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

Authors: Jun Li; Guilian Gao
Journal: Journal of Function Spaces
 (2014)

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## Abstract

The oscillatory hyper-Hilbert transform along curves is of the following form:Hn,α,βf(x)=∫01‍f(x-Γ(t))eit-βt-1-αdt,  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 for Hn,α,β have been given by Chen et al. (2008 and 2010). In this paper, for some α, β,  and p, the boundedness of Hn,α,β on Sobolev spaces Lsp(Rn) and the boundedness of this operator from Ls2(Rn) to L2(Rn) are obtained.


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## Body

## 1. Introduction

In the paper, we mainly discuss singular integrals in the following form:(1)Hn,α,βf(x)=∫01‍f(x-Γ(t))eit-βt-1-αdt,
where α≥0,  β≥0, and  Γ(t)=(tp1,tp2,…,tpn) denotes a curve in then-dimensional spaces.Operators of this kind originate from the significant Hilbert transform:(2)Hf(x)=p.v.∫R‍f(x-t)tdt.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:
(3)TΩ(f)(x)=p.v.∫Rn‍f(x-y)Ω(y/|y|)|y|ndy=12∫Sn-1‍Ω(y′)Hy′(f)(x)dσ(y′),
where Ω is odd, and the directional Hilbert transform is
(4)Hy′(f)(x)=p.v.∫R‍f(x-ty′)dtt.In order to generalize the rotation method, Fabes and Rivière [2] introduced the Hilbert transform along curves:
(5)HΓf(x)=p.v.∫-∞+∞‍f(x-Γ(t))dtt.Afterwards, the research ofHΓf(x) attracted many scholars, among which Wainger and his fellows contributed to it quite remarkably.Another development derived from Hilbert transform is hypersingular Hilbert transforms:(6)Hαf(x)=p.v.∫-11‍f(x-t)dtt|t|α,0<α<1.
As such operator has more singularity, f is required to have some smoothness. It can be proved that Hα is bounded from Lαp(Rn) to Lp(Rn), where 1<p<∞.A natural question is how to balance the more singularity due to|t|α, without extra smoothness of f. Since Hilbert transform is essentially “oscillatory,” we can bring in an oscillatory factor eit-β in Hα. So is the oscillatory hypersingular integral along curves in the following form:
(7)Hn,α,βf(x)=∫-11‍f(x-Γ(t))ei|t|-βdtt|t|α,
where α≥0,  β≥0, and  Γ(t)=(tp1,tp2,…,tpn) denotes a curve in then-dimensional spaces.In this direction, the thesis of Zielinski [3] was pioneering. In the case n=2,   Γ(t)=(t,t2), he proved
(8)∥H2,α,βf∥L2(R2)⪯∥f∥L2(R2)⟺β≥3α.Later on, Chandarana [4] generalized the result of Zielinski into more common curves, showing the corresponding boundedness on L2(R3) and Lp(R3). 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]).
Letθ=(θ1,θ2,…,θn)∈Rn and  Γθ(t)=(θ1|t|p1,θ2|t|p2,…,θn|t|pn). Define Hn,α,β as
(9)Hn,α,βf(x)=∫-11‍f(x-Γθ(t))ei|t|-β|t|-1-αdt,β>α.
If p1,p2,…,pn,α,β are all positive, (1)
∥Hn,α,βf∥Lp(Rn)⪯∥f∥Lp(Rn),as long asβ>(n+1)α  and2β/(2β-(n+1)α)<p<2β/(n+1)α;(2)
