of simplicity. Applying Hölder’s inequality and using that and Lemma 4 we obtain the conclusion that the expression above is bounded by
We next prove Theorem 1 under assumption (ii). It was proven in [6, page 136] that condition (6) is invariant under the adjoints; that is, it is also valid for the symbols of the dual operators . To prove the required assertion, by duality, it is enough to prove that and are bounded from to . We may assume that . Since , we see , . Hence, . Since and , we deduce that ; then . It is obvious that . Since , . That is, ; then . Therefore, we take a positive number such that , and such that and . We have Similarly, we have
This concludes the proof of Theorem 1.
3. The Proof of Theorem 2
We begin with some lemmas which will be used in the proof of Theorem 2.
Lemma 6 (see ). Let and and let be a weight in . Then, there exists (depending on the constant of ) such that for all function for which the left-hand side is finite.
Lemma 7 (see ). Let , and . Let be a multiplier satisfying for , , and ; then is bounded from to .
Remark 8. It should be pointed out that Lemma 7 can be extended to the case .
Lemma 9 (see ). Let , , and . Then there exists a constant such that
for all with .
Next, we give a pointwise control of which becomes very useful in the proof of Theorem 2.
Lemma 10. Let . Assume that which satisfies for and + + + . For any , one has .
Proof. For simplicity, we only prove for the case , since there is no essential difference for the general case. Fix an and a cube with side length , such that . Let , where and for and . Since , we have
We first consider . By Kolmogorov’s inequality, Hölder’s inequality, and Lemma 7, we have
where with and , .
Next we deal with . We choose , where
We may split as , where
Now we estimate first. We decompose as
Let , where with supp and , . Thus, we have
Applying Hölder’s inequality we have
Let and . Then we havewhere the last inequality holds due to Lemma 9. Suppose that . Since , we have
On the other hand
where . Since , we havewhere in the last inequality Lemma 9 was used again and hence
Combining the above arguments we have
Thus, we obtain . What remain to be considered are and . We just estimate since the same arguments can be applied to :