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

AbstractIn this paper we establish a restriction estimate for a class of oscillatory integral operators along a paraboloid, 
			Pd−1:={(x1,…,xd):xd=x12+⋯+xd−12}.$$ {\mathbb{P}^{d-1}:=\bigl\{ (x_{1},\ldots ,x_{d}):x_{d}=x_{1}^{2}+ \cdots +x_{d-1}^{2}\bigr\} .} $$ Specifically, we consider the oscillatory integral operators defined by 
			1Tm,nf(x)=∫Rdei(x1mξ1n+⋯+xdmξdn)f(ξ)dξ,$$ T_{m,n}f(x)= \int _{\mathbb{R}^{d}}e^{i(x_{1}^{m} \xi _{1}^{n}+\cdots +x _{d}^{m}\xi _{d}^{n})}f(\xi )\,d\xi , $$ where n, m are integers satisfying 2≤d<n≤2md$2\leq d< n\leq 2md$, then 
			∥Tm,nf∥L2(dσ,Pd−1∩Bd(0,1))≤Cm,n,d∥f∥Lp(Rd)$$ \Vert T_{m,n}f \Vert _{L^{2} (d\sigma , \mathbb{P}^{d-1}\cap B^{d}(0,1) )} \leq C_{m,n,d} \Vert f \Vert _{L^{p}(\mathbb{R}^{d})} $$ holds for 1<p≤4md4md−n$1< p\leq \frac{4md}{4md-n}$. A necessary condition is also given to ensure this boundedness.


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 dσ is the surface measure on S induced by the Lebesgue measure on R n , then holds for 1 < p < 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 dσ , one may get the nontrivial estimate It should be noted that our paper replaces the Fourier decay of dσ by an oscillatory integral decay estimate. In [1], Tomas improved the estimate (4) to The endpoint case was due to Stein. The inequality (3) for 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 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 [4][5][6][7][8].
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, 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 where φ(x, ξ ) is a smooth function and ψ(x, ξ ) is a cut-off function. For this kind of operator, we show the decay estimate for suitable p, q, θ . Since the Fourier restriction operator is a Hörmander-type oscillatory integral operator, if letting λ → +∞ 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) It is obvious that the mixed Hessian matrix has rank zero at (x, ξ ) = (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.
where dσ is the surface measure on paraboloid and B d (0, 1) is the unit ball in n-dimensional Euclidean space.

Proof of Theorem 1
Proof TT * transforms our boundedness estimate to scalar oscillatory integral estimate.
Note that the phase function of I(λ) is a polynomial of degree 2m with variable coefficients λ 1 , . . . , λ d-1 . To establish the decay estimate of I(λ) with respect to |λ|, we should precisely describe the relation between decay estimate and these variable coefficients. The following lemma is exactly what we want. Then This uniform multidimensional Van der Corput lemma can be found in [13] and [14]. By applying this lemma to I(λ), we can see that where c m > 1 is a positive constant only depending on m. A pigeonholing argument yields Returning to our original argument, the decay estimate of |I(λ)| naturally gives rise to an upper bound of |K(ξ , η)|, Define a norm by |ξ | n = |ξ 1 | n + · · · + |ξ d | n 1 n . Thus Consequently, For positive a, b, define a fractional integration operator by The boundedness of R b a on Lebesgue spaces was essentially established in [15]; it states the following. Obviously, The Hölder inequality implies We use (2) and set Simple computation gives p = 4md 4mdn .
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.

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. (7) holds, the exponent p must obey

Theorem 2 If
Proof If the inequality holds, then, for sufficiently small δ > 0, we have where where 0 < c < 1 is a small absolute constant determined later. Let where Now, we give the lower bound of F k (x). In fact, for 1 ≤ k ≤ d -1, we have if c m c n < 2 -10 .
With the same assumptions on c and c , we also have |F d (x d )| ≥ c δ -m n . By plugging the constructed function (11) into the inequality (10), we obtain From (8), the Jacobi of dσ can be bounded from below and above on the unit ball. Thus We finish our proof of necessity.
Our result is not sharp since there is a gap between sufficient and necessary conditions.

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).