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A restriction estimate for a class of oscillatory integral operators along paraboloid

Journal of Inequalities and Applications20192019:18

https://doi.org/10.1186/s13660-019-1963-4

  • Received: 12 June 2018
  • Accepted: 10 January 2019
  • Published:

Abstract

In this paper we establish a restriction estimate for a class of oscillatory integral operators along a paraboloid,
$$ {\mathbb{P}^{d-1}:=\bigl\{ (x_{1},\ldots ,x_{d}):x_{d}=x_{1}^{2}+ \cdots +x_{d-1}^{2}\bigr\} .} $$
Specifically, we consider the oscillatory integral operators defined by
$$ T_{m,n}f(x)= \int _{\mathbb{R}^{d}}e^{i(x_{1}^{m} \xi _{1}^{n}+\cdots +x _{d}^{m}\xi _{d}^{n})}f(\xi )\,d\xi , $$
(1)
where n, m are integers satisfying \(2\leq d< n\leq 2md\), then
$$ \Vert T_{m,n}f \Vert _{L^{2} (d\sigma , \mathbb{P}^{d-1}\cap B^{d}(0,1) )} \leq C_{m,n,d} \Vert f \Vert _{L^{p}(\mathbb{R}^{d})} $$
holds for \(1< p\leq \frac{4md}{4md-n}\). A necessary condition is also given to ensure this boundedness.

Keywords

  • Restriction estimate
  • Oscillatory integral operator
  • Multidimensional Van der Corput
  • Knapp’s counterexample

MSC

  • 42B20
  • 47G10

1 Introduction

The Fourier restriction conjecture has attracted a lot of attention in the development of modern harmonic analysis. It was raised by Stein in the 1960s and can be stated as follows (in its dual form).

