Linear Restriction Estimates for the Wave Equation with Inverse Square Potential

LINEAR RESTRICTION ESTIMATES FOR THE WAVE EQUATION WITH INVERSE SQUARE POTENTIAL

JUNYONG ZHANG AND JIQIANG ZHENG

Abstract. In this paper, we study a modiﬁed linear restriction estimates associated with the wave equation with inverse square potential. In particular, we show the classical linear restriction estimates hold in their almost sharp range when the initial data is radial. Key Words: Linear restriction estimate, Spherical harmonics, Bessel function, Inverse square potential AMS Classiﬁcation: 42B37, 35Q40, 47J35.

Contents 1. Introduction and Statement of Main Result 1 2. Preliminary 4 2.1. Spherical harmonic expansions and the Bessel functions 4 3. Proof of the Main Theorem 6 3.1. Hankel transform and the solution 7 3.2. Estimates of Hankel transforms 8 3.3. Conclude the Proof of Theorem 1.2 13 4. Appendix: Vector valued inequality 15 References 17 References 17

1. Introduction and Statement of Main Result In this paper, we study a modified restriction estimate associated with the wave equation perturbed by an inverse square potential. More precisely, we consider the following wave equation with a singular potential: { ∂2u − ∆u + a u = 0 (t, x) ∈ R × Rn, a ∈ R, t |x|2 (1.1) u(t, x)|t=0 = 0, ∂tu(t, x)|t=0 = f(x). − a The scale-covariance elliptic operator Pa := ∆ + |x|2 appearing in (1.1) plays a key role in many problems of physics and geometry. The heat and Schrödingerflows for the elliptic operator Pa have been studied in the theory of combustion (see [8]), and in quantum mechanics (see [6]). The wave equation (1.1) arises in the study of the wave propagation on conic manifolds [5]. There has been a lot of interest in developing Strichartz estimates both for the Schrödingerand wave equations with the inverse square potential, we refer the reader to N. Burq etc.[2, 3, 12, 13] and the authors [11]. However, as far as we know, there is little result about the 1 2 JUNYONG ZHANG AND JIQIANG ZHENG restriction estimates associated with the operator Pa, which arises in the study of eigenfunctions estimate of Pa. Here we aim to address some restriction issues in the special settings associated with the operator Pa. In the case of the linear wave equation with no potential, i.e. a = 0, we can solve the equation by the Fourier transform formula √ ∫ sin(t −∆) 1 ( ) dξ u(t, x) = √ f = e2πix·ξ e2πit|ξ| − e−2πit|ξ| fˆ(ξ) , (1.2) −∆ 2i Rn |ξ| where the Fourier transform is defined by ∫ fˆ(ξ) = e−2πix·ξf(x)dx. Rn It is well known that the spacetime norm estimate of u(t, x) is connected with the following linear adjoint cone restriction estimate ∨ ∥(F dσ) ∥ q R×Rn ≤ C ∥F ∥ p , (1.3) Lt,x( ) p,q,n,S L (S,dσ) where F is a Schwartz function and the inverse space-time Fourier transform of the measure F dσ defined by ∫ ∫ ∨ 2πi(x·ξ+tτ) | | 2πi(x·ξ+t|ξ|) dξ (F dσ) (t, x) = F (τ, ξ)e dσ(ξ) = F ( ξ , ξ)e | |. S Rn ξ Here the set S is an non-empty smooth compact subset of the cone, {(τ, ξ) ∈ R × Rn : τ = |ξ|} with n ≥ 2, dξ the canonical measure dσ is the pull-back of the measure |ξ| under the projection map (τ, ξ) 7→ ξ. By the decay of (dσ)∨ and the Knapp counterexample, the two 2n n+1 ≤ n−1 necessary conditions for (1.2) are q > n−1 and q p′ , see [16, 22]. The corresponding linear adjoint restriction conjecture for cone asserts that Conjecture 1.1. The inequality (1.3) holds with constants depending on n, p, q 2n n+1 ≤ n−1 and S if and only if q > n−1 and q p′ . Even though there is a large amount of literature focused on this problem, this problem remains open for n ≥ 4. For the progress on the conjecture, we refer the readers to [1, 19, 22, 24, 25, 26, 28]. Shao [15] provided two simple and novelty arguments to prove the Conjecture 1.1 holds true for the spatial rotation invariant functions which are supported on the cone. Motivated by [15], the authors [9] utilized the spherical harmonics expansion and analyzed the asymptotic behavior of the Bessel function to generalize Shao’s [15] result for cone cases by establishing ∨ ∥ ∥ q q ≤ ∥ ∥ (F dσ) L (R;L L2 (Sn−1)) Cp,q,n,S F Lp(S,dσ), (1.4) t rn−1dr θ 2n n+1 ≤ n−1 where q > n−1 and q p′ . In the situation that a ≠ 0, the spacetime Fourier transform becomes no longer so useful, and so one may establish an approximate parametrix for the fundamental solution and then obtain a good control on it. However, we have to go around this to resort to the spherical harmonics expansion and Hankel transforms due to some technique reasons about the singular potential. With the motivation of viewing the potential term as a perturbation on angular in [2, 12, 11], we solve the equation by a solution in form of harmonic expansion series. Even though the harmonic expansion expression leads to some loss of angular regularity in the LINEAR RESTRICTION ESTIMATES FOR WAVE WITH INVERSE SQUARE POTENTIAL 3 restriction estimates, it allows us to show the restriction estimates when q is close to 2n/(n − 1). A key ingredient in this process is to explore the oscillatory properties of the Bessel function and eit|ξ| to overcome the difficulties, which comes from the low decay of the Bessel function Jν (r) when ν ∼ r. Finally, by using the property of the hypergeometric function showed in [12], we prove a vector valued inequality for the Hankel transform to obtain the results. Our main theorem is stated as: Theorem 1.2. Let n ≥ 2 and let u be the solution of the equation (1.1) with ′ − − 2 p (n+1) 2n a > (n 2) /4. Suppose q = n−1 > n−1 and p > 1, then there exists a constant only depending on p, q, n, a such that 1). if f is a radial Schwartz function, then

