Estimates for Marcinkiewicz integral operators and extrapolation
© Ali; licensee Springer. 2014
Received: 2 October 2013
Accepted: 27 June 2014
Published: 23 July 2014
In this article, we establish estimates for parametric Marcinkiewicz integral operators with rough kernels. These estimates and extrapolation arguments improve and extend some known results on Marcinkiewicz integrals.
MSC:40B20, 40B15, 40B25.
Throughout this article, let , be the unit sphere in which is equipped with the normalized Lebesgue surface measure . Also, we let for and denote the exponent conjugate to p; that is .
If , we denote by . The operators have their roots in the classical Marcinkiewicz integral operators which were introduced by Stein in  in which he studied the Boundedness of when (). More precisely, he proved that is of type for and of weak type .
The Marcinkiewicz integral operators play an important role in many fields in mathematics such as Poisson integrals, singular integrals and singular Radon transforms. They have received much attention from many authors (we refer the readers to [1–6], as well as , and the references therein).
Define , where the infimum is taken over the whole q-block decomposition of Ω, then is a norm on the space , and the space is a Banach space.
The study of parametric Marcinkiewicz integral operator was initiated by Hörmander in  in which he showed that is bounded on for when and with . However, the authors of  proved that is bounded on for when and with . This result was improved in  in which the authors established that is bounded on if and , where is the collection of all measurable functions satisfying .
On the other hand, Al-Qassem and Al-Salman in  found that if with , then is bounded on for . Furthermore, they proved that is sharp on .
Recently, it was proved in  that if for some and for some , then is bounded on for any p satisfying , where ϕ is , a convex and increasing function with . Very recently, Al-Qassem and Pan established in  that if for some and for some , then is bounded on for any p satisfying , where is a polynomial mapping and each is a real valued polynomial on .
Our main concern in this work is in dealing with Marcinkiewicz operators under very weak conditions on the singular kernels. In fact, we establish certain estimates for , and then we apply an extrapolation argument to obtain and improve some results on Marcinkiewicz integrals. Our approach in this work provides an alternative way in dealing with such kind of operators. Our main result is described in the following theorem.
Throughout this paper, the letter C denotes a bounded positive constant that may vary at each occurrence but independent of the essential variables.
2 Definitions and lemmas
In this section, we present and establish some lemmas used in the sequel. Let us start this section by introducing the following.
where is defined in the same way as , but with replacing Ω, h by , , respectively. We write for the total variation of σ.
In order to prove Theorem 1.1, it suffices to prove the following lemmas.
hold for all . The constant C is independent of k, ξ and ϕ.
The proof is complete. □
Following a similar argument to the one used in [, Lemma 2.7], we achieve the following lemma.
for any with .
where is the Hardy-Littlewood maximal function of f in the direction of y. By this, and since is bounded in with bounded independent of y, we obtain our desired result. □
By using Hölder’s inequality plus Lemma 2.3, we finish the proof. □
holds for arbitrary functions on .
where ς is a function in with . Thus, by (2.6) and (2.7), our estimate holds for ; and therefore the proof of Lemma 2.5 is complete. □
In the same manner, we prove the following lemma.
holds for arbitrary functions on .
3 Proof of the main result
Consequently, by (3.2) and (3.5), we get our result for the case for some .
The proof of our theorem for the case for some is obtained by following the above argument, except that we need to invoke Lemma 2.6 instead of Lemma 2.5. Therefore, the proof of Theorem 1.1 is complete.
4 Further results
The power of our theorem is in applying the extrapolation method on it (see ). In particular, Theorem 1.1 and extrapolation lead to the following theorem.
- (i)If for some , then
- (ii)If , then
for , where
The author would like to thank Dr. Hussain Al-Qassem for his suggestions and comments on this note.
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