Journal of Function Spaces

Volume 2014 (2014), Article ID 797956, 10 pages

http://dx.doi.org/10.1155/2014/797956

Research Article

Weighted Estimates for Bilinear Operators

Hua Zhu¹ and Heping Liu²

¹Beijing International Studies University, Beijing 100024, China

²LMAM, School of Mathematical Science, Peking University, Beijing 100871, China

Received 31 May 2013; Accepted 26 November 2013; Published 6 February 2014

Academic Editor: Dashan Fan

Copyright © 2014 Hua Zhu and Heping Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the boundedness of weighted multilinear operators given by products of finite vectors of Calderón-Zygmund operators. We also investigate weighted estimates for bilinear operators related to Schrödinger operator.

1. Introduction

Bilinear (or multilinear) operators have attracted many researchers’ attention, due to their relations closely connected to the Cauchy integral along with Lipschitz curves, Calderón commutators, and compensated compactness. In [1–3] and references therein, we can see an extensive study on the Hardy space estimate of bilinear operators. In [4], we can see the bilinear operators related to a Schrödinger operator and estimates of them with respect to the Hardy type space associated with the Schrödinger operator under some general conditions. In our paper, we study weighted estimates for bilinear operators which have the same expression as the operators in [4].

In this paper, and will denote fixed integers ≥2. Given a matrix of convolution Calderón-Zygmund kernels on , we define as the associated Calderón-Zygmund operators. We denote by the -linear operator:

which originally is defined for smooth compactly supported functions . For , , we denote by the usual weighted Hardy space as defined in [5, 6], that is, the set of all distributions on for which the maximal function is in , where and is smooth, nonzero, and compactly supported, and we also denote by the weak as defined in [7], that is, the set of all distributions on for which the maximal function is in weak . If , the weak Hardy space was introduced by Feerman and Soria in [8]. first appeared in [9] (see also [10]).

Theorem 1. Assume that are given and let be their harmonic mean. We also assume that the harmonic mean of any proper subset of the ’s is greater than 1. If, for all , the -linear operator satisfies

Then, for , one has following conclusions.

(1) If , maps .(2) If , maps .(3) If , maps .

Let be a Schrödinger operator on , , where is a fixed nonnegative potential. We will assume that belongs to reverse Hölder class for some ; that is, there exists such that

for every ball . In what follows, denotes the ball centered at and of the radius . Trivially, provided . It is well known that if for some , then there exists , which depends only on and the constant in (3), such that (see [11]). Throughout this paper, we always assume that . Thus, for some .

Let be the semigroup of linear operators generated by and let be their kernels; that is,

By the Trotter product formula (cf. [12]),

The maximal function with respect to the semigroup is defined by

The weighted Hardy-type space related to is naturally defined by (see [13])

Following [14], we define the auxiliary function by

The auxiliary function plays an important role in studying the boundedness of singular integral operators related to the Schrödinger operator as well as the atomic decomposition of and (see [13–15]).

In our paper, we also consider the following bilinear operators:

where , with and , are Calderón-Zygmund operators related to and satisfy the following two conditions.

(i)There exist parallel Calderón-Zygmund operators related to the Laplacian and a constant such that

where and denote the kernels of and , respectively.(ii)One of the parallel bilinear operators

has the vanishing moment; that is, either or satisfies

We will show that either or is bounded from to .

Theorem 2. Suppose that the bilinear operators are defined as above. Let and . Then either or (but not both), which corresponds to the parallel bilinear operator satisfying (12), maps into and there exists a constant such that

This paper is organized as follows. In Section 2, we give some notation and preliminary estimates on and the kernel which have been proved in [14–16]. In Section 3, we prove Theorem 1, and in Section 4 we prove Theorem 2.

Throughout this paper, we will use to denote a positive constant, which is not necessarily the same at each occurrence. By , we mean that there exists a constant , such that . For a given ball , we denote by the concentric ball with twice radius, and .