− 1 ∥u(t, x)∥ q R×Rn ≤ C ∥|ξ| p fˆ∥ p Rn ; (1.5) Lt,x( ) p,q,n,a L ( ) 2). if f is a Schwartz function (may not be radial) and p ≥ 2, then

− 1 ∥ ∥ q q ≤ ∥| | p ˆ∥ u(t, x) L (R;L L2 (Sn−1)) Cp,q,n,a ξ f Lp(Rn). (1.6) t rn−1dr θ Remarks: i). This result extends the classical restriction estimate associated with the − a Laplace operator to the restriction estimate associated with ∆ + |x|2 . We obtain much more estimates than the Strichartz estimates [2, 12] which focus on p = 2. This result also can be viewed as a extension of the result about the operator − a 2 ∞ n−1 ∆ + r2 acting on L ((0, ); r dr) in [4]. ii). This result means that we almost show the classical linear restriction estimates holds for radial functions in the conjecture range, even though for some negative a. iii). When a = 0, we recover the cone restriction result in Shao [14]. When supp fˆ is compact, we can extend the result for q ≥ p′(n + 1)/(n − 1), which is of the same range as in the cone restriction conjecture. iv). (1.5) gives a Strichartz-type estimate

− 1 ∥ ∥ − ≤ ∥|∇| 2 ∥ u(t, x) 2(n+1)/(n 1) R×Rn C f L2(Rn) Lt,x ( ) for the radial solution. The method used here can generalize the result for the radial initial data to a linear ﬁnite combination of products of the Hankel transform of radial functions and spherical harmonics. We hopefully remove the whole radial assumption in (1.5) in future, at least for q ≥ 2(n + 3)/(n + 1). v). If fˆ ⊂ {ξ : N ≤ |ξ| ≤ 2N} and f is radial, the method here can be employed to obtain the Strichartz estimate n−2 − n+1 ∥u(t, x)∥ q R×Rn ≤ CN 2 q ∥f∥ 2 Rn for q > 2n/(n − 1). Lt,x( ) L ( ) We remark that the pair (q, q) is allowed to be out of the admissible requirement in [12], it is however consistent with the admissible range due to the authors [11]. vi). We heavily resort to the harmonic expansion formula to give the expression of the solution due to the potential, which causes the restriction p ≥ 2. The resolvent expression may be more ﬂexible to remove this restriction. 4 JUNYONG ZHANG AND JIQIANG ZHENG

Now we introduce some notations. We use A ≲ B to denote the statement that A ≤ CB for some large constant C which may vary from line to line and depend on various parameters, and similarly use A ≪ B to denote the statement A ≤ C−1B. We employ A ∼ B to denote the statement that A ≲ B ≲ A. If the constant C depends on a special parameter other than the above, we shall denote it explicitly by subscripts. For instance, Cϵ should be understood as a positive constant not only depending on p, q, n and S, but also on ϵ. Throughout this paper, pairs of ′ 1 1 ≤ ≤ ∞ conjugate indices are written as p, p , where p + p′ = 1 with 1 p . This paper is organized as follows: In the section 2, we present some simple facts about the Hankel transforms, and the Bessel functions and also recall the Van der Corput lemma. Section 3 is devoted to the proof of Theorem 1.2 via the spherical harmonics expansion and analyzing the asymptotic behavior of the Bessel function. In the appendix section, we ﬁnally show a vector valued inequality used in Section 3. Acknowledgments: The authors were partly supported by the Fundamental Research Foundation of B. I. T.(20111742015).

2. Preliminary In this section, we provide some simple and standard facts about the Hankel transform and the Bessel functions. We conclude this section by recalling the Van der Corput lemma.

2.1. Spherical harmonic expansions and the Bessel functions. We begin with recalling the expansion formula with respect to the spherical harmonics. For more details, we refer to Stein-Weiss [18]. For the sake of convenience, let ξ = ρω and x = rθ with ω, θ ∈ Sn−1. (2.1) For any g ∈ L2(Rn), the expansion formula with respect to the spherical harmonics yields ∑∞ d∑(k) g(x) = ak,ℓ(r)Yk,ℓ(θ) k=0 ℓ=1 where

{Yk,1,...,Yk,d(k)} is the orthogonal basis of the spherical harmonic space of degree k on Sn−1, called Hk, with the dimension 2k + n − 2 d(0) = 1 and d(k) = Ck−1 ≃ ⟨k⟩n−2. k n+k−3 We remark that for n = 2, the dimension of Hk is a constant, which is independent of k. Obviously, we have the orthogonal decomposition ⊕∞ L2(Sn−1) = Hk. k=0 By orthogonality, it gives

∥g(x)∥ 2 = ∥ak,ℓ(r)∥ 2 . (2.2) Lθ ℓk,ℓ LINEAR RESTRICTION ESTIMATES FOR WAVE WITH INVERSE SQUARE POTENTIAL 5

For our purpose, we need to recall the Fourier transform of ak,ℓ(r)Yk,ℓ(θ). Then Theorem 3.10 in [18] asserts the Hankel transforms formula

∑∞ d∑(k) ∫ ∞ k − n−2 n gˆ(ρω) = 2πi Yk,ℓ(ω)ρ 2 J n−2 (2πrρ)ak,ℓ(r)r 2 dr. (2.3) k+ 2 k=0 ℓ=1 0

