Boundedness of Oscillatory Hyper-Hilbert Transform along Curves on Sobolev Spaces

t dt, where α ≥ 0, β ≥ 0, and Γ(t) = (tp1 , tp2 , . . . , tpn ). The study on this operator is motivated by the hyper-Hilbert transform and the strongly singular integrals. The Lp bounds forH n,α,β have been given by Chen et al. (2008 and 2010). In this paper, for some α, β, and p, the boundedness ofH n,α,β on Sobolev spaces Lp s (Rn) and the boundedness of this operator from L2 s (Rn) to L2(Rn) are obtained.

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 Ω 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  Γ () 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 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  ≥ 0,  ≥ 0, and Γ() = (  1 ,   2 , . . .,    ) denotes a curve in the n-dimensional spaces.In this direction, the thesis of Zielinski [3] was pioneering.In the case  = 2, Γ() = (, Later on, Chandarana [4] generalized the result of Zielinski into more common curves, showing the corresponding boundedness on  2 (R 3 ) and   (R 3 ).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.

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  ∈ S  and let  be multiple index.Define If  is a function, then   , the derivative of , in the meaning of distribution, is called weak derivative.
It is easy to see that    (R  ) is a proper subspace of   (R  ).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 1 <  < ∞.The inhomogeneous Sobolev spaces    (R  ) consisted of all the elements  of S  , which satisfies the following property: And the corresponding norm is given below: For the definition, there are some observations: (1) if  = 0,    (R  ) =   (R  ); (2) for every ,    (R  ) is subset of   (R  ); (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 1 <  < ∞.We define homogeneous Sobolev spaces L   (R  ) as follows: and, for the distributions in L   (R  ), we can define What should be noticed is that the elements of homogeneous Sobolev spaces L   (R  ) may not belong to   (R  ).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  ∈ N. If | () ()| ≥ 1 for all  ∈ (, ) and one of the two below conditions are satisfied: ( The main results of the paper are as follows. Theorem

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 Φ is a  ∞ function, supported on [1/2, 2].By normalization, it can be assumed that is true for all  > 0. So we can decomposite  ,, as follows: On account of the support of Φ, we only need to consider the case where  ≥ 0. By Minkowski's inequality, it is easy to obtain the boundedness of   on  1 (R  ): Taking Fourier transform, we get the multiple form of   : where In [5], the authors proved So, To make sure  ,, is bounded on  2  (R  ) (for all ), it is only needed that  > ( + 1), which is the same as the requirement of the boundedness on  2 (R  ).Roughly speaking, the operators preserve the smoothness of the functions.
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.