Conjecture 1

Suppose \(S_{0}\) is a compact subset of a smooth \((d-1)\)-dimensional manifold S with nonvanishing Guassian curvature and is the surface measure on S induced by the Lebesgue measure on \({\mathbb{R}}^{n}\), then
$$ \int _{S_{0}} \bigl\vert \hat{f}(\xi ) \bigr\vert \,d \boldsymbol{\sigma } \leq C \Vert f \Vert _{L^{p}( \mathbb{R}^{d})} $$
(2)
holds for \(1< p<\frac{2d}{d+1}\).
A natural idea to attack this problem is to establish a stronger estimate, for example the \(L^{2}\) estimate, of the left hand side of (2). Therefore, the Hölder inequality and the finite measure of \(S_{0}\) yield the corresponding inequality. Combining the \(TT^{*}\) method with the Fourier decay of surface measure , one may get the nontrivial estimate
$$ \Vert \hat{f} \Vert _{L^{2}(S,d\sigma )} \leq C \Vert f \Vert _{L^{p}(\mathbb{R}^{d})}, $$
(3)
for
$$ 1< p\leq \frac{4d}{3d+1}. $$
(4)
It should be noted that our paper replaces the Fourier decay of by an oscillatory integral decay estimate. In [1], Tomas improved the estimate (4) to
$$ 1< p< \frac{2d+2}{d+3} $$
up to the endpoint
$$ p=\frac{2d+2}{d+3}. $$
The endpoint case was due to Stein. The inequality (3) for
$$ 1< p\leq \frac{2n+2}{n+3} $$
(5)
is now known as the Tomas–Stein estimate; it is also sharp in view of Knapp’s counterexample; see [2]. Now, we return to the conjecture 1. Obviously, the Tomas–Stein estimate gives rise to a highly nontrivial improvement on this conjecture. However, the case
$$ \frac{2d+2}{d+3}< p< \frac{2d}{d+1} $$
was still unknown until the breakthrough of Bourgain [3] in 1991. Bourgain first improved the exponent in (5) by connecting the Fourier restriction with the behavior of Kakeya maximal operators. Later, a bilinear approach was developed by Wolff, Tao and other mathematicians, which refers to the interaction between two different surfaces. Since a bilinear restriction implies a linear restriction in some sense, improvements on a bilinear restriction naturally lead to improvements on linear restriction; this work can be found in [48]. In the first decade of the 21st century, the concentration on Fourier restriction turns to multilinear case. Some remarkable results were obtained in [9] by Bennett, Carbery and Tao; they proved the near-optimal multilinear Kakeya and restriction conjectures. But to what extent multilinear restriction can imply linear restriction was not clear until the breakthrough by Bourgain and Guth [10]. Later, Guth applied the polynomial partitioning method to the restriction problem and obtained the range for p,
$$ 1< p< \frac{13}{9}, $$
in the 3-dimensional case; see [11].
The purpose of our paper is to extend the restriction estimate from Fourier transform to oscillatory integral operators. Xu and Yan have established some corresponding results on the \((d-1)\)-dimensional unit sphere in [12]. The definition of the oscillatory integral operators is given by
$$ T_{\lambda }(f) (\xi )= \int _{\mathbb{R}^{d-1}}e^{i\lambda \phi (x, \xi )}\psi (x,\xi )f(x)\,dx $$
where \(\phi (x,\xi )\) is a smooth function and \(\psi (x,\xi )\) is a cut-off function. For this kind of operator, we show the decay estimate
$$ \Vert T_{\lambda } \Vert _{L^{p}\rightarrow L^{q}}\leq C\lambda ^{-\theta }, $$
(6)
for suitable p, q, θ. Since the Fourier restriction operator is a Hörmander-type oscillatory integral operator, if letting \(\lambda \to +\infty \) and some limiting arguments, the decay estimate will imply the restriction conjecture. Details can be found in [2]. The Hörmander-type oscillatory integral operator requires that the mixed Hessian matrix of phase function ϕ should have maximal rank \(d-1\) on the support of ψ. Consider the dual operator of (1)
$$ T^{*}_{m,n}g(\xi )= \int _{\mathbb{R}^{d-1}}e^{i [x_{1}^{m} \xi _{1}^{n}+\cdots +(x_{1}^{2}+\cdots +x_{d-1}^{2})^{m}\xi _{d}^{n} ]} \tilde{\psi }(x)g(x)\,dx_{1} \cdots dx_{d-1}. $$
It is obvious that the mixed Hessian matrix
[ 2 ϕ x j ξ k ] ( x , ξ ) = m n ( x 1 m 1 ξ 1 n 1 0 2 x 1 | x | 2 ( m 1 ) ξ d n 1 0 0 0 x d 1 m 1 ξ d 1 n 1 2 x d 1 | x | 2 ( m 1 ) ξ d n 1 )
has rank zero at \((x,\xi )=(0,0)\) since \(n>2\). Therefore, (1) is not of Hörmander-type. For such non-Hörmander-type oscillatory integral operators we obtain the following restriction estimate.

Theorem 1

Suppose that n, m, d are integers satisfying \(2\leq d< n\leq 2md\). Define the oscillatory integral operators \(T_{m,n}\) as (1). Then, for all \(1< p\leq \frac{4md}{4md-n}\),
$$ \Vert T_{m,n}f \Vert _{L^{2}(d\sigma , \mathbb{P}^{d-1}\cap B^{d}(0,1) {)}}\leq C_{m,n,d} \Vert f \Vert _{L^{p}(\mathbb{R}^{d})}, $$
(7)
where is the surface measure on paraboloid and \(B^{d}(0,1)\) is the unit ball in n-dimensional Euclidean space.