Here the Bessel function Jk(r) of order k is defined by ∫ k 1 (r/2) isr − 2 (2k−1)/2 −1 Jk(r) = 1 e (1 s ) ds with k > and r > 0. Γ(k + 2 )Γ(1/2) −1 2 A simple computation gives the rough estimates Crk ( 1 ) |J (r)| ≤ 1 + , (2.4) k k 1 2 Γ(k + 2 )Γ(1/2) k + 1/2 where C is an absolute constant and these estimates will be mainly used when r ≲ 1. Another well known asymptotic expansion about the Bessel function is √ 2 kπ π J (r) = r−1/2 cos(r − − ) + O (r−3/2), as r → ∞ k π 2 4 k but with a constant depending on k (see [18]). As pointed out in [17], if one seeks − 1 a uniform bound for large r and k, then the best one can do is |Jk(r)| ≤ Cr 3 . To investigate the behavior of asymptotic on k and r, we recall Schläfli’sintegral representation [27] of the Bessel function: for r ∈ R+ and k > − 1 ∫ ∫ 2 π ∞ 1 ir sin θ−ikθ sin(kπ) −(r sinh s+ks) Jk(r) = e dθ − e ds 2π −π π 0 (2.5)

:= J˜k(r) − Ek(r). + We remark that Ek(r) = 0 when k ∈ Z . A simple computation gives that for r > 0 ∫ ∞ sin(kπ) −(r sinh s+ks) −1 |Ek(r)| = e ds ≤ C(r + k) . (2.6) π 0 Next, we recall the properties of Bessel function Jk(r) in [17, 21], we refer the readers to [10] for the detail proof.

Lemma 2.1 (Asymptotics of the Bessel function). Assume k ≫ 1. Let Jk(r) be the Bessel function of order k deﬁned as above. Then there exist a large constant C and small constant c independent of k and r such that: • ≤ k when r 2 −c(k+r) |Jk(r)| ≤ Ce ; (2.7) • k ≤ ≤ when 2 r 2k − 1 − 1 − 1 |Jk(r)| ≤ Ck 3 (k 3 |r − k| + 1) 4 ; (2.8) • when r ≥ 2k ∑ − 1 ir Jk(r) = r 2 a(r)e + E(r), (2.9)

−1 where |a(r)| ≤ C and |E(r)| ≤ Cr . 6 JUNYONG ZHANG AND JIQIANG ZHENG

For our purpose, we define n − 2 √ µ(k) = + k, and ν(k) = µ2(k) + a with a > −(n − 2)2/4. (2.10) 2 For the sake of simplicity, we sometimes briefly write ν as ν(k). Let f be Schwartz function defined on Rn, we define the Hankel transform of order ν: ∫ ∞ − n−2 n−1 (Hν f)(ξ) = (rρ) 2 Jν (rρ)f(rω)r dr, (2.11) 0 where ρ = |ξ|, ω = ξ/|ξ| and Jν is the Bessel function of order ν. Specifically, if the function f is radial, then ∫ ∞ − n−2 n−1 (Hν f)(ρ) = (rρ) 2 Jν (rρ)f(r)r dr. (2.12) 0

∑∞ d∑(k) We remark that if f(x) = ak,ℓ(r)Yk,ℓ(θ), it follows from (2.3) that k=0 ℓ=1

∑∞ d∑(k) ( ) ˆ k f(ξ) = 2πi Yk,ℓ(ω) Hµ(k)ak,ℓ (ρ). (2.13) k=0 ℓ=1 The following properties of the Hankel transform are obtained in [2, 12]: [ Lemma 2.2. Let H be deﬁned above and A := −∂2 − n−1 ∂ + ν2(k) − ( ) ] ν ν(k) r r r n−2 2 −2 2 r . Then H H−1 (i) ν = ν , H H H∗ (ii) ν is self-adjoint, i.e. ν = ν , 2 (iii) Hν is an L isometry, i.e. ∥Hν ϕ∥ 2 = ∥ϕ∥ 2 , Lξ Lx 2 2 (iv) Hν (Aν ϕ)(ξ) = |ξ| (Hν ϕ)(ξ), for ϕ ∈ L . We conclude this section by recalling the Van der Corput’s lemma [17].

Lemma 2.3. Let ϕ be a smooth real valued function deﬁned on an interval [a, b], and |ϕ(k)(x)| ≥ 1 for all x ∈ [a, b]. Then ∫ b iλϕ(x) − 1 e dx ≤ ckλ k (2.14) a holds when: • k ≥ 2, or • k = 1 and ϕ′(x) is monotonic. The bound ck is independent of ϕ and λ.

3. Proof of the Main Theorem In this section, we will use the asymptotic properties of the Bessel function and the stationary phase argument to establish two estimates of Hankel transform. A key ingredient is to eﬀectively explore the oscillatory property of the Bessel function and eit|ξ| to obtain more decay. LINEAR RESTRICTION ESTIMATES FOR WAVE WITH INVERSE SQUARE POTENTIAL 7

