Multilinear Fourier multipliers on variable Lebesgue spaces
© Ren and Sun; licensee Springer. 2014
Received: 4 August 2014
Accepted: 4 December 2014
Published: 18 December 2014
In this paper, we study the properties of a bilinear multiplier space. We give a necessary condition for a continuous bounded function to be a bilinear multiplier on variable exponent Lebesgue spaces, and we prove the localization theorem of multipliers on variable exponent Lebesgue spaces. Moreover, we present a Mihlin-Hörmander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces and give some applications.
MSC:42B15, 42B20, 42B25.
where and .
where denotes the set of points in Ω on which .
We refer to  for an introduction to variable exponent Lebesgue spaces.
for , .
for all , where is an integer and s is a sufficiently large integer.
Tomita  gave a Hörmander type theorem for multilinear multipliers. Specifically, is bounded from to for all with and in (1.1). Furthermore, Grafakos and Si studied the case in . The boundedness of multilinear Calderón-Zygmund operators with multiple weights was achieved by Grafakos et al. .
Under the Hörmander conditions, Fujita and Tomita  obtained some weighted estimates of for classical weights. Then Li et al.  got some weighted results of multilinear multipliers by considering the end-point cases, using weighted Carleson measure theory and employing multilinear interpolation theory. In , Chen and Lu proved a Hörmander type multilinear theorem on weighted Lebesgue spaces when the Fourier multipliers were only assumed with limited smoothness. In , the boundedness of with multiple weights satisfying condition (1.1) was given by Bui and Duong. In , Li and Sun got some weighted estimates of with multiple weights under the Hörmander conditions in terms of the Sobolev regularity. Huang and Xu  obtained the boundedness of multilinear Calderón-Zygmund operators on variable exponent Lebesgue spaces.
In this paper, we study the weighted estimates of with nearly the same conditions as in , but on variable exponent Lebesgue spaces.
The theory of bilinear multipliers was first studied by Coifman and Meyer . They considered the ones with smooth symbols. Then, Muscalu et al. achieved some new results for non-smooth symbols in .
The study of bilinear multipliers has experienced a big progress since Lacey and Thiele [16, 17] proved that are -multipliers for each triple such that , and each . In , Kulak and Gürkanlı first studied some properties of the bilinear multiplier space. In , Fan and Sato proved the DeLeeuw type theorems for the transference of multilinear operators on Lebesgue and Hardy spaces from to . In , Blasco gave the transference theorems from to . We also refer to [21, 22] for details.
We first give some definitions.
Definition 1.1 ()
for all f and .
We call m a bilinear multiplier on of type if there exists some such that for all f and , i.e., extends to a bounded bilinear operator from to .
We write for the space of bilinear multipliers of type . Let .
A similar function space is defined in the following.
if for all f and can be extended to a bounded bilinear operator from to .
Definition 1.3 ()
where is independent of x or y.
We simply write instead of if there is no confusion. We also use to represent the collection of all continuous functions on . By C etc., we denote various positive constants which may have different values even in the same line.
2 Some results on the space
Some properties of the bilinear multiplier space on variable spaces were given by Kulak and Gürkanlı . Here we give some other properties.
First, we introduce the standard singular kernel.
Definition 2.1 ()
for , ;
Theorem 2.2 (Localization)
and , where C is independent of Q.
Let denote the space of multipliers which correspond to bounded operators from to .
To prove Theorem 2.2, we need the following results in the theory of variable Lebesgue spaces.
Lemma 2.3 ([, Theorem 5.39])
Let T be a singular integral operator with a standard singular kernel K. Given such that , if the Hardy-Littlewood maximal operator ℳ is bounded on , then for all functions f that are bounded and have compact support, , and T extends to a bounded operator on .
Then we get the result. □
The following is an explicit example.
and . □
Next we show that the space is invariant under certain operators.
