- 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 .
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.
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 .
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 .
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 .
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 .
2. Main Lemmas
We need two lemmas given by Tolsa.
Lemma 2.1 (see ).
Lemma 2.2 (see ).
To prove our theorems, we prepare another two key lemmas.
Proof of Lemma 2.3.
The proof of Lemma 2.3 is finished.
Proof of Lemma 2.4.
3. Proof of Theorems
Proof of Theorem 1.4.
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 .
then we finish the proof of Theorem 1.4.
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.
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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