3.1. Hankel transform and the solution. Let us consider the equation (1.1) in polar coordinates. Write v(t, r, θ) = u(t, rθ) and g(r, θ) = f(rθ) = f(x). Then v(t, r, θ) satisfies that { ∂ v − ∂ v − n−1 ∂ v − 1 ∆ v + a v = 0 tt rr r r r2 θ r2 (3.1) v(0, r, θ) = 0, ∂tv(0, r, θ) = g(r, θ). We use the spherical harmonic expansion to write ∑∞ d∑(k) g(r, θ) = ak,ℓ(r)Yk,ℓ(θ). (3.2) k=0 ℓ=1 Using separation of variables, we can write v as a superposition ∑∞ d∑(k) v(t, r, θ) = vk,ℓ(t, r)Yk,ℓ(θ), (3.3) k=0 ℓ=1 where v satisfies the following equation k,ℓ { − ∂ v − ∂ v − n−1 ∂ v + k(k+n 2)+a v = 0 tt k,ℓ rr k,ℓ r r k,ℓ r2 k,ℓ (3.4) vk,ℓ(0, r) = 0, ∂tvk,ℓ(0, r) = ak,ℓ(r) for each k, ℓ ∈ N , 1 ≤ ℓ ≤ d(k). Define ( ) − 2 n − 1 ν2(k) − n 2 A : = −∂2 − ∂ + 2 . (3.5) ν(k) r r r r2 Then it reduces to consider{ ∂ v + A v = 0 tt k,ℓ ν(k) k,ℓ (3.6) vk,ℓ(0, r) = 0, ∂tvk,ℓ(0, r) = ak,ℓ(r). Applying the Hankel transform to (3.6), we have by Lemma 2.2 { ∂ v˜ + ρ2v˜ = 0 tt k,ℓ k,ℓ (3.7) v˜k,ℓ(0, ξ) = 0, ∂tv˜k,ℓ(0, ξ) = bk,ℓ(ρ), where v˜k,ℓ(t, ρ) = (Hν vk,ℓ)(t, ρ), bk,ℓ(ρ) = (Hν ak,ℓ)(ρ). (3.8) Solving this ODE and using the Hankel transform, we obtain ∫ ∞ − n−2 n−2 vk,ℓ(t, r) = (rρ) 2 Jν(k)(rρ)˜vk,ℓ(t, ρ)ρ dρ 0 ∫ ∞ ( ) 1 − n−2 itρ −itρ n−2 = (rρ) 2 Jν(k)(rρ) e − e bk,ℓ(ρ)ρ dρ. 2i 0 Therefore we get u(x, t) = v(t, r, θ) ∑∞ d∑(k) ∫ ∞ − n−2 n−2 = Yk,ℓ(θ) (rρ) 2 Jν(k)(rρ) sin(tρ)bk,ℓ(ρ)ρ dρ k=0 ℓ=1 0 (3.9) ∑∞ d∑(k) [ ] −1 = Yk,ℓ(θ)Hν(k) ρ sin(tρ)bk,ℓ(ρ) (r). k=0 ℓ=1 8 JUNYONG ZHANG AND JIQIANG ZHENG

3.2. Estimates of Hankel transforms. In this subsection, we prove some estimates for Hankel transforms of the order ν(k). These are key estimates for proving the main theorem. Proposition 3.1. Let R ≫ 1, φ be a smooth function supported in the interval I := [1, 2]. Then

∞ ∫ ( ∑ d∑(k) ∞ ) 1 −itρ 2 2 e Jν(k)(rρ)bk,ℓ(ρ)φ(ρ)dρ 2 R 2 0 Lt ( ;Lr ([R,2R])) k=0 ℓ=1 (3.10) ( ∑∞ d∑(k) ) 1 2 2 ≤ C |bk,ℓ(ρ)| , L2 (I) k=0 ℓ=1 ρ where C is a constant independent of R. Proof. Using the Plancherel theorem in t, we have

∞ ( ∑ d∑(k) ) 1 2 ≲ 2 L.H.S of (3.10) Jν(k)(rρ)bk,ℓ(ρ)φ(ρ) 2 . Lρ L2([R,2R]) k=0 ℓ=1 r Along with this, it is easy to verify (3.10), if we could prove ∫ 2R 2 |Jk(r)| dr ≤ C (3.11) R where R ≫ 1 and the constant C is independent of k and R. To prove (3.11), we write ∫ ∫ ∫ ∫ 2R 2 2 2 2 |Jk(r)| dr = |Jk(r)| dr + |Jk(r)| dr + |Jk(r)| dr R I1 I2 I3 ∩ k ∩ k ∩ ∞ where I1 = [R, 2R] [0, 2 ],I2 = [R, 2R] [ 2 , 2k] and I3 = [R, 2R] [2k, ]. By using (2.7) and (2.9) in Lemma 2.1, we have ∫ ∫ 2 −cr −cR |Jk(r)| dr ≤ C e rdr ≤ Ce , (3.12) I1 I1 and ∫ 2 |Jk(r)| dr ≤ C. (3.13) I3 On the other hand, ∫ ∫ 2 − 2 − 1 − 1 |Jk(r)| dr ≤ C k 3 (1 + k 3 |r − k|) 2 dr ≤ C. k k [ 2 ,2k] [ 2 ,2k] Observe [R, 2R] ∩ [ k , 2k] = ∅ unless R ∼ k, therefore 2 ∫ 2 |Jk(r)| dr ≤ C. (3.14) I2 Together with (3.12) and (3.13), this yields (3.11). □