2 Proof of Theorem 1

Proof

\(TT^{*}\) transforms our boundedness estimate to scalar oscillatory integral estimate. For any Schwartz function f, we now compute the \(L^{2}\) norm of \(T_{m,n}(f)\) on the compact subset of \((n-1)\)-dimensional paraboloid. It follows that
$$\begin{aligned} & \bigl\Vert T_{m,n}(f) \bigr\Vert ^{2}_{L^{2}(\mathbb{P}^{d-1}\cap B^{d}(0,1))} \\ &\quad = \bigl\langle T_{m,n}(f),T_{m,n}(f) \bigr\rangle _{\mathbb{P} ^{d-1}\cap B^{d}(0,1)} \\ &\quad = \int _{\mathbb{P}^{d-1}\cap B^{d}(0,1)}T_{m,n}(f) (x) \overline{T_{m,n}(f) (x)}\,d\sigma (x) \\ &\quad = \int _{\mathbb{R}^{d}} \int _{\mathbb{R}^{d}} \int _{\mathbb{P}^{d-1}\cap B^{d}(0,1)}e^{i [x_{1}^{m} (\xi _{1} ^{n}-\eta _{1}^{n})+\cdots +x_{d}^{m}( \xi _{d}^{n}-\eta _{d}^{n}) ]}\,d\sigma (x) f(\xi )\overline{f(\eta )}\,d\xi \,d\eta \\ &\quad \leq \int _{\mathbb{R}^{d}} \int _{\mathbb{R}^{d}} \biggl\vert \int _{\mathbb{P} ^{d-1}\cap B^{d}(0,1)}e^{i [(x_{1}^{m} (\xi _{1}^{n}-\eta _{1}^{n})+ \cdots +x_{d}^{m}( \xi _{d}^{n}-\eta _{d}^{n}) ]}\,d\sigma (x) \biggr\vert \bigl\vert f(\xi )\overline{f(\eta )} \bigr\vert \,d\xi \,d\eta . \end{aligned}$$
Denoting the inner integral in the last inequality by \(K(\xi ,\eta )\), then
$$ \bigl\Vert T_{m,n}(f) \bigr\Vert ^{2}_{L^{2}(\mathbb{P}^{d-1}\cap B^{d}(0,1))} \leq \int _{\mathbb{R}^{d}} \int _{\mathbb{R}^{d}} \bigl\vert K(\xi ,\eta ) \bigr\vert \bigl\vert f( \xi )f(\eta ) \bigr\vert \,d\xi \,d\eta . $$
Set
$$ I(\lambda )= \int _{\mathbb{P}^{d-1}\cap B^{d}(0,1)}e^{i(\lambda _{1}x _{1}^{m} +\cdots +\lambda _{d}x_{d}^{m})}\,d\sigma (x), $$
where \(\lambda =(\lambda _{1},\ldots ,\lambda _{d})\). We desire to establish the decay estimate of \(|I(\lambda )|\) with respect to \(|\lambda |\gg 1\). In fact,
$$\begin{aligned} I(\lambda ) &= \int _{\mathbb{P}^{d-1}\cap B^{d}(0,1)}e^{i(\lambda _{1}x _{1}^{m} +\cdots +\lambda _{d}x_{d}^{m})}\,d\sigma (x) \\ &= \int _{U}e^{i(\lambda _{1}x_{1}^{m} +\cdots +\lambda _{d}(x_{1}^{2}+ \cdots +x_{d-1}^{2})^{m})}\varphi (x_{1},\ldots ,x_{d-1})\,dx_{1}\cdots dx_{d-1}, \end{aligned}$$
where \(U\subset \mathbb{R}^{d-1}\) is the projection of \(\mathbb{P} ^{d-1}\cap B^{d}(0,1)\) on \((d-1)\)-dimensional Euclidean space and
$$ \varphi (x_{1},\ldots ,x_{d-1})=\sqrt {1+4\bigl(x_{1}^{2}+\cdots +x_{d-1} ^{2}\bigr)}. $$
(8)
Note that the phase function of \(I(\lambda )\) is a polynomial of degree 2m with variable coefficients \(\lambda _{1},\ldots ,\lambda _{d-1}\). To establish the decay estimate of \(I(\lambda )\) with respect to \(|\lambda |\), we should precisely describe the relation between decay estimate and these variable coefficients. The following lemma is exactly what we want.

