Abstract
Let T be the multilinear Calderón-Zygmund operator with non-smooth kernel and let be its corresponding maximal operator. In this paper, vector-valued weighted norm inequalities for T and are established. As applications, weighted strong type estimates for vector-valued commutators associated with T and are deduced respectively.
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1 Introduction and main results
Let T be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values in the space of tempered distributions,
Following [1], we say that T is a multilinear Calderón-Zygmund operator if, for some , it extends to a bounded multilinear operator from to , where , and if there exists a function K, defined off the diagonal in , satisfying
for all ;
and
for some and all , whenever . Such kernels are called m-linear Calderón-Zygmund kernels, and the collection of such functions is denoted by m- in [1]. As in [2], we define the maximal multilinear operator by
where is the smooth truncation of T given by
The vector-valued multilinear Calderón-Zygmund operator and vector-valued maximal multilinear operator associated with T are defined and studied in [3, 4].
where for .
Theorem A [3]
Assume that T is a multilinear Calderón-Zygmund operator. Let , and such that , . If , then there exists a constant such that
Theorem B [4]
Assume that T is a multilinear Calderón-Zygmund operator. Let , and such that , .
-
(i)
If and , then there exists a constant such that
-
(ii)
If at least one and , then there exists a constant such that
It is worth noting that similar results hold for in Theorems A and B.
We will replace (1.3) by a weaker regularity condition on the kernel K. Assume that operators are associated with kernels in the sense that
for every function , , and satisfy the following size condition:
where s is a positive fixed constant and h is a positive, bounded, decreasing function satisfying that for some ,
The j th transpose of T is defined via
for all , g in . It is easy to check that the kernel of is related to the kernel K of T via the identity
To maintain uniform notation, we may occasionally denote and .
Assumption (H0) We always assume that there exist some and some with such that both and T map to .
Assumption (H1) Assume that there exist operators with kernels that satisfy the conditions (1.4) and (1.5) with constants s and η for each and that for every , there exists kernel such that
for all in with . Also assume that there exist a non-negative function with and a constant so that for every and every , all and all , we have
whenever .
Kernels K that satisfy the size estimate (1.2) and Assumption (H1) with parameters m, A, s, η, ε are called generalized Calderón-Zygmund kernels, and their collection is denoted by m-. We say that T is of class m- if T has an associated kernel K in m-.
Assumption (H2) Assume that there exist operators with kernels that satisfy conditions (1.4) and (1.5) with constants s and η, and there exist kernels such that the representation is valid
and that there exist a non-negative function and and a positive constant ε such that
for some , whenever . Moreover, assume that for all ,
whenever , and for all ,
whenever and .
The commutators associated with T and are defined respectively by
and
Here and subsequently, we often write and .
For simplicity of notation, we often write with . For the sequence of vector functions, the commutators associated with vector-valued and can be defined by
From now on, we always assume that T is a multilinear operator in m- and its kernel satisfies Assumption (H2). Recently, if , Peng et al. [5] obtained the following weighted strong type estimates for and with multiple weights (see Definition 2.1).
Theorem C [5]
Let , with , . Then we have
-
(i)
There exists a constant C such that
-
(ii)
If each , then there exists a constant C such that
where . Similar results still hold for , which extend the results in [6]significantly.
In this work, we first pursue results parallel to Theorems A and B, then extend Theorem C to a vector-valued version. The main results can be stated as follows.
Theorem 1.1 Let , and such that , . If , then there exists a constant such that
Moreover, similar estimates hold for .
Theorem 1.2 Let , and such that , .
-
(i)
If and , then there exists a constant such that
-
(ii)
If at least one and , then there exists a constant such that
Moreover, similar estimates hold for .
Theorem 1.3 Let , with , , and with . Suppose that , and . Then we have
-
(i)
There then exists a constant such that
-
(ii)
If , then there exists a constant such that
Moreover, similar estimates hold for .
Remark 1.4 If and , Theorem 1.3 can be seen as the vector-valued extension of Theorem 4.5 in [6] and Theorem C, respectively.
2 Proofs of Theorem 1.1 and Theorem 1.2
Let us begin with the definition of Hardy-Littlewood maximal operator, that is
The sharp maximal function is defined by
For , we also need the maximal function and .
The new maximal function ℳ can be defined by
For , as in [7], a modified maximal function is given by
For exponents , we will often write p for the number given by , and for the vector . Let us recall the definition of weights.
Definition 2.1 Let . Given , set . We say that satisfies the condition if
when , is understood as .
We will use the following lemmas in the proof of Theorem 1.1 and Theorem 1.2.
Lemma 2.2 [3]
Let be an m-linear operator, and let and be fixed indices such that . For , the following estimate holds:
Then, for all indices, and satisfy , and such that , and all . Then the following inequality holds:
Lemma 2.3 [7]
Suppose that for some , and satisfying , T maps to . Let , , and . Then
-
(i)
can be extended to a bounded operator from to if all the exponents are strictly greater than 1.
