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Multilinear Fourier multipliers on variable Lebesgue spaces
Journal of Inequalities and Applications volume 2014, Article number: 510 (2014)
Abstract
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.
1 Introduction
Given a non-empty open set , we denote by the set of exponent functions such that
where and .
Let be the set of exponent functions such that
Given a measurable function f on Ω for , we define the modular functional associated with by
where denotes the set of points in Ω on which .
The variable exponent Lebesgue space is defined to be the set of Lebesgue measurable functions f on Ω satisfying for some . The norm of f in the space is defined by
In the case that , it is defined to be the set of all functions f satisfying , for some (see [1]). A quasi-norm in the space is defined by
We refer to [2] for an introduction to variable exponent Lebesgue spaces.
Similarly, for and a weight function w, the weighted variable exponent Lebesgue space (see [3]) is defined to be the set of Lebesgue measurable functions f on Ω that satisfies
In this paper, we study some properties of the space of bilinear Fourier multipliers and the Mihlin-Hörmander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces. Specifically, let m satisfy certain conditions. We discuss the N-linear Fourier multiplier operator defined by
for , [4].
The multilinear Fourier multipliers have been studied for a long time. In [4], Coifman and Meyer proved that is bounded from to for all with and satisfying
for all , where is an integer and s is a sufficiently large integer.
Tomita [5] 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 [6]. The boundedness of multilinear Calderón-Zygmund operators with multiple weights was achieved by Grafakos et al. [7].
Under the Hörmander conditions, Fujita and Tomita [8] obtained some weighted estimates of for classical weights. Then Li et al. [9] got some weighted results of multilinear multipliers by considering the end-point cases, using weighted Carleson measure theory and employing multilinear interpolation theory. In [10], Chen and Lu proved a Hörmander type multilinear theorem on weighted Lebesgue spaces when the Fourier multipliers were only assumed with limited smoothness. In [11], the boundedness of with multiple weights satisfying condition (1.1) was given by Bui and Duong. In [12], Li and Sun got some weighted estimates of with multiple weights under the Hörmander conditions in terms of the Sobolev regularity. Huang and Xu [13] 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 [12], but on variable exponent Lebesgue spaces.
The theory of bilinear multipliers was first studied by Coifman and Meyer [14]. They considered the ones with smooth symbols. Then, Muscalu et al. achieved some new results for non-smooth symbols in [15].
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 [18], Kulak and Gürkanlı first studied some properties of the bilinear multiplier space. In [19], Fan and Sato proved the DeLeeuw type theorems for the transference of multilinear operators on Lebesgue and Hardy spaces from to . In [20], Blasco gave the transference theorems from to . We also refer to [21, 22] for details.
We first give some definitions.
Definition 1.1 ([18])
Let , , and be a bounded function on . Define
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.
Definition 1.2 Given a function M on , let . We say that
if for all f and can be extended to a bounded bilinear operator from to .
Definition 1.3 ([2])
A function is said to belong to the class if
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 [18]. Here we give some other properties.
First, we introduce the standard singular kernel.
Definition 2.1 ([2])
Given a function , it is called a standard singular kernel if there exists a constant such that:
-
1.
, ;
-
2.
, ;
-
3.
for , ;
-
4.
exists.
Theorem 2.2 (Localization)
Suppose that
Q is a rectangle in and that the Hardy-Littlewood maximal operator ℳ is bounded on , where , . Then
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 ([[2], 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 .
Theorem 2.4 Suppose that , and . Then we have
Proof For any f and , we have
Therefore,
Then we get the result. □
The following is an explicit example.
Example 2.5 Suppose that , and , where and . Then
Proof For any f and , we have
By Hölder’s inequality [2], we have
Thus . By Theorem 2.4, we have
□
Proof of Theorem 2.2 We only consider the case . Other cases can be proved similarly. Suppose that . Then, for any f and ,
Note that by (3.9) of [23], we have , where denotes the operator and H denotes the Hilbert transform operator. Since the Hilbert transform has a standard singular kernel, by Lemma 2.3 we have
So
Similarly we can prove that
Hence by Theorem 2.4, we get
and . □
Next we show that the space is invariant under certain operators.
