Commutators of intrinsic square functions on generalized Morrey spaces
© Wu and Zheng; licensee Springer. 2014
Received: 25 September 2013
Accepted: 13 March 2014
Published: 28 March 2014
In this paper, we obtain the boundedness of intrinsic square functions and their commutators generated with BMO functions on generalized Morrey spaces. Our theorems extend some well-known results.
where . Denote .
We refer for details to .
In , Wilson proved the following result.
Theorem A Let , , then is bounded from to itself.
were first defined by Guliyev in . When , . It is the classical Morrey space which was first introduced by Morrey in . There are many papers discussed the conditions on to obtain the boundedness of operators on the generalized Morrey spaces. For example, in , the function φ is supposed to be a positively growth function and satisfy the double condition: for all , , where is a constant independent of r. This type of conditions on φ is studied by many authors; see, for example, [11, 12]. In , the following statement was proved by Nakai for the Calderón-Zygmund singular integral operators T.
where c does not depend on x and r. Then the operator T is bounded on for and from to for .
where c does not depend on x and t. Then the operator T is bounded from to for and from to for .
By an easy computation, we can check that if the pair satisfies double condition, then it will satisfy condition (1). Moreover, if satisfies condition (1), it will also satisfy condition (2). But the opposite is not true. We refer to  and Remark 4.7 in  for details.
Our main results in this paper are stated as follows.
Theorem 1.1 Let , , let satisfy condition (2), then is bounded from to .
Theorem 1.2 Let , , let satisfy condition (2), then for , we have is bounded from to .
Theorem 1.3 Let , , , let satisfy condition (3), then is bounded from to .
Theorem 1.4 Let , , , let satisfy condition (3), then for , is bounded from to .
In , the author proved that the functions and are pointwise comparable. Thus, as a consequence of Theorem 1.1 and Theorem 1.3, we have the following results.
Corollary 1.5 Let , , let satisfy condition (2), then is bounded from to .
Corollary 1.6 Let , , , and let satisfy condition (3), then is bounded from to .
Throughout this paper, we use the notation to mean that there is a positive constant C (≥1) independent of all essential variables such that . Moreover, C maybe different from place to place.
2 Proofs of main theorems
Before proving the main theorems, we need the following lemmas.
Lemma 2.1 ()
where is the Hardy operator , .
- (1)For ,
- (2)Let , , then
Then, by an easy computation, we get Lemma 2.3.
By a similar argument as in , we can easily get the following lemma.
Lemma 2.4 Let , , then the commutators is bounded from to itself whenever .
Now we are in a position to prove the theorems.
Since , by (7), (8) and (12), we have the desired theorem. □
Using an argument similar to the above proofs and that of Theorem 1.2, we can also show the boundedness of . □
This work was completed with the support of Scientific Research Fund of Zhejiang Provincial Education Department No. Y201225707.
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