Proposition 3.2. Suppose R ≫ 1. Let φ be a smooth function supported in the interval I := [1, 2]. Then LINEAR RESTRICTION ESTIMATES FOR WAVE WITH INVERSE SQUARE POTENTIAL 9

i) if K is finite, there exists a constant CK independent of R such that ∫ ( ∑K d∑(k) ∞ ) 1 −itρ 2 2 e Jν(k)(rρ)bk,ℓ(ρ)φ(ρ)dρ ∞ R ∞ 0 Lt ( ;Lr ([R,2R])) k=0 ℓ=1 (3.15) ( ∑K d∑(k) ) 1 − 1 2 2 ≤ CK R 2 |bk,ℓ(ρ)| ; L1 (I) k=0 ℓ=1 ρ ii) if K is infinite, there exists a constant C independent of R such that ∫ ( ∑K d∑(k) ∞ ) 1 −itρ 2 2 e Jν(k)(rρ)bk,ℓ(ρ)φ(ρ)dρ ∞ R ∞ 0 Lt ( ;Lr ([R,2R])) k=0 ℓ=1 (3.16) ( ∑K d∑(k) ) 1 − 1 2 2 ≤ CR 2 |bk,ℓ(ρ)| . L2 (I) k=0 ℓ=1 ρ Proof. We first prove (3.15). Recalling the well known asymptotic expansion about the Bessel function √ 2 kπ π J (r) = r−1/2 cos(r − − ) + O (r−3/2), as r → ∞, k π 2 4 k − 1 we have |Jν(k)(r)| ≤ CK r 2 when r ≫ 1. By the Minkowski inequality and the Hausdorff-Young inequality in t, there exists a constant CK independent of R such that ∫ ( ∑K d∑(k) ∞ ) 1 −itρ 2 2 e Jν(k)(rρ)bk,ℓ(ρ)φ(ρ)dρ L∞(R;L∞([R,2R])) k=0 ℓ=1 0 t r ∑K d∑(k) 1 ( ) 1 − 2 2 ≤ CK R 2 bk,ℓ(ρ) . L1 (I) k=0 ℓ=1 ρ Thus this proves (3.15). When K is infinite, we need show a precise estimate which is uniform in K. We utilize the Schläfli’sintegral representation of the Bessel func- ˜ tion (2.5) to write Jν(k)(rρ) = Eν(k)(rρ) + Jν(k)(rρ). By (2.6), the Minkowski inequality and the Hausdorff-Young inequality in t, there exists a constant C independent of K,R such that ∫ ( ∑K d∑(k) ∞ ) 1 −itρ 2 2 e Eν(k)(rρ)bk,ℓ(ρ)φ(ρ)dρ L∞(R;L∞([R,2R])) k=0 ℓ=1 0 t r ∑K d∑(k) ( ) 1 −1 2 2 ≤ CR bk,ℓ(ρ) . L1 (I) k=0 ℓ=1 ρ ˜ Thus it remains to prove (3.16) replacing Jν(k) by Jν(k). We decompose [−π, π] into three partitions as follows

[−π, π] = I1 ∪ I2 ∪ I3 where π π I = {θ : |θ| ≤ δ},I = [−π, − −δ]∪[ +δ, π],I = [−π, π]\(I ∪I ), (3.17) 1 2 2 2 3 1 2 10 JUNYONG ZHANG AND JIQIANG ZHENG with 0 < δ ≪ 1. We deﬁne

Φr,k(θ) = sin θ − kθ/r, (3.18) and χδ(θ) is a smooth function given by { 1, θ ∈ [−δ, δ]; χδ(θ) = 0, θ ̸∈ [−2δ, 2δ].

Then split J˜k(r) into three pieces and write ∫ 1 π irΦr,k(θ) J˜k(r) = e dθ 2π −π ( ∫ ∫ ∫ ) 1 π irΦr,k(θ) irΦr,k(θ) irΦr,k(θ) = e χδ(θ)dθ + e dθ + e (1 − χδ(θ))dθ 2π −π I2 I3 ˜1 ˜2 ˜3 := Jk (r) + Jk (r) + Jk (r). (3.19) ∈ ′ − When θ I2, the function Φr,k(θ) = cos θ k/r is monotonic in the intervals − − π − π [ π, 2 δ] and [ 2 + δ, π] respectively and satisﬁes that | ′ | ≥ | | ≥ Φr,k(θ) k/r + cos θ sin δ. Then by Lemma 2.3, we have uniformly in k ∫ 1 irΦr,k(θ) −1 e dθ ≤ cδr . (3.20) 2π I2 ∈ | ′′ | ≥ When θ I3, then Φr,k(θ) sin δ, the Lemma 2.3 again yields that ∫ 1 irΦr,k(θ) −1/2 e (1 − χδ(θ))dθ ≤ cδr , (3.21) 2π I3 uniformly in k. Using the similar arguments as above, it follows from (3.20) and (3.21) that

∞ ∫ ( ∑ d∑(k) ∞ ( ) ) 1 2 −itρ ˜2 ˜3 2 e Jν(k)(rρ) + Jν(k)(rρ) bk,ℓ(ρ)φ(ρ)dρ L∞(R;L∞([R,2R])) k=0 ℓ=1 0 t r ∑∞ d∑(k) 1 ( ) 1 − 2 2 ≲ R 2 bk,ℓ(ρ) . L1 (I) k=0 ℓ=1 ρ (3.22)

To establish (3.16) replacing J by J˜1 , we need eﬀectively use the oscillation ν(k) ν(k) ∑ itρ j i π ρj j of e . To this end, we write Fourier series as b (ρ) = b e 2 where b = ∫ ∑ k,ℓ j k,ℓ k,ℓ 4 − π j 1 i 2 ρj | |2 ∥ ∥2 e bk,ℓ(ρ)dρ. Then b = bk,ℓ(ρ) 2 . For simplicity, we use the 4 0 j k,ℓ Lρ(I) scaling argument to reduce the problem by replacing t, r by 2πt, 2πr respectively and deﬁne ∫ ∞ ∫ 1 − − j − k 2πi(t 4 )ρ 2πiρr sin θ iν(k)θ ψt− j (r) = e e χδ(θ)dθφ(ρ)dρ. (3.23) 4 2π 0 R LINEAR RESTRICTION ESTIMATES FOR WAVE WITH INVERSE SQUARE POTENTIAL 11