Lemma 1

Let \(P(x)=\sum_{|\alpha |\leq D}c_{\alpha }x^{\alpha }\) be a polynomial of degreeD, φ be a smooth function in the unit ball \(B^{d}(0,1)\), and Ω be any convex subset of the unit ball. Then
$$ \biggl\vert \int _{\varOmega }e^{iP(x)}\varphi (x)\,dx \biggr\vert \leq C_{D,d} \biggl(\sum_{0< \vert \alpha \vert \leq D} \vert c_{\alpha } \vert \biggr)^{-\frac{1}{D}} \sup_{x\in B^{d}(0,1)} \bigl( \bigl\vert \varphi (x) \bigr\vert + \bigl\vert \nabla \varphi (x) \bigr\vert \bigr). $$
This uniform multidimensional Van der Corput lemma can be found in [13] and [14]. By applying this lemma to \(I(\lambda )\), we can see that
$$ \bigl\vert I(\lambda ) \bigr\vert \leq C_{m,d} \bigl( \vert \lambda _{1} \vert +\cdots +c_{m} \vert \lambda _{d} \vert \bigr)^{-\frac{1}{2m}}, $$
where \(c_{m}>1\) is a positive constant only depending on m. A pigeonholing argument yields
$$ \bigl\vert I(\lambda ) \bigr\vert \leq C_{m,d} \vert \lambda \vert ^{-\frac{1}{2m}}. $$
Returning to our original argument, the decay estimate of \(|I(\lambda )|\) naturally gives rise to an upper bound of \(|K(\xi ,\eta )|\),
$$\begin{aligned} \bigl\vert K(\xi ,\eta ) \bigr\vert &\leq C_{m,d} \bigl( \bigl\vert \xi _{1}^{n}-\eta _{1}^{n} \bigr\vert + \cdots + \bigl\vert \xi _{d}^{n}-\eta _{d}^{n} \bigr\vert \bigr)^{- \frac{1}{2m}} \\ &\leq C_{m,d}\big\vert \vert \xi _{1} \vert ^{n}- \vert \eta _{1} \vert ^{n}+\cdots +|\xi \bigl\vert _{d} ^{n}- \vert \eta _{d} \vert ^{n} \bigr\vert ^{-\frac{1}{2m}}. \end{aligned}$$
Define a norm by
$$ \vert \xi \vert _{n}= \bigl( \vert \xi _{1} \vert ^{n}+\cdots + \vert \xi _{d} \vert ^{n} \bigr)^{ \frac{1}{n}}. $$
Thus
$$ \bigl\vert K(\xi ,\eta ) \bigr\vert \leq C_{m,d} \bigl\vert \vert \xi \vert _{n}^{n}- \vert \eta \vert _{n}^{n} \bigr\vert ^{- \frac{1}{2m}}. $$
Consequently,
$$\begin{aligned} & \bigl\Vert T_{m,n}(f) \bigr\Vert ^{2}_{L^{2}(\mathbb{P}^{d-1}\cap B^{d}(0,1))} \\ &\quad \leq C_{m,d} \int _{\mathbb{R}^{d}} \int _{\mathbb{R}^{d}} \bigl\vert \vert \xi |_{n} ^{n}- \vert \eta \vert _{n}^{n} \bigr\vert ^{-\frac{1}{2m}}\big|f(\xi )f(\eta )\big|\,d\xi \,d \eta . \end{aligned}$$
For positive a, b, define a fractional integration operator by
$$ R_{a}^{b}(f) (\eta )= \int _{\mathbb{R}^{d}} \bigl\vert \vert \xi |_{a}^{a}- \vert \eta \vert _{a} ^{a} \bigr\vert ^{-\frac{d}{b}}f(\xi )\,d\xi . $$
The boundedness of \(R_{a}^{b}\) on Lebesgue spaces was essentially established in [15]; it states the following.

Lemma 2

For \(n< a\leq b\),
  1. 1.

    \(R_{a}^{b}\) is of weak type \((1,b/a)\).

     
  2. 2.

    \(R_{a}^{b}\) is bounded from \(L^{p}(\mathbb{R}^{d})\) to \(L^{q}( \mathbb{R}^{d})\), whenever \(\frac{1}{p}=\frac{1}{q}+\frac{b-a}{b}\) and \(1< p<\frac{b}{b-a}\).

     
Obviously,
$$ \bigl\Vert T_{m,n}(f) \bigr\Vert ^{2}_{L^{2}(\mathbb{P}^{d-1}\cap B^{d}(0,1))} \leq C_{m,d} \int _{\mathbb{R}^{d}}R^{2md}_{n}\bigl( \vert f \vert \bigr) (\xi ) \bigl\vert f(\eta ) \bigr\vert \,d \eta . $$
The Hölder inequality implies
$$ \bigl\Vert T_{m,n}(f) \bigr\Vert ^{2}_{L^{2}(\mathbb{P}^{d-1}\cap B^{d}(0,1))} \leq C_{m,d} \bigl\Vert R^{2md}_{n}\bigl( \vert f \vert \bigr) \bigr\Vert _{L^{p'}} \Vert f \Vert _{L^{p}}. $$
We use (2) and set
$$ \frac{1}{p}=\frac{1}{p'}+\frac{2md-n}{2md}. $$
Simple computation gives
$$ p=\frac{4md}{4md-n}. $$
By the density of Schwartz functions in \(L^{p}\), a limiting argument yields the same restriction phenomena on \(L^{p}\). The proof of Theorem 1 is completed.  □

Remark 1

In fact, as Xu and Yan have done in [12], Theorem 1 can be extended to oscillatory integral operators with more general phase functions.

