Two-Weight Extrapolation on Lorentz Spaces

Abstract

We give several two-weight extrapolation theorems on Lorentz spaces which extend the results of Cruz-Uribe and Pérez (2000) and some applications for the singular integral operators, the potential type operators, and commutators.

1. Introduction and Main Results

The classical extrapolation theorem is due to Rubio de Francia [1], who showed that if sublinear operator is bounded on for some , and every , then is bounded on for every and every . Cruz-Uribe et al. [2] and Curbera et al. [3] generalized the extrapolation theorem of Rubio de Francia and got a lot of new extrapolation theorems associated with weights. These theorems have proved to be the key to solving many problems in harmonic analysis.

Cruz-Uribe and Pérez [4] gave several other extrapolation theorems for pairs of weights of the forms and , where is any nonnegative function, , and is the th iteration of the Hardy-Littlewood maximal operator. The purpose of this paper is to extend the extrapolation theorems in [4] and derive some extrapolation results on Lorentz spaces.

Let , be a nonnegative, locally integrable weight function. For a measurable function , is the distribution function of with respect to the measure . For , the weighted Lorentz spaces are the collection of all functions , such that , where

For , denote the conjugate exponent of : .

In the case when either and , or , or , is a Banach space. Furthermore, we have the relationship

Our main results are the following.

Theorem 1. Let and be operators (not necessarily linear), and let be a function in a suitable test class for both and . Suppose that there exist positive constants and and a positive integer such that for all weights Then, for all , , there exists a constant depending only on , and , such that, for all weights , where is the largest integer less than or equal to .

Theorem 2. Let and be operators (not necessarily linear), and let be a function in a suitable test class for both and . Suppose that there exist positive constants and and a positive integer such that for all weights Then, for all , , there exists a constant depending only on , and , such that, for all weights ,

2. Applications

Let be the Hardy-Littlewood maximal operator; Fefferman and Stein [5] showed that, for every , every nonnegative function , and every function ,

Using Theorem 1, we can get the following result.

Proposition 3. For , , there exists a constant , such that, for all weights and function ,

Let be a Calderón-Zygmund operator. Wilson [6] and Pérez [7] showed that if , then, for any weight and every , where the exponent is sharp. Using Theorem 1 we can get the following result.

Proposition 4. Let be a Calderón-Zygmund operator, , ; then, for any weight and every ,

For the commutators of singular integral operators with a BMO function , Pérez [8] showed the analogous weighted inequalities as in (10). By Theorem 1, we can give similar results as in Proposition 4; details are omitted.

Lerner [9] establishes for any Calderón-Zygmund operator and any arbitrary weight . Using Theorems 1 and 2, we can get the following results.

Proposition 5. Let be a Calderón-Zygmund operator, , ; then, for any weight and all ,

Now, we consider the potential type operators and commutators. For a nonnegative, locally integrable function on , assume that satisfies the following weak growth condition: there are constants , with the property that, for all , The basic examples are provided by the Riesz potential of order , , defined by the kernel , .

Define the potential type operator and the maximal operator by where For the Riesz potential of order , . Pérez [10] studies the two-weight strong type inequalities for , .

Lemma 6 (see [10]). Let be the potential type operator with satisfying (14), let and be bounded functions with compact support, and let ; then, there exist a family of cubes and a family of pairwise disjoint subsets , , with for all , such that where .

By Lemma 6, we can easily prove the following result.

Lemma 7. Let be the potential type operator with satisfying (14); then, there is a constant such that, for any weight and all ,

Using (18) and Theorems 1 and 2, we can get the following results.

Proposition 8. Let be the potential type operator with satisfying (14), , ; then, for any weight and every ,

For and , the commutators of potential type integral operator with a BMO function are defined by Li [11] gave the two-weight strong type inequalities for , . We can easily get similar results as in Lemmas 6 and 7 from [11]; by Theorem 1, we can obtain some weighted inequalities for the commutators of potential type integral operators; details are omitted.

3. Proofs of Theorems

We need some notations and facts. Let be a Young function, that is, a continuous, convex, and increasing function with such that as . Given a Young function , we define the -average of a function over a cube by In particular, for the Young function , the -average of a function over a cube is denoted by . Define the maximal function associated with the Young function as Pérez [8] obtained that .

Proof of Theorem 1. Fix , , and let . Then, by duality, The last inequality follows since and are associate spaces. Fixing one of these ’s, we use (4) to continue with where is an appropriate maximal operator to be chosen soon. To conclude, we just need to show that or equivalently where To do this, we choose pointwise bigger than ; then, we trivially have Therefore, by Marcinkiewicz’s interpolation theorem for Lorentz spaces in [12], it will be enough to show that for some which amounts to proving But this result follows from [13]: it is shown there that, for and , We finally choose the appropriate parameters and weight. Let , , and pick the weight This shows that We conclude the proof of (5).

Proof of Theorem 2. The proof of this theorem proceeds exactly as that of (5) with minor changes. At inequality (23), fixing one of these ’s, we use (6) to continue with For and , we have , where is a constant depending only on the dimension and is the unweighted centered Hardy-Littlewood maximal operator: in which is the ball of radius centered at . Furthermore, where is the weighted centered maximal operator. Then, for , To conclude, we just need to show that or equivalently where We notice that for each . With this, we trivially have Therefore, by Marcinkiewicz’s interpolation theorem for Lorentz spaces, it will be enough to show that for some which amounts to proving Taking and using the well-known fact that is bounded on , , with a constant that depends only on and , we get (43).
This shows that Taking , we get (7). This ends 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

The authors would like to express their gratitude to the referee for his very valuable comments and suggestions. This study is supported by the Natural Science Foundation of Hebei Province (A2014205069 and A2015403040) and Postdoctoral Science Foundation of Hebei Province (B2013003007).