Let m = t − j , then we write 4 ∫ k 1 2πiρ(r sin θ−m) −iν(k)θ ψm(r) = e e χδ(θ)φ(ρ)dρdθ 2π 2 ∫R (3.24) 1 −iν(k)θ = φˇ(r sin θ − m)e χδ(θ)dθ. 2π R For our purpose, we need to investigate the asymptotic behavior of the function k ψm(r). We consider the following two cases. • Subcase (a): 4R ≤ |m|. Since R ≥ 1, hence |m| ≥ 4. Sinceφ ˇ is a Schwartz function, then

−N |φˇ(r sin θ − m)| ≤ CN (1 + |r sin θ − m|) ∀N > 0. On the other hand, we have |r sin θ − m| ≥ |m| − r| sin θ| ≥ |m|/100, since r ≤ 2R ≤ |m| and |θ| ≤ 2δ. Thus, (3.24) gives | k | ≤ | | −N ψm(r) Cδ,N (1 + m ) . (3.25) − j Hence keeping in mind m = t 4 , we have ∞ ( ∑ d∑(k) ∑ ) 1 j k 2 2 b ψ j (r) k,ℓ − ∞ t 4 L (R;L∞([R,2R])) k=0 ℓ=1 { ≤| − j |} t r j:4R t 4 ∞ ( ∑ d∑(k) ∑ ) 1 j 2 ≤ −N | j | | − | −N 2 Cδ,N R bk,ℓ (1 + t ) . 4 L∞(R;L∞([R,2R])) k=0 ℓ=1 { ≤| − j |} t r j:4R t 4 By the Cauchy-Schwarz inequality and choosing N large enough, the above is bounded by

( ∑∞ d∑(k) ∑ ) 1 j 2 −N | j |2 | − | −N Cδ,N R bk,ℓ (1 + t ) 4 L∞(R;L∞([R,2R])) k=0 ℓ=1 j t r (3.26) ∞ ∞ ( ∑ d∑(k) ∑ ) 1 ( ∑ d∑(k) ) 2 2 1 ≤ −N | j |2 ≲ −N 2 Cδ,N R bk,ℓ R bk,ℓ(ρ) . L2 (I) k=0 ℓ=1 j k=0 ℓ=1 ρ • Subcase (b): |m| < 4R. Again sinceφ ˇ is a Schwartz function, then

−N |φˇ(r sin θ − m)| ≤ CN (1 + |r sin θ − m|) ∀N > 0. By (3.24), it gives ∫ ( k ≤ CN ψm(r) dθ 2π {θ:|θ|<2δ,|r sin θ−m|≤1} ∫ ) + (1 + |r sin θ − m|)−N dθ . {θ:|θ|<2δ,|r sin θ−m|≥1} Let y = r sin θ − m, then we have ∫ ∫ ( ) k ≤ CN | | −N ≲ 1 ψm(r) dy + (1 + y ) dy . (3.27) 2πr {y:|y|≤1} {y:|y|≥1} r 12 JUNYONG ZHANG AND JIQIANG ZHENG

{ ∈ Z | − j | ≤ } For ﬁxed t, R, we deﬁne the set A = j : t 4 4R . It is easy to see the cardinality of A is O(R). Thus it follows from (3.27) and the Cauchy-Schwarz inequality that

∞ ( ∑ d∑(k) ∑ ) 1 j k 2 2 b ψ j (r) k,ℓ − ∞ t 4 L (R;L∞([R,2R])) k=0 ℓ=1 j∈A t r

( ∞ d(k) ) 1 ∞ d(k) ∑ ∑ ∑ ( ∑ ∑ ) 1 − 1 j 2 2 − 1 2 ≤ 2 | | ≲ 2 2 Cδ,N R bk,ℓ R bk,ℓ(ρ) . L2 (I) k=0 ℓ=1 j k=0 ℓ=1 ρ (3.28) □

Proposition 3.3. Let φ be a smoothing function supported on I = [1, 2] and R is ′ (n+1)p 2n dyadic number. Then for q = n−1 > n−1 , we have that i) when K is ﬁnite, there exists a constant CK independent of R such that ∫ ( ∑K d∑(k) ∞ ) 1 − n−2 −itρ 2 2 r 2 e Jν(k)(rρ)bk,ℓ(ρ)φ(ρ)dρ Lq (R;Lq ([R,2R])) k=0 ℓ=1 0 t rn−1dr

{ } ( ∑K d∑(k) ) 1 n − n−1 − 2n 2 (1 − ) 2 ≤ CK min R q ,R 2 q(n 1) |bk,ℓ(ρ)| ; Lp(I) k=0 ℓ=1 ρ (3.29)

ii) when K is inﬁnite, there exists a constant C independent of R such that ∫ ( ∑K d∑(k) ∞ ) 1 − n−2 −itρ 2 2 r 2 e Jν(k)(rρ)bk,ℓ(ρ)φ(ρ)dρ Lq (R;Lq ([R,2R])) k=0 ℓ=1 0 t rn−1dr

{ } ( ∑K d∑(k) ) 1 n − n−1 − 2n 2 (1 − ) 2 ≤ C min R q ,R 2 q(n 1) |bk,ℓ(ρ)| . L2 (I) k=0 ℓ=1 ρ (3.30)

Proof. To prove (3.29) and (3.30), we ﬁrst consider the case R ≲ 1. The Minkowski inequality and the Hausdorﬀ-Young inequality in t show that