3 The necessary condition to guarantee (7)

In this part, we will construct an example in the spirit of Knapp’s counterexample to show the necessary condition of (7). Then we have following.

Theorem 2

If (7) holds, the exponent p must obey
$$ p\leq \frac{2m(d+1)}{2m(d+1)-(d-1)n}. $$
(9)

Proof

If the inequality
$$ \biggl[ \int _{\mathbb{P}^{d-1}\cap B^{d}(0,1)} \bigl\vert T_{m,n}f(x) \bigr\vert ^{2}\,d\sigma \biggr]^{\frac{1}{2}}\leq C_{m,n,d} \Vert f \Vert _{L^{p}(\mathbb{R}^{d})} $$
holds, then, for sufficiently small \(\delta >0\), we have
$$\begin{aligned} \biggl[ \int _{R_{\delta }\cap \mathbb{P}^{d-1}\cap B^{d}(0,1)} \bigl\vert T_{m,n}f(x) \bigr\vert ^{2}\,d\sigma \biggr]^{\frac{1}{2}}&\leq \biggl[ \int _{\mathbb{P}^{d-1}\cap B^{d}(0,1)} \bigl\vert T_{m,n}f(x) \bigr\vert ^{2}\,d\sigma \biggr]^{\frac{1}{2}} \\ &\leq C_{m,n,d} \Vert f \Vert _{L^{p}(\mathbb{R}^{d})}, \end{aligned}$$
(10)
where
$$ R_{\delta }= \bigl\{ (x_{1},\ldots ,x_{d}): \vert x_{j} \vert \leq c\delta ^{\frac{1}{2}}, x_{d}\leq c \delta , j=1,\ldots ,d-1 \bigr\} , $$
where \(0< c<1\) is a small absolute constant determined later. Let
$$ f(\xi )=\prod_{k=1}^{d}\chi _{I_{k}} (\xi _{k} ) $$
(11)
where \(I_{d}=[-c'\delta ^{-\frac{m}{n}}, c'\delta ^{-\frac{m}{n}}]\) and \(I_{k}=[-c'\delta ^{-\frac{m}{2n}}, c'\delta ^{-\frac{m}{2n}}]\) (\(1 \leq k\leq d-1\)) are intervals and \(0< c'<1\) is a sufficiently small constant, determined later. For a fixed point \(x\in R_{\delta }\cap \mathbb{P}^{d-1}\cap B^{d}(0,1)\), we have
$$\begin{aligned} & \bigl\vert T_{m,n}f(x) \bigr\vert \end{aligned}$$
(12)
$$\begin{aligned} &\quad = \Biggl\vert \int _{\mathbb{R}^{d}}e^{i (x_{1}^{m}\xi _{1}^{n}+\cdots +x_{d}^{m}\xi _{d}^{n} )}\prod_{k=1}^{d} \chi _{I_{k}} (\xi _{k} )\,d\xi _{1}\cdots d\xi _{d} \Biggr\vert \\ &\quad =\prod_{k=1}^{d} \biggl\vert \int _{\mathbb{R}}e^{ix_{k}^{m}\xi _{k}^{n}} \chi _{I_{k}}(\xi _{k})\,d\xi _{k} \biggr\vert \\ &\quad :=\prod_{k=1}^{d} \bigl\vert F_{k}(x_{k}) \bigr\vert . \end{aligned}$$
(13)
Now, we give the lower bound of \(F_{k}(x)\). In fact, for \(1\leq k \leq d-1\), we have
$$\begin{aligned} & \bigl\vert F_{k}(x_{k}) \bigr\vert \\ &\quad = \biggl\vert \int _{-\infty }^{+\infty }e^{ix_{k}^{m}\xi _{k}^{n}}\chi _{I _{k}}( \xi _{k})\,d\xi _{k} \biggr\vert \\ &\quad = \biggl\vert \int _{-c'\delta ^{-\frac{m}{2n}}}^{c'\delta ^{-\frac{m}{2n}}}e ^{ix_{k}^{m}\xi _{k}^{n}}\,d\xi _{k} \biggr\vert \\ &\quad = \biggl\vert \int _{-c'\delta ^{-\frac{m}{2n}}}^{c'\delta ^{-\frac{m}{2n}}} \bigl[\cos \bigl(x_{k}^{m} \xi _{k}^{n}\bigr)+ i\sin \bigl(x_{k}^{m} \xi _{k}^{n}\bigr) \bigr]\,d\xi _{k} \biggr\vert \\ &\quad \geq \biggl\vert \int _{-c'\delta ^{-\frac{m}{2n}}}^{c'\delta ^{- \frac{m}{2n}}}\cos \bigl(x_{k}^{m} \xi _{k}^{n}\bigr)\,d\xi _{k} \biggr\vert . \end{aligned}$$
