Abstract
The oscillatory hyper-Hilbert transform along curves is of the following form: , where , , and . The study on this operator is motivated by the hyper-Hilbert transform and the strongly singular integrals. The bounds for have been given by Chen et al. (2008 and 2010). In this paper, for some , , and , the boundedness of on Sobolev spaces and the boundedness of this operator from to are obtained.
1. Introduction
In the paper, we mainly discuss singular integrals in the following form: where , , and denotes a curve in the n-dimensional spaces.
Operators of this kind originate from the significant Hilbert transform:
In [1], Calderón and Zygmund brought in the rotation method, shifting the study of the homogeneous singular integral operators to that of directional Hilbert transforms: where is odd, and the directional Hilbert transform is
In order to generalize the rotation method, Fabes and Rivière [2] introduced the Hilbert transform along curves:
Afterwards, the research of attracted many scholars, among which Wainger and his fellows contributed to it quite remarkably.
Another development derived from Hilbert transform is hypersingular Hilbert transforms: As such operator has more singularity, is required to have some smoothness. It can be proved that is bounded from to , where .
A natural question is how to balance the more singularity due to , without extra smoothness of . Since Hilbert transform is essentially “oscillatory,” we can bring in an oscillatory factor in . So is the oscillatory hypersingular integral along curves in the following form: where , , and denotes a curve in the n-dimensional spaces.
In this direction, the thesis of Zielinski [3] was pioneering. In the case , , he proved
Later on, Chandarana [4] generalized the result of Zielinski into more common curves, showing the corresponding boundedness on and . However, as the complexity of his method with the dimension increases, he did not reach a general result.
After years’ exploration, the authors in [5] solved the question completely.
Theorem C (see [5]). Let and . Define as If are all positive, (1), as long as and;(2).
Further on, the authors [6] proved that if are mutually different, then
In [5], it is showed that we only need to consider the part of , and could be reduced to . That is the operator which is given at the very start: and so is what we will discuss in the next section. Just under the bases of [5, 6], we probe into the boundedness of on Sobolev spaces.
2. Preliminary and Main Results
As we know, smoothness is a crucial property of functions, and it is common to use high-ordered continuity to describe it. Yet an arbitrary function is not always differentiable. Due to this, Sobolev spaces are introduced to measure the differentiability of some more common functions. These spaces are widely used in both harmonic analysis and PDE.
There are several equivalent definitions of such spaces. Let us start with the classical definition. Firstly, we need to recall the concept of generalized derivatives.
Definition 1. Let and let be multiple index. Define If is a function, then , the derivative of , in the meaning of distribution, is called weak derivative.
Definition 2 (see [7]). Let be a nonnegative integer and . We can define the Sobolev spaces as follows: And the norm is given as Where .
It is easy to see that is a proper subspace of . The indice characterizes the smoothness of the function spaces, and we have the following inclusion relations:
In the above definition, should be an integer. Further on, we can extend the definitions, without assuming to be an integer.
Definition 3 (see [7]). Let be real and . The inhomogeneous Sobolev spaces consisted of all the elements of , which satisfies the following property: And the corresponding norm is given below:
For the definition, there are some observations:(1)if , ;(2)for every , is subset of ;(3)if is a nonnegative integer, the two definitions coincide.
Along with inhomogeneous Sobolev spaces, we can give the definition of the homogeneous Sobolev spaces.
Definition 4 (see [7]). Let be a real number and . We define homogeneous Sobolev spaces as follows: and, for the distributions in , we can define
What should be noticed is that the elements of homogeneous Sobolev spaces may not belong to . Actually, these elements are equivalent classes of the temper distributions. For more details, please refer to chapter 6 of [7].
We also need the following Van der Corput Lemma, which is the most important lemma to estimate the oscillating integrals.
Van der Corput Lemma. Let and be smooth real functions in , and . If for all and one of the two below conditions are satisfied: , is monotone in ; , then
The main results of the paper are as follows.
Theorem 5. For the operator , in the definition of , are all positive. If and , then
Theorem 6. For the operator , in the definition of , are all positive. If and is the biggest integer not more than , then
3. Proof of the Main Results
Proof of Theorem 5. To deal with the singularity on the denominator of the operator , a dyadic decomposition is introduced.
Suppose is a function, supported on . By normalization, it can be assumed that
is true for all . So we can decomposite as follows:
On account of the support of , we only need to consider the case where .
By Minkowski’s inequality, it is easy to obtain the boundedness of on :
Taking Fourier transform, we get the multiple form of :
where
In [5], the authors proved
Thus, by Plancherel’s theorem, we have
So,
To make sure is bounded on (for all ), it is only needed that , which is the same as the requirement of the boundedness on . Roughly speaking, the operators preserve the smoothness of the functions.
To get the boundedness on (), we will use the interpolation between (25) and (29). It can be shown that
As is arbitrary, it suffices to show that ; that is,
So is bounded on .
By duality argument, it is finally proved that if , then is bounded on , where and is arbitrary.
Theorem 5 indicates that the operator can sustain the “smoothness” of functions. If what we care about is not the boundedness from Sobolev spaces to Sobolev spaces, but the boundedness from Sobolev spaces to spaces, then the lifting of the smoothness of can reduce the restriction of , , which would be explained in the next theorem.
Proof of Theorem 6. Here we will follow the notations and calculations in Theorem 5; that is,
Let be the largest integer, not exceeding . For Sobolev spaces , by Plancherel’s theorem, when ,
and, for an element of ,
The case , will be used later.
We will make a more accurate estimation of . Notice that is a function, supported on . By substitution of variables , it is shown that
where we extend the upper limit of the integral into infinity. Considering the support of and , this extension will not make essential difference to the result.
In [5], the authors use Van der Corput Lemma and an elementary statement to prove
After thoughtful investigation of the proof in [5], it is unearthed that the part will only contribute to the control constant in the inequality above, without any effect on the order of the index.
In the subsequent calculation, we will substitute the part with notation . Afterwards, always means a function, supported on . With the process, will represent different functions, which will not do harm to the final result. That is, if is a function supported on , then
using integration by parts:
Notice that indicates different functions in different places; still, they are all functions supported on .
By (38), the absolute value of every integral above can be dominated by . Along with Cauchy’s inequality, we have
Repeating integration by parts, it is suggested, for any , that
So an estimation to the norm of could be made. Recall that represents the largest integer not exceeding :
Further on, to guarantee is bounded from to , it is only needed that
that is, .
When , ; that is, , which is the result in [5].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This research was supported by PSF of Zhejiang province (BSH1302046).