( ∑K d∑(k) ) 1 − n−2 2 2 L.H.S of (3.29), (3.30) ≲ r 2 Jν(k)(rρ)bk,ℓ(ρ)φ(ρ) q′ . Lρ Lq ([R,2R]) k=0 ℓ=1 rn−1dr Then there exists a constant C independent of K such that by (2.4) L.H.S of (3.29), (3.30) ∫ ( ( K d(k) ) q ) 1 2R − ∑ ∑ ν(k) 2 − (n 2)q (8πr) 2 2 n−1 q ≤ C r 2 φ(ρ)bk,ℓ(ρ) q′ r dr 2ν(k)Γ(ν(k) + 1 )Γ(1/2) Lρ R k=0 ℓ=1 2 ( ∑K d∑(k) ) n 2 1 q 2 ≤ CR ∥ φ(ρ)bk,ℓ(ρ) ∥ q′ . Lρ (I) k=0 ℓ=1 LINEAR RESTRICTION ESTIMATES FOR WAVE WITH INVERSE SQUARE POTENTIAL 13

Secondly, we consider the case R ≫ 1. By Proposition 3.1 and Proposition 3.2, we use interpolation to obtain ∫ ∑K d∑(k) ∞ n−2 ( ) 1 − itρ 2 2 r 2 Jν(k)(rρ)e bk,ℓ(ρ)φ(ρ)dρ Lq (R;Lq ([R,2R])) 0 t rn−1dr k=0 ℓ=1 (3.31) ∑K d∑(k) − n−1 − 2n ( 2) 1 2 (1 q(n−1) ) 2 ≤ CK R φ(ρ)bk,ℓ(ρ) q′ . Lρ (I) k=0 ℓ=1 and when K is inﬁnite ∫ ∑∞ d∑(k) ∞ n−2 ( ) 1 − itρ 2 2 r 2 Jν(k)(rρ)e bk,ℓ(ρ)φ(ρ)dρ Lq (R;Lq ([R,2R])) 0 t rn−1dr k=0 ℓ=1 (3.32) ( ∑∞ d∑(k) ) − n−1 (1− 2n ) 2 1 ≤ 2 q(n−1) 2 CR φ(ρ)bk,ℓ(ρ) 2 . Lρ(I) k=0 ℓ=1

′ (n+1)p 2n ⊂ Therefore, since q = n−1 > n−1 and supp φ [1, 2], we show (3.29) and (3.30). □

3.3. Conclude the Proof of Theorem 1.2. Now we turn to conclude the proof Theorem 1.2. We have known that if f is radial, so is u in (3.9). To prove Theorem 1.2, we need to estimate the following by (3.9)

∑K d∑(k) [ ] H −1 (3.33) Yk,ℓ(θ) ν(k) ρ sin(tρ)bk,ℓ(ρ) (r) Lq (R;Lq L2 (Sn−1)), t rn−1dr θ k=0 ℓ=1 in the cases of K = 0 and K = ∞, which are responding to the radial case and the general case respectively. To this end, we use the orthogonality and break (3.33) into the following dyadic decomposition form to estimate ∫ ( ∑ ( ∑ ( ∑K d∑(k) ∞ − n−2 itρ ρ (3.33) ≤C (rρ) 2 J (rρ)e χ( ) ν(k) M R M k=0 ℓ=1 0 ) ) (3.34) ) 1 q 1/q n−2 2 2 × bk,ℓ(ρ)ρ dρ , Lq (R;Lq ([R,2R])) t rn−1dr where R,M are dyadic numbers and χ is a smoothing function supported on [1, 2]. By the scaling argument, we have ∫ ( ∑ ( ∑ ( ∑K d∑(k) ∞ (n−1)− n+1 − n−2 R.H.S of (3.34) ≤ C M q (rρ) 2 Jν(k)(rρ) R M k=0 ℓ=1 0 ) ) ) 1 q 1/q itρ n−2 2 2 × e χ(ρ)bk,ℓ(Mρ)ρ dρ . Lq (R;Lq ([MR,2MR])) t rn−1dr 14 JUNYONG ZHANG AND JIQIANG ZHENG

n −1 Applying Proposition 3.3 with φ(ρ) = χ(ρ)ρ 2 to the above, one can see that when K is ﬁnite ( ∑ ( ∑ { } n − n−1 − 2n (1 − ) R.H.S of (3.34) ≤ CK min (RM) q , (RM) 2 q(n 1) R M ∑K d∑(k) ) ) n+1 ( n ) 1 q 1/q (n−1)− −1 2 2 × M q χ(ρ)ρ 2 bk,ℓ(Mρ) , Lp k=0 ℓ=1 ρ (3.35) and when K is inﬁnite ( ∑ ( ∑ { } n − n−1 (1− 2n ) R.H.S of (3.34) ≤ C min (RM) q , (RM) 2 q(n−1) R M ∑∞ d∑(k) ) ) n+1 ( n ) 1 q 1/q (n−1)− −1 2 2 × M q χ(ρ)ρ 2 bk,ℓ(Mρ) . L2 k=0 ℓ=1 ρ (3.36)