Theorem 2.6 Given , , if
then
and .
Proof For any f and , we have
By Minkowski’s inequality,
□
Theorem 2.7 Suppose that , and . Then
and .
Proof For any f and , we have
By Young’s inequality, we have
Thus, we get the conclusion. □
Finally, we consider the necessary condition of this kind of multipliers. The bilinear classical counterpart was obtained by Hörmander [[24], Theorem 3.1] and Blasco [25]. The multilinear classical one was proved by Grafakos and Torres, see [[26], Proposition 5] and [[27], Proposition 2.1]. And the one for multipliers on Lorentz spaces was given by Villarroya [[28], 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)
Suppose that there is a non-zero continuous bounded function M such that . Then
To prove the theorem, we need the following results.
Proposition 2.9 ([[2], Corollary 2.22])
Fix Ω and . If , then ; if , then .
Proposition 2.10 ([[2], Corollary 2.23])
Given Ω and , suppose . If , then
If , then
Lemma 2.11 Let . If , then there exists some such that
when λ is sufficiently large.
Proof Let . Define by . By a simple change of variable, one gets that
where we use the fact that [[29], Example 2.2.9].
Observe that
where .
Similarly we have
By Proposition 2.9, we get , when λ is sufficiently large. Thus by Proposition 2.10, we have
So
where .
Similarly we can get
All the inequalities above are established when the λ is sufficiently large.
By the assumption, we have
Now combining (2.1), (2.2), (2.3) and (2.4), we get
Hence
when λ is sufficiently large. □
We are now ready to prove Theorem 2.8.
Proof of Theorem 2.8 Assume that . By a simple calculation, we obtain that
where . Thus . Applying Lemma 2.11 to , we get
Observe that and M is continuous. By letting , we have
Since y is arbitrary, we have . This is a contradiction. Thus
□
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, [[3], Theorem 4.5, Theorem 4.7], [30] and [31].
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 [32].
We give another way to get the Mihlin-Hörmander conditions for multilinear Fourier multipliers on weighted variable spaces.
First we use Q to denote a cube in . Recall that the Hardy-Littlewood maximal operator is defined by
And the sharp maximal function is defined by
For , we also define
For and , we define
Definition 3.1 ([33])
Given with and . Let . Set
We say that satisfies the condition if
When , then is understood as .
We now give a Mihlin-Hörmander type theorem for multilinear Fourier multipliers on weighted variable exponent Lebesgue spaces.
Theorem 3.2 Suppose that , and
Set , a series of variable indexes , and , such that , where , . Suppose that there are , such that the Hardy-Littlewood maximal operator ℳ is bounded on and , where , , . Then there exists some such that
Before proving the theorem, we present some preliminary results. The following inequality is a classical result of Fefferman and Stein [34].
Proposition 3.3 ([34])
Let and . Then there exists some constants such that
The next result comes from Lemma 2.6 in [12]. For our purpose, we restate it in the proper way.
Proposition 3.4 ([12])
Let such that , . If , then under the assumption of Theorem 3.2, there exists some such that for all , , we have
Proposition 3.5 ([3])
Let X be a metric measure space and Ω be an open set in X. Assume that for some and satisfying
and for every weight , there holds the inequality
for all in a given family ℱ. Let the variable exponent be defined by
Let the exponent and the weight ϱ satisfy that and ℳ is bounded on .
Then, for all with , the inequality
is valid with a constant .
Remark 3.6 Note that the condition in the extrapolation theorem of [3] can be released to with nearly no modification to the proof.