2. Preliminaries

Throughout this paper, we will denote for any set . For , denote by the adjoint number of ; that is, .

We review some needed background about Muckenhoupt weights. A weight is a nonnegative locally integrable function on . We say that , if there exists a constant such that

for every ball . The class is defined replacing the above inequality by

where denotes the standard Hardy-Littlewood operator.

Let and let be a locally integrable nonnegative function. We denote the weighted space by and set

We also denote the weak by and set

We first sum up some properties of weights in the following results.

Lemma 3. One has the following properties.

(i) for .(ii)Let . Then if and only if  .(iii)If , then there exists such that .(iv)If , then the measure is doubling: precisely, for all and all cubes one has

where denotes the Muckenhoupt characteristic constant and denotes the cube with the same center as Q and side length times the side length of .(v).

Proof. Properties are standard; see, for instance, [17–19].

Lemma 4. Let be a ball of and let be any measurable subset of . Let . Then, there exist constants , and such that

Proof. For the proof, we refer to [17].

Next we recall some basic properties of the auxiliary function . It is known that for any (cf. [14]). Therefore,

where

Lemma 5 (see [15]). There exist a constant and a sequence of points such that the family of critical balls , defined by , satisfies

(i),(ii) for every .

Furthermore, there exists a family of functions such that

(iii),(iv) and for all ,(v).

Lemma 6 (see [14]). There exist and so that, for all ,

In particular, when and .

To prove Theorem 2, we need the following estimates for the kernel .

Lemma 7 (see [15, 16]). For every , there is a constant such that

Also, there exists a constant such that

where .

3. The Proof of Theorem 1

We fix and we let be their harmonic mean. For , we can get the result just by Hölder’s inequality and the boundedness of Calderón-Zygmund operators. Then we just need to study the case . Fix a smooth compactly supported function in , an , and define . Without loss of generality we may assume that is supported in . We need to show that is in when and in when . We also fix a smooth cutoff such that    on and is supported in . We call for simplicity and . We now decompose , where

In each above exactly functions among the ’s are multiplied by and the remaining are left intact.

For any fixed and any such that we have

where is the Hardy-Littlewood maximal function of at the point . We denote the maximal truncated operator of by . We begin with the :

Then we have

Define by . By Hölder inequality and the boundedness of and , the norm in of is bounded by

We obtain that

For , we write where

Using the same reason as before, we have the following estimate of :

where each is either or and therefore . Define by . For each , Hölder inequality implies that

Then we get

For , by the similar method, we get the desired conclusions. Now we turn to the term , and we use two inequalities which have been proved in [3] as follows:

where .

Then, for , we get

where we use , and we obtain conclusion (2) of Theorem 1.

For conclusion (3), let , , and be arbitrary; then, by using (36) and the weak result for the Hardy-Littlewood maximal function, we can get

and the expression minimizes when all the terms that appear in the sum are equal.

Finally, we get the weak type estimate:

and we complete the proof.

4. The Proof of Theorem 2

We give the proof for the bilinear operator only and the proof for is similar. Let and . Assume that the parallel Calderón-Zygmund operators and the parallel bilinear operator satisfy (10) and (12). We split into three parts:

where

It follows from Theorem 1 that and

Since , we have and

It suffices to show ; that is,

For simplicity we write . Choose a sequence of points and a family of functions as in Lemma 5. Then

For , by using (23), we get

and for and , Lemma 6 yields and ; then we have

For the inner integral we have

where . Then we obtain

and hence

To estimate , we write

For , we apply (23) and Lemma 6 to obtain

and hence

To estimate , we denote

For and , by Lemma 6 we have ; then the estimate (24) gives

For the inner integral, by properties of weight, we have

which yield

Now let us estimate . For , let and . Then

where denotes the characteristic function of the set . Let with satisfy for and for . Set and . We split the operator into four parts:

We first estimate

Write

where

Then we have

where we have used (5) in the last inequality. Since are Calderón-Zygmund operators, their kernels satisfy the standard kernel estimate for some . For ,

where is the Hardy-Littlewood maximal operator. Thus,

and hence

A similar argument shows that

Finally we consider the term . By using (5) and (10) we have

where . For , , and , we have . It follows from Lemma 6 that for a fixed constant . Hence, a well-known result for fractional integrals gives

where . Since , we are always able to choose and such that and . Then we get

Therefore, we have

Similarly, we get

Then we obtain

and we complete the proof of Theorem 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Hua Zhu is supported by Seed Project of Beijing International Studies University. Heping Liu is supported by National Natural Science Foundation of China under Grant no. 11371036 and the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant no. 2012000110059. Both authors would like to thank the referees for some very valuable suggestions which made this paper more readable.

References

R. Coifman and L. Grafakos, “Hardy space estimates for multilinear operators,” I, Revista Matemática Iberoamericana, vol. 8, pp. 45–67, 1992.

C. Coifman, P. L. Lion, Y. Meyer, and S. Semmes, “Compensated compactness and Hardy spaces,” Journal de Mathématiques Pures et Appliquées, vol. 72, pp. 247–286, 1993.

L. Grafakos, “Hardy space estimates for multilinear operators,” Revista Matemática Iberoamericana, vol. 8, pp. 69–92, 1992.

C.-C. Lin, Y.-C. Lin, H. Liu, and Y. Liu, “Bilinear operators associated with Schrödinger operators,” Studia Mathematica, vol. 205, no. 3, pp. 281–295, 2011.

J. Garca-Cuerva and K. S. Kazarian, “Calderón-Zygmund operators and unconditional bases of weighted Hardy spaces,” Studia Mathematica, vol. 109, no. 3, pp. 255–276, 1994.

J. Stromberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math, Springer, Berlin, Germany, 1989.

T. Quek and D. Yang, “Calderón-zygmund-type operators on weighted weak hardy spaces over ℝn,” Acta Mathematica Sinica, English Series, vol. 16, no. 1, pp. 141–160, 2000.

R. Feerman and F. Soria, “The space weak H¹,” Studia Mathematica, vol. 85, pp. 1–16, 1987.

H. Liu, The Weak Hp Spaces on Homogeneous Groups, Lecture Notes in Math, Springer, Berlin, Germany, 1991.

J. Alvarez, “ and weak continuity of calderón-zygmund type operators, fourier analysis: analytic and geometric aspects,” Marcel Dekker, pp. 17–34, 1994.

F. W. Gehring, “The Lp-integrability of the partial derivatives of A quasiconformal mapping,” Acta Mathematica, vol. 130, no. 1, pp. 265–277, 1973.

J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, NY, USA, 1985.

H. Liu, L. Tang, and H. Zhu, “Weighted Hardy spaces and BMO spaces associated with Schrödinger operators,” Mathematische Nachrichten, vol. 285, no. 17-18, pp. 2173–2207, 2012.

Z. Shen, “Lp estimates for Schrödinger operators with certain potentials,” Annales de l'institut Fourier, vol. 45, no. 2, pp. 513–546, 1995.

J. Dziubański and J. Zienkiewicz, “Hardy space H¹ associated to Schrödinger operator with potential satisfying reverse holder inequality,” Revista Matematica Iberoamericana, vol. 15, no. 2, pp. 279–296, 1999.

J. Dziubański, G. Garrigós, T. Martínez, J. L. Torrea, and J. Zienkiewicz, “BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality,” Mathematische Zeitschrift, vol. 249, no. 2, pp. 329–356, 2005.

J. Duoandikoetxea, Fourier Analysis, Graduate Studies in Mathematics, vol. 29, American Mathematical Society, Providence, RI, USA, 2000.

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Edu, New Jersey, NJ, USA, 2004.

E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, USA, 1993.