Since \(x\in {R_{\delta }\cap \mathbb{P}^{d-1}\cap B^{d}(0,1)}\), we have
$$ \bigl\vert F_{k}(x_{k}) \bigr\vert \geq c'\delta ^{-\frac{m}{2n}} $$
if
$$ c^{m}\bigl(c'\bigr)^{n}< 2^{-10}. $$
With the same assumptions on c and \(c'\), we also have \(|F_{d}(x_{d})| \geq c'\delta ^{-\frac{m}{n}}\).
By plugging the constructed function (11) into the inequality (10), we obtain
$$\begin{aligned} C_{m,n,d} \Vert f \Vert _{L^{p}(\mathbb{R}^{d})}&=C_{m,n,d} \bigl(2c'\bigr)^{\frac{d}{p}}\delta ^{-\frac{m(d+1)}{2np}} \\ &\geq \biggl[ \int _{R_{\delta }\cap \mathbb{P}^{d-1}\cap B^{d}(0,1)} \bigl\vert T_{m,n}f(x) \bigr\vert ^{2}\,d\sigma \biggr]^{\frac{1}{2}} \\ &= \Biggl[ \int _{R_{\delta }\cap \mathbb{P}^{d-1}\cap B^{d}(0,1)} \Biggl\vert \prod_{k=1}^{d} \big\vert F_{k}(x_{k}) \big\vert \Biggr\vert ^{2}\,d\sigma \Biggr]^{\frac{1}{2}} \\ &\gtrsim \bigl[\sigma \bigl(R_{\delta }\cap \mathbb{P}^{d-1}\cap B^{d}(0,1) \bigr) \bigr]^{\frac{1}{2}}\delta ^{-\frac{m(d+1)}{2n}}. \end{aligned}$$
From (8), the Jacobi of can be bounded from below and above on the unit ball. Thus
$$ \sigma \bigl(R_{\delta }\cap \mathbb{P}^{d-1}\cap B^{d}(0,1) \bigr)\approx \delta ^{\frac{d-1}{2}}. $$
It means that
$$ 1\gtrsim \delta ^{\frac{m(d+1)}{2np}-\frac{m(d+1)}{2n}+\frac{d-1}{4}} $$
holds for all sufficiently small δ. This requires
$$ \frac{m(d+1)}{2np}-\frac{m(d+1)}{2n}+\frac{d-1}{4}\geq 0, $$
which equals
$$ p\leq \frac{2m(d+1)}{2m(d+1)-(d-1)n}. $$
We finish our proof of necessity. □

Remark 2

It is easy to verify that
$$ \frac{4md}{4md-n}< \frac{2m(d+1)}{2(d+1)m-(d-1)n}. $$
Our result is not sharp since there is a gap between sufficient and necessary conditions.

Declarations

Funding

The second author is supported by National Natural Science Foundation of China (No. 11601427), The Fundamental Research Funds for the Central Universities (No. 3102017zy035).

Authors’ contributions

All authors worked in coordination and contributed equally. All authors carried out the proof, read and approved the final version of the manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
(2)
School of Natural and Applied Sciences, Northwestern Polytechnical University, Xi’an, China
(3)
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, USA

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