Proposition 3.7 ([[11], Proposition 2.3])
Let and for and . Then the inequality
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
Proof of Theorem 3.2 For any , , and , by Proposition 3.3 and Proposition 3.4, we have
where is defined as in Proposition 3.4.
Since the maximal operator ℳ is bounded on , by Proposition 3.5, we have
By Hölder’s inequality,
where
Since , we can choose such that . Thus by Proposition 3.7, we get that
is valid for all , . Using the boundedness of ℳ again, we see from Proposition 3.5 that
It follows from (3.3) that
By (3.2), we obtain the desired conclusion as follows:
□
As an application of Theorem 3.2, we now consider the case when weight functions are defined by
where are fixed points in , .
Corollary 3.9 Suppose that , and
Let the variable exponents and satisfy that , where , , and . Suppose that there exists some and such that for , , and that
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 [[3], Definition 2.7].
Definition 3.10 ([3])
Let and there exists and such that for all . A weight function w of the form
is said to belong to the class if
and
We first give some lemmas that are needed to prove Corollary 3.9.
Lemma 3.11 ([[2], Proposition 2.3])
Given a domain Ω, if , then is equivalent to assuming .
Lemma 3.12 ([[3], Remark 2.10])
For every , there hold the implications
where .
Lemma 3.13 ([[3], 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.
Proof of Corollary 3.9 Fix some . Let , and be defined as in Theorem 3.2. By the assumption, we have
So . By Lemma 3.14, ℳ is bounded on . Again, by the assumption, we get
Note that the left-hand sides of (3.5) and (3.6) are equal to and , respectively. So .
By Lemma 3.11, we know . Therefore, . Thus . Now by Lemma 3.14, ℳ is bounded on . By Theorem 3.2, there exists some such that
□
References
Tao X, Yu X, Zhang H: Multilinear Calderón Zygmund operators on variable exponent Morrey spaces over domains. Appl. Math. J. Chin. Univ. Ser. B 2011,26(2):187-197. 10.1007/s11766-011-2704-8
Cruz-Uribe DV, Fiorenza A Applied and Numerical Harmonic Analysis: Foundations and Harmonic Analysis. In Variable Lebesgue Spaces. Birkhäuser, Heidelberg; 2013.
Kokilashvili VM, Samko SG: Operators of harmonic analysis in weighted spaces with non-standard growth. J. Math. Anal. Appl. 2009,352(1):15-34. 10.1016/j.jmaa.2008.06.056
Coifman RR, Meyer Y Astérisque 57. In Au Delà des Opérateurs Pseudo-différentiels. Société Mathématique de France, Paris; 1978. With an English summary
Tomita N: A Hörmander type multiplier theorem for multilinear operators. J. Funct. Anal. 2010,259(8):2028-2044. 10.1016/j.jfa.2010.06.010
Grafakos L, Si Z: The Hörmander type multiplier theorem for multilinear operators. J. Reine Angew. Math. 2012, 668: 133-147.
Grafakos L, Liu L, Maldonado D, Yang D: Multilinear analysis on metric spaces. Diss. Math. 2014., 497: Article ID 121
Fujita M, Tomita N: Weighted norm inequalities for multilinear Fourier multipliers. Trans. Am. Math. Soc. 2012,364(12):6335-6353. 10.1090/S0002-9947-2012-05700-X
Li W, Xue Q, Yabuta K: Weighted version of Carleson measure and multilinear Fourier multiplier. Forum Math. 2012. 10.1515/forum-2012-0083
Chen J, Lu G: Hörmander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness. Nonlinear Anal. 2014, 101: 98-112.
