- Research Article
- Open Access
Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures
© Q. Xue and J. Zhang. 2009
- Received: 18 April 2009
- Accepted: 18 August 2009
- Published: 28 September 2009
Let be a positive Radon measure on which may be nondoubling. The only condition that satisfies is for all , , and some fixed constant . In this paper, we introduce the operator related to such a measure and assume it is bounded on . We then establish its boundedness, respectively, from the Lebesgue space to the weak Lebesgue space , from the Hardy space to and from the Lesesgue space to the space . As a corollary, we obtain the boundedness of in the Lebesgue space with .
- Singular Integral
- Classical Operator
- Singular Integral Operator
- Fractional Integral Operator
- Atomic Block
for all , where is some fixed number with . For example, In 2001, Tolsa in [1, 2] investigated the weak (1,1) inequality for singular integrals, the Littlewood-Paley theory and the theorem with nondoubling measures. In 2002, García-Cuerva and Gatto  investigated the boundedness properties of fractional integral operators associated to nondoubling measures. In 2005, Hu et al.  studied the multilinear commutators of singular integrals with nondoubling measures. Since 2007, Hu et al.  have proved some boundedness results of Marcinkiewicz integrals with nondoubling measures on some function spaces.
On the other hand, let be a function on such that there exist positive constants , , , and satisfying
(a) and ,
(c) for .
where and .
Note that if we replace by in the above definition and when is the Poisson kernel, we obtain classical -function defined and studied by Stein  and later by Fefferman , where the weak (1,1) with and weak with boundedness of function were obtained. In the same paper, Fefferman  also established the bounds of for and . For the more generalized -function defined by (1.1), the boundedness is also well known (see, e.g., [8, pages 309–318]). On the other hand, inspired by the works of Sakamoto and Yabuta in 1999, the first author in this paper studied parametric -function systematically in his PhD thesis . Later, in 2008, Lin and Meng  gave some results on parametric -function with nondoubling measures. But their result only valid for , one cannot obtain the results for classical operators even for or in the classical case studied by Stein in 1961 .
In this paper, we will study the properties of operator with nondoubling measures on some function spaces under the conditions (a)–(c).
First, before stating our main results, we give some notation and definitions, let be a closed cube with sides parallel to the axes. Denote its side length by and its center by . Given and , we say is -doubling if , where is the cube concentric with with side length . If are not specified, by a doubling cube we mean a -doubling cube. For any cube , we denote by the smallest doubling cube which contains and has the same center as .
with , where is the first integer such that .
For a fixed , a function is called an atomic block if
(1)there exists some cube such that ;
where the infimum is taken over all the possible decompositions of in atomic blocks.
It was shown by Tolsa that the space was proved to be the Hardy space in  with equivalent norms. We will denote the space and the norm , respectively, by and for convenience. He also proved that the dual space of is the following space .
where denotes the mean value of over cube . The minimal constant above is defined to be the norm of in the space and denoted by .
The minimal constant as above is defined to be the norm of in the space , we denote it by .
In this paper, we always assume that and are considered as they are defined at the beginning of this paper. Our main results are as follows.
Let be a function on , satisfying (a)–(c), , . If is bounded on , then it is also bounded from to .
Let be a function on , satisfying (a)–(c), , . If is bounded on , then it is also bounded from to .
Let be a function on , satisfying (a)–(c), , . If is bounded on , then it is also bounded on for any .
It is natural to consider the similar problems with more general rough kernels. However, even in the doubling measure case, if we take in (1.1) (in this case, is defined and studied by ), from the results in , we know that it is impossible to give similar results as above for Littlewood-Paley function even for . In fact, by the counter example in , even the boundedness does not hold. In this sense, the condition we assumed on is necessary and reasonable. On the other hand, in 2008, Lin and Meng  gave some results on parametric -function with nondoubling measures. In fact the results in  are only valid for . By the same reason as above, one cannot obtain the result when which in this case, the operator coincides with the classical operator studied by Torchinsky and Wang in  and it is a generalization of the classical operators studied by Stein and Fefferman.
Even in the classical case, the index is sharp for weak boundedness; see  for detail.
We arrange our paper as follows, in Section 2, we give and prove some key lemmas. The proof of our main theorems will be given in Section 3. Throughout this paper, the letter will denote a positive constant that may vary at each occurrence but is independent of the essential variables. will always denote that there exists a constant , such that .
We need two lemmas given by Tolsa.
Lemma 2.1 (see ).
where depends only on , , , .
Lemma 2.2 (see ).
For any and any then we have one has the following:
(a)there exists a family of almost disjoint cubes (that means , depends only on ) such that
(a.2) for any ;
(a.3) a.e. on ,
(b)for each , let be the smallest ( )-doubling cube of the form , , with that , and let , then there exists a family of functions with and with constant sign satisfying
(b.2) , where is some constant;
To prove our theorems, we prepare another two key lemmas.
Proof of Lemma 2.3.
Note that and , then , . These two inequalities will always be used in the following proof, so we will not mention them every time.
The proof of Lemma 2.3 is finished.
Proof of Lemma 2.4.
Since and , we have that , .
Since , we can choose small enough such that and .
It is easy to see that .
Proof of Theorem 1.4.
To prove Theorem 1.4, we will choose and .
Let and . Applying Lemma 2.2 to and , we obtain a family of almost disjoint cubes . With the notation , , the same as in Lemma 2.2, we can decompose , with that and . And can be decomposed as . By (a.1) of Lemma 2.2, we have .
Since and , we only need to show that both and satisfy the inequality (2.4).
then we finish the proof of Theorem 1.4.
Choose small enough such that , and .
and by Lemma 2.2, we conclude that .
Therefore we finish the proof of Theorem 1.4.
Proof of Theorem 1.5.
Combining (3.27) and (3.37), we finish the proof of Theorem 1.5.
Proof of Theorem 1.6.
The method to prove (3.43) is quite similar as to prove (2.4) in the proof of Theorem 1.4 and we omit it.
This can be obtained from the same idea used before, see also the main step in , here we omit the proof of it.
provided that is a -doubling cube.
for any two -doubling cubes .
which completes the proof of Theorem 1.6.
Using Theorems 1.4 and 1.5 and [4, Theorem 3.1], Corollary 1.7 is obvious.
The first author was supported partly by NSFC (Grant: 10701010), NSFC (Key program Grant: 10931001), CPDRFSFP (Grant: 200902070) and SRF for ROCS, SEM.
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