Thus, we get the conclusion. □
Finally, we consider the necessary condition of this kind of multipliers. The bilinear classical counterpart was obtained by Hörmander [, Theorem 3.1] and Blasco . The multilinear classical one was proved by Grafakos and Torres, see [, Proposition 5] and [, Proposition 2.1]. And the one for multipliers on Lorentz spaces was given by Villarroya [, Proposition 3.1]. Some of their proofs used the translation-invariant property of the classical spaces, which is, however, no longer valid on . In the following, we prove the variable version of the necessary condition.
Theorem 2.8 (Necessary condition)
To prove the theorem, we need the following results.
Proposition 2.9 ([, Corollary 2.22])
Fix Ω and . If , then ; if , then .
Proposition 2.10 ([, Corollary 2.23])
when λ is sufficiently large.
where we use the fact that [, Example 2.2.9].
All the inequalities above are established when the λ is sufficiently large.
when λ is sufficiently large. □
We are now ready to prove Theorem 2.8.
3 The Mihlin-Hörmander type estimate for multilinear multipliers on weighted variable exponent Lebesgue spaces
Roughly speaking, in the linear case, by adding the condition that the Hardy-Littlewood maximal operator is bounded on weighted variable spaces, the results of multipliers on weighted variable spaces can be derived from the weighted multiplier theorem on classical Lebesgue spaces and the extrapolation theorem on weighted variable spaces. See, for example, [, Theorem 4.5, Theorem 4.7],  and .
However, in the multilinear case, the method faces some challenges. One problem is that we have no multilinear extrapolation theorem on spaces with variable exponents yet, though the counterpart on classical Lebesgue spaces appeared early, see .
We give another way to get the Mihlin-Hörmander conditions for multilinear Fourier multipliers on weighted variable spaces.
Definition 3.1 ()
When , then is understood as .
We now give a Mihlin-Hörmander type theorem for multilinear Fourier multipliers on weighted variable exponent Lebesgue spaces.
Before proving the theorem, we present some preliminary results. The following inequality is a classical result of Fefferman and Stein .
Proposition 3.3 ()
The next result comes from Lemma 2.6 in . For our purpose, we restate it in the proper way.
Proposition 3.4 ()
Proposition 3.5 ()
Let the exponent and the weight ϱ satisfy that and ℳ is bounded on .
is valid with a constant .
Remark 3.6 Note that the condition in the extrapolation theorem of  can be released to with nearly no modification to the proof.
Proposition 3.7 ([, Proposition 2.3])
holds if and only if , where .
Remark 3.8 When , the conclusion above is valid. Specifically, let and , then holds if and only if .
We are now ready to prove Theorem 3.2
where is defined as in Proposition 3.4.
where are fixed points in , .
for . Then is bounded from to .
To prove Corollary 3.9, we need to define a class of weight functions, which is a special case of [, Definition 2.7].
Definition 3.10 ()
We first give some lemmas that are needed to prove Corollary 3.9.
Lemma 3.11 ([, Proposition 2.3])
Given a domain Ω, if , then is equivalent to assuming .
Lemma 3.12 ([, Remark 2.10])
Lemma 3.13 ([, Theorem 2.10])
Suppose that Ω is an unbounded open set of . Let satisfy , and let there exists some and such that for . If , then ℳ is bounded on the space .
Then we have the following lemma.
Lemma 3.14 Let satisfy . Suppose that there exists some and such that for . If , then ℳ is bounded on the space for all , where .
Proof If , then . By Lemma 3.11, we have . And since , by Lemma 3.12 we know . Then it follows from Lemma 3.13 that ℳ is bounded on . □
Now we are ready to prove Corollary 3.9.
Note that the left-hand sides of (3.5) and (3.6) are equal to and , respectively. So .
This work was partially supported by the National Natural Science Foundation of China (11371200) and the Research Fund for the Doctoral Program of Higher Education (20120031110023). The authors thank Kangwei Li for very useful discussions and suggestions.
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