-
(ii)
can be extended to a bounded operator from to if some exponents equal 1.
Similar results hold for T.
Note that if each , then and this inclusion is strict (see [8] for details). This fact together with Lemma 2.3 yields the following weighted estimates.
Lemma 2.4 Consider an m-tuple , where and satisfy . Then there exists a constant C such that
and
Proof of Theorem 1.1 and Theorem 1.2 As a consequence of Lemma 2.2 and Lemma 2.4, we obtain Theorem 1.1 (see the proof of Corollary 3 in [3]). From [7] we know that for all , and , the following two inequalities hold:
So, we get . A similar inequality still holds for . Theorem 1.2 follows by repeating the same steps as in Corollary 3.3 in [4]. In fact, we apply Theorem 2.1 in [4] to the families
Hölder’s inequality and the normal inequalities for the maximal operator yield the desired results. □
3 Proof of Theorem 1.3
We begin with some lemmas which will be used in the proof of Theorem 1.3.
Lemma 3.1 [9]
Let and let ω be a weight in . Then there exists (depending upon the condition of ω) such that
for every function such that the left-hand side is finite.
Lemma 3.2 Let , and with . Then there exists a constant such that
for any smooth vector function for any .
Proof Fix a point and a cube Q centered at x. Set , where . Let and . It is easy to see
where and in the last sum each or ∞ and in each term there is at least one . Since , it follows that
where .
Applying Kolmogorov’s inequality and Theorem 1.2 to , we have
We proceed to the estimate for . We can take . If , we have
Since and , we get for all . Applying Assumption (H2), we obtain
Since , . Note that , for all , hence . Similarly, we get .
Now we estimate the typical representative of , that is, and .
Thus, we get
For , by Assumption (H2), we have
By a similar argument, we deduce that . In other case, we can also deduce the same estimate with minor modifications on the above arguments. We have thus proved Lemma 3.2. □
Lemma 3.3 Let , and with . Suppose that . There then exists a constant depending only on δ and ε such that
for any smooth vector function for any , where .
Proof For simplicity of notation, we write instead of the product of m functions and let , for . Let and Q be a cube centered at x. Then we have
Noting that . Then we get
Since , it follows that
where . We can choose with . Since , Hölder’s inequality gives
Let us estimate term III. Set , where . Let and . Taking , we have
where in the last sum each or ∞ and in each term there is at least one .
If , applying Kolmogorov’s inequality and Theorem 1.2, we get
If , we have
Consider now the term . Taking , we have by Assumption (H2)
Similarly, we get . Now we consider only the typical representative of III. Similar to the estimates of in Lemma 3.2, we have
Then Lemma 3.3 is proved. □
Lemma 3.4 Let , , and with and let . Suppose that . Then there exists a constant such that
for any smooth function with compact support.
Proof We assume that the right-hand side of (3.2) is finite, since otherwise there is nothing to be proved. For , by using Lemma 3.1 and Lemma 3.3, we obtain
For the general case , similarly to the case for , we have
To apply the Fefferman-Stein inequality in (3.3), one needs to verify now that and . We will only show the first one since the proof of another one is very similar but easier.
Suppose that the symbols and the weight ω are bounded functions. Since has compact support, we may assume . Then we have
We choose and such that . Theorem 1.1 and Hölder’s inequality imply
Consider now the term . Since , we have by Assumption (H2)
Thus, .
For the general case, we can check the limit (see the proof of [10], Theorem 1.1, we omit the details here). Thus, (3.2) is proved. □
Proof of Theorem 1.3 (i) Since , Lemma 3.4 gives
It follows from [8] that there exists such that . On the other hand, since for all , the generalized Jensen inequality yields that
for all j. One sees immediately that
Since , it follows from Hölder’s inequality and the well-known inequality of Fefferman-Stein [9] that
We then have
This is the desired conclusion.
-
(ii)
Since , there exists such that . Analysis similar to that in the proof of Theorem 1.3 (i) shows that
The Hölder inequality implies
To prove Theorem 1.3 holds for , it suffices to prove Lemmas 3.2-3.4 hold for . The proof follows from similar steps in [11] and combines the argument we used in the above lemmas. The key for tackling the new complexities is a very careful deal with the supremum, we refer the reader to [11]. This concludes the proof of the theorem. □
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Acknowledgements
This work was supported by the Mathematical Tianyuan Foundation of China (No. 11226102). The author thanks the referees for carefully reading the manuscript and providing many valuable suggestions, which have improved this article.
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Si, Z. Weighted estimates for vector-valued multilinear operators with non-smooth kernels. J Inequal Appl 2013, 250 (2013). https://doi.org/10.1186/1029-242X-2013-250
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DOI: https://doi.org/10.1186/1029-242X-2013-250