Bui TA, Duong XT: Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers. Bull. Sci. Math. 2013,137(1):63-75. 10.1016/j.bulsci.2012.04.001
Li, K, Sun, W: Weighted estimates for multilinear Fourier multipliers. (2012). arXiv:1207.5111 [math.CA]
Huang A, Xu J: Multilinear singular integrals and commutators in variable exponent Lebesgue spaces. Appl. Math. J. Chin. Univ. Ser. B 2010,25(1):69-77. 10.1007/s11766-010-2167-3
Coifman RR, Meyer Y: Fourier analysis of multilinear convolutions, Calderón’s theorem, and analysis of Lipschitz curves. Lecture Notes in Math. 779. In Euclidean Harmonic Analysis. Springer, Berlin; 1980:104-122. (Proc. Sem., Univ. Maryland, College Park, Md., 1979)
Muscalu C, Tao T, Thiele C: Multi-linear operators given by singular multipliers. J. Am. Math. Soc. 2002,15(2):469-496. 10.1090/S0894-0347-01-00379-4
Lacey M, Thiele C: Estimates on the bilinear Hilbert transform for . Ann. Math. (2) 1997,146(3):693-724. 10.2307/2952458
Lacey M, Thiele C: On Calderón’s conjecture. Ann. Math. (2) 1999,149(2):475-496. 10.2307/120971
Kulak Ö, Gürkanlı AT: Bilinear multipliers of weighted Lebesgue spaces and variable exponent Lebesgue spaces. J. Inequal. Appl. 2013., 2013: Article ID 259
Fan D, Sato S: Transference on certain multilinear multiplier operators. J. Aust. Math. Soc. 2001,70(1):37-55. 10.1017/S1446788700002263
Blasco O: Bilinear multipliers and transference. Int. J. Math. Math. Sci. 2005,2005(4):545-554. 10.1155/IJMMS.2005.545
Auscher P, Carro MJ:On relations between operators on , and . Stud. Math. 1992,101(2):165-182.
Blasco O: Notes in transference of bilinear multipliers. III. In Advanced Courses of Mathematical Analysis. World Scientific, Hackensack; 2008:28-38.
Duoandikoetxea J Graduate Studies in Mathematics. In Fourier Analysis. Am. Math. Soc., Providence; 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe
Hörmander L:Estimates for translation invariant operators in spaces. Acta Math. 1960, 104: 93-140. 10.1007/BF02547187
Blasco O: Notes on the spaces of bilinear multipliers. Rev. Unión Mat. Argent. 2009,50(2):23-37.
Grafakos L, Torres RH: Multilinear Calderón-Zygmund theory. Adv. Math. 2002,165(1):124-164. 10.1006/aima.2001.2028
Grafakos L, Soria J: Translation-invariant bilinear operators with positive kernels. Integral Equ. Oper. Theory 2010,66(2):253-264. 10.1007/s00020-010-1746-2
Villarroya F: Bilinear multipliers on Lorentz spaces. Czechoslov. Math. J. 2008,58(133)(4):1045-1057.
Grafakos L Graduate Texts in Mathematics 249. In Classical Fourier Analysis. 2nd edition. Springer, Heidelberg; 2008.
Kurtz DS:Littlewood-Paley and multiplier theorems on weighted spaces. Trans. Am. Math. Soc. 1980,259(1):235-254.
Kurtz DS, Wheeden RL: Results on weighted norm inequalities for multipliers. Trans. Am. Math. Soc. 1979, 255: 343-362.
Grafakos L, Martell JM: Extrapolation of weighted norm inequalities for multivariable operators and applications. J. Geom. Anal. 2004,14(1):19-46. 10.1007/BF02921864
Lerner AK, Ombrosi S, Pérez C, Torres RH, Trujillo-González R: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. Math. 2009,220(4):1222-1264. 10.1016/j.aim.2008.10.014
Fefferman C, Stein EM: Spaces of several variables. Acta Math. 1972,129(3-4):137-193.
Acknowledgements
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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Ren, J., Sun, W. Multilinear Fourier multipliers on variable Lebesgue spaces. J Inequal Appl 2014, 510 (2014). https://doi.org/10.1186/1029-242X-2014-510
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DOI: https://doi.org/10.1186/1029-242X-2014-510