∥Hn,α,βf∥L2(Rn)⪯∥f∥L2(Rn),ifβ=(n+1)α.Further on, the authors [6] proved that if pi are mutually different, then
(10)∥Hn,α,βf∥L2(Rn)⪯∥f∥L2(Rn)⟺β≥(n+1)α.In [5], it is showed that we only need to consider the part of t≥0, and Γθ(t) could be reduced to  Γ(t)=(tp1,tp2,…,tpn). That is the operator which is given at the very start:
(11)Hn,α,βf(x)=∫01‍f(x-Γ(t))eit-βt-1-αdt,
and so is what we will discuss in the next section. Just under the bases of [5, 6], we probe into the boundedness of  Hn,α,β 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.
Letu∈S′ and let α be multiple index. Define
(12)〈∂αu,f〉=(-1)|α|〈u,∂|α|f〉.
If u is a function, then ∂αu, the derivative of u, in the meaning of distribution, is called weak derivative.Definition 2 (see [7]).
Letk be a nonnegative integer and 1<p<∞. We can define the Sobolev spaces Lkp(Rn) as follows:
(13)Lkp(Rn)={f∈Lp(Rn)∣∂αf∈Lp(Rn),∀α,|α|≤k}.
And the norm is given as
(14)∥f∥Lkp(Rn)=∑|α|≤k‍∥∂αf∥Lp(Rn),
Where  ∂(0,…,0)f=f.It is easy to see thatLkp(Rn) is a proper subspace of Lp(Rn). The indice k characterizes the smoothness of the function spaces, and we have the following inclusion relations:
(15)Lp(Rn)⊃L1p(Rn)⊃L2p(Rn)⊃L3p(Rn)⊃⋯.In the above definition,k should be an integer. Further on, we can extend the definitions, without assuming k to be an integer.Definition 3 (see [7]).
Lets be real and 1<p<∞. The inhomogeneous Sobolev spaces Lsp(Rn) consisted of all the elements u of S′, which satisfies the following property:
(16)((1+|ξ|2)s/2u^)ˇ∈Lp(Rn).
And the corresponding norm is given below:
(17)∥u∥Lsp(Rn)=∥((1+|ξ|2)s/2u^)ˇ∥Lp(Rn).For the definition, there are some observations:(1)
ifs=0, Lsp(Rn)=Lp(Rn);(2)
for everys, Lsp(Rn) is subset of Lp(Rn);(3)
ifs=k 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]).
Lets be a real number and 1<p<∞. We define homogeneous Sobolev spaces L˙sp(Rn) as follows:
(18)L˙sp(Rn)={u∣u∈S′(Rn)P,(|ξ|su^)ˇ∈Lp(Rn)},
and, for the distributions in L˙sp(Rn), we can define
(19)∥u∥L˙sp(Rn)=∥(|·|su^)ˇ∥Lp(Rn).What should be noticed is that the elements of homogeneous Sobolev spacesL˙sp(Rn) may not belong to Lp(Rn). 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 ψ and ϕ be smooth real functions in (a,b), and k∈N. If |ψ(k)(t)|≥1 for all t∈(a,b) and one of the two below conditions are satisfied: (1)  k=1, ψ′(t) is monotone in (a,b); (2)k≥2, then
(20)|∫ab‍eiλψ(t)ϕ(t)dt|≤Ckλ-1/k(|ϕ(b)|+∫ab‍|ϕ′(t)|dt).The main results of the paper are as follows.Theorem 5.
For the operatorHn,α,β, in the definition of Γθ(t), p1,p2,…,pn,α,β are all positive. If β>(n+1)α and 2β/(2β-(n+1)α)<p<2β/(n+1)α, then
(21)∥Hn,α,βf∥Lsp(Rn)⪯∥f∥Lsp(Rn).Theorem 6.
For the operatorHn,α,β, in the definition of Γθ(t), p1,p2,…,pn,α,β are all positive. If α<β(N+1/(n+1)) and N is the biggest integer not more than s, then
(22)∥Hn,α,βf∥L2(Rn)⪯∥f∥Ls2(Rn).


## 3. Proof of the Main Results

Proof of Theorem5.
To deal with the singularity on the denominator of the operatorHn,α,β, a dyadic decomposition is introduced.
SupposeΦ is a C∞ function, supported on [1/2,2]. By normalization, it can be assumed that
(23)∑j=-∞+∞‍Φ(2jt)≡1
is true for all t>0. So we can decomposite Hn,α,β as follows:
(24)Hn,α,βf(x)=∑j=0∞‍∫01‍Φ(2jt)f(x-Γ(t))eit-βt-1-αdt=∑j=0∞‍Hjf(x).
On account of the support of Φ, we only need to consider the case where j≥0.
By Minkowski’s inequality, it is easy to obtain the boundedness ofHj on L1(Rn):
(25)∥Hjf∥L1(Rn)⪯2jα∥f∥L1(Rn).
Taking Fourier transform, we get the multiple form ofHjf:
(26)Hjf^(ξ)=mj(ξ)f^(ξ),
where
(27)mj(ξ)=mj(ξ1,ξ2,…,ξn)=∫01‍Φ(2jt)t-1-αei(t-β-∑k=1nξktpk)dt.
In [5], the authors proved
(28)∥mj∥L∞(Rn)⪯2j(α-β/(n+1)).
Thus, by Plancherel’s theorem, we have(29)∥Hjf∥Ls2(Rn)=∥((1+|ξ|2)s/2Hjf^(ξ))ˇ∥L2(Rn)=∥(1+|ξ|2)s/2Hjf^(ξ)∥L2(Rn)=∥(1+|ξ|2)s/2f^(ξ)mj(ξ)∥L2(Rn)⪯2j(α-β/(n+1))∥(1+|ξ|2)s/2f^∥L2(Rn)=2j(α-β/(n+1))∥f∥Ls2(Rn).
So,(30)∥Hn,α,βf∥Ls2(Rn)⪯∑j=0∞‍2j(α-β/(n+1))∥f∥Ls2(Rn).
To make sure Hn,α,β is bounded on Ls2(Rn) (for all s), it is only needed that β>(n+1)α, which is the same as the requirement of the boundedness on L2(Rn). Roughly speaking, the operators preserve the smoothness of the functions.
To get the boundedness onLsp(Rn)(p>1), we will use the interpolation between (25) and (29). It can be shown that
(31)∥Hj∥L2s(1-1/p)p(Rn)→L2s(1-1/p)p⪯2j(α-2β(1-1/p)/(n+1)).
As s is arbitrary, it suffices to show that α-2β(1-1/p)/(n+1)<0; that is,
(32)2β2β-(n+1)α<p≤2.
So Hn,α,β is bounded on Lsp(Rn).
By duality argument, it is finally proved that ifβ>(n+1)α, then Hn,α,β is bounded on Lsp(Rn), where  2β/(2β-(n+1)α)<p<2β/(n+1)α and s is arbitrary.Theorem5 indicates that the operator Hn,α,β 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 Lp spaces, then the lifting of the smoothness of f can reduce the restriction of α, β, which would be explained in the next theorem.Proof of Theorem6.
Here we will follow the notations and calculations in Theorem5; that is,
(33)Hn,α,βf(x)=∑j=0∞‍Hjf(x);Hjf^(ξ)=mj(ξ)f^(ξ);mj(ξ)=∫01‍Φ(2jt)t-1-αei(t-β-∑k=1nξktpk)dt.
LetN be the largest integer, not exceeding s. For Sobolev spaces Ls2(Rn), by Plancherel’s theorem, when s1>s2,
(34)Ls12(Rn)⊂Ls22(Rn),
and, for an element f of Ls12,
(35)∥f∥Ls22(Rn)<∥f∥Ls12(Rn).
The case s1=s, s2=N will be used later.
We will make a more accurate estimation ofmj. Notice that Φ is a C∞ function, supported on [1/2,2]. By substitution of variables 2jt→t, it is shown that
(36)mj(ξ)=2jα∫0∞‍Φ(t)t1+αei(2jβt-β-∑k=1nξk2-jpktpk)dt=2jαmα,β,
where we extend the upper limit of the integral into infinity. Considering the support of Φ and j≥0, 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
(37)∥mα,β∥L∞(Rn)⪯2-jβ/(n+1).
After thoughtful investigation of the proof in [5], it is unearthed that the part Φ(t)/t1+α 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 partΦ(t)/t1+α with notation Ψ(t). Afterwards, Ψ(t) always means a C∞ function, supported on [1/2,2]. With the process, Ψ(t) will represent different functions, which will not do harm to the final result. That is, if Ψ(t) is a C∞ function supported on [1/2,2], then
(38)∥∫0∞‍Ψ(t)ei(2jβt-β-∑k=1nξk2-jpktpk)dt∥L∞(Rn)⪯2-jβ/n+1,(39)mα,β(ξ)=i2jβ∫0∞‍e-i(∑k=1nξk2-jpktpk)Ψ(t)dei2jβt-β;
using integration by parts:
(40)mα,β(ξ)=-i2jβ∫0∞‍ei(2jβt-β-∑k=1nξk2-jpktpk)Ψ(t)dt+∑k=1n‍ξkpk2-jpk2jβ∫0∞‍ei(2jβt-β-∑k=1nξk2-jpktpk)Ψ(t)dt.
Notice that Ψ indicates different functions in different places; still, they are all C∞ functions supported on [1/2,2].
By (38), the absolute value of every integral above can be dominated by 2-jβ/(n+1). Along with Cauchy’s inequality, we have
(41)|mα,β(ξ)|⪯2-jβ/(n+1)2-jβ(1+|ξ|2)1/2.
Repeating integration by parts, it is suggested, for any M, that
(42)|mα,β(ξ)|⪯2-jβ/(n+1)2-jMβ(1+|ξ|2)M/2.
So an estimation to theL2 norm of Hjf could be made. Recall that N represents the largest integer not exceeding s:
(43)∥Hjf∥L2(Rn)=∥Hjf^∥L2(Rn)=∥mjf^∥L2(Rn)⪯2jα-jNβ-jβ/(n+1)∥(1+|ξ|2)N/2f^∥L2(Rn)=2jα-jNβ-jβ/(n+1)∥f∥LN2(Rn)≤2jα-jNβ-jβ/(n+1)∥f∥Ls2(Rn).
Further on, to guarantee Hn,α,β is bounded from Ls2(Rn) to L2(Rn), it is only needed that
(44)α-Nβ-βn+1<0;
that is, α<β(N+1/(n+1)).
Whens=0, N=0; that is, β>(n+1)α, which is the result in [5].



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