2n By q > n−1 , one has ∑ n − n−1 (1− 2n ) sup min{(RM) q , (RM) 2 q(n−1) } < ∞ R M and ∑ n − n−1 (1− 2n ) sup min{(RM) q , (RM) 2 q(n−1) } < ∞. M R p → q 2n Then by Schur’s test lemma and ℓ , ℓ with q > n−1 > p, we have that in the case when K is ﬁnite R.H.S of (3.34) ( ∑ ∑K d∑(k) ) n+1 ( ) 1 p 1/p [(n−1)− ]p 2 2 ≤ CK M q χ(ρ) |bk,ℓ(Mρ)| Lp M k=0 ℓ=1 ρ ( ∑ ∑K d∑(k) ) n−1 − n+1 ρ ( ) 1 n−2 p 1/p (3.37) ( ′ q )p 2 2 ≤ CK M p χ( ) |bk,ℓ(ρ)| ρ p M Lp M k=0 ℓ=1 ρ ∑K d∑(k) ( ) 1 2 2 ≤ CK bk,ℓ(ρ) ; Lp (R+) k=0 ℓ=1 ρn−2dρ and in the case when K is inﬁnite and p ≥ 2 R.H.S of (3.34) ( ∞ d(k) ) ∑ − ( ∑ ∑ ) n 1 − n+1 ρ 1 n−2 q 1/q ( ′ q )q 2 2 ≤ C M p χ( ) |bk,ℓ(ρ)| ρ 2 M Lp M k=0 ℓ=1 ρ (3.38) ∑∞ d∑(k) ( ) 1 2 2 ≤ C bk,ℓ(ρ) . Lp (R+) k=0 ℓ=1 ρn−2dρ [ ] By Lemma 2.2, we have bk,ℓ(ρ) = Hν(k)Hµ(k) Hµ(k)ak,ℓ (ρ). By the condition in 2n Theorem 1.2, one has 1 < p < n−1 which implies (4.1) below. By using Proposition LINEAR RESTRICTION ESTIMATES FOR WAVE WITH INVERSE SQUARE POTENTIAL 15

4.1 in the appendix, we obtain ∑K d∑(k) ( [ ] ) 1 2 2 R.H.S of (3.34) ≤ C Hµ(k)ak,ℓ (ρ) . Lp (R+) k=0 ℓ=1 ρn−2dρ By (2.2) and (2.13), under the condition of Theorem 1.2, we further have − 1 − 1 p ˆ p ˆ R.H.S of (3.34) ≤ C∥|ξ| f(ξ)∥Lp (R+;L2 (Sn−1)) ≤ C∥|ξ| f(ξ)∥Lp(Rn). ρn−1dρ ω ξ Therefore, we prove Theorem 1.2.

4. Appendix: Vector valued inequality For our purpose, we consider the measure space (R+, dw(ρ)) with dw(ρ) = ρn−2dρ. By Lp(w), 1 ≤ p ≤ ∞, we denote the corresponding Lebesgue space equipped with the norm ∫ ( ∞ ) 1 p p ∥f∥p = |f| dw . 0 H H { }∞ ∈ Proposition 4.1. Let ν , µ be Hankel transform deﬁned above and fk k=0 Lp(w; ℓ2). Suppose 1 < p < ∞ such that for λ = (n − 2)/2 n − 1 λ − ν(0) < < λ + µ(0) + 2. (4.1) p Then there exists a constant C such that ∑ ∑ ( ) 1 ( ) 1 2 2 2 2 |Hν(k)Hµ(k)fk| ≤ C |fk| . (4.2) Lp(w) Lp(w) k k

Proof. Let Tk = Hµ(k)Hν(k), we first follow the argument of proving Theorem 3.1 in [12] to show ∥Tkfk∥Lp(w) ≤ C∥fk∥Lp(w). (4.3) By the argument in [12], we can write ∫ ∞ 0 n−1 (Tkfk)(ρ) = kν(k),µ(k)(ρ, s)fk(s)s ds, 0 where the kernel{ − sβ λ α+β β−α 2 0 Aα,β ρλ+β+2 F ( 2 + 1, 2 + 1; β + 1; (s/ρ) ) for s < ρ; k (ρ, s) = − α,β sα λ β+α α−β 2 Aβ,α ρλ+α+2 F ( 2 + 1, 2 + 1; α + 1; (ρ/s) ) for s > ρ; with F (a, b; c; d) is the hypergeometric function and the coefficient α+β 2Γ( 2 + 1) Aα,β = β−α . Γ( 2 )Γ(β + 1) 0 When s is near ρ, the kernel kν,µ(ρ, s) behaves like c(ρ − s)−1 + O(− log |ρ − s|). Let us define ∫ ∞ n−1 [ n−1 ]ds f p g0 p (Tk[s fk(s)])(ρ) := kν,µ(ρ, s) s fk(s) , (4.4) 0 s where n−1 − n−1 g0 p 0 n p kν,µ(ρ, s) = ρ kν,µ(ρ, s)s . (4.5) 16 JUNYONG ZHANG AND JIQIANG ZHENG

Then ∫ ∞ [ ] n−1 n−1 − n−1 n−1 ds f p p 0 n p p (Tk[s fk(s)])(ρ) = ρ kν,µ(ρ, s)s s fk(s) 0 s n−1 = ρ p (Tkfk)(ρ). Note that n−1 p (Tkfk)(ρ) Lp(w) = ρ (Tkfk)(ρ) Lp , ρ−1dρ to prove (4.3), it suﬃces to show f ∥Tkfk∥Lp(ρ−1dρ) ≤ C∥fk∥Lp(ρ−1dρ). (4.6) By the argument in [12] again, one has { O(ρ−λ−µ−2+ϵs−λ+µ−ϵ) for s < ρ; |k0 (ρ, s)| = ν,µ O(ρν−λ−ϵs−λ−ν−2+ϵ) for s > ρ, then { λ+µ+2− n−1 −ϵ O((s/ρ) p ) for s < ρ; |g0 | kν,µ(ρ, s) = λ−ν− n−1 +ϵ O((s/ρ) p ) for s > ρ. Since n − 1 λ − ν < < λ + µ + 2, p

g0 1 dρ then the kernel kν,µ is bounded in L ( ρ ). By the logarithmic coordinates, the g0 ∼ operator Tk is a convolution operator with the kernel kν,µ. When s ρ, we recall g0 that the kernel kν,µ is a Calderon-Zygmund kernel behaved like c(ρ − s)−1 + O(− log |ρ − s|). Applying the Young’s inequality to the region away from ρ ∼ s and the Calderon- Zygmund theory to the region ρ ∼ s, we obtain (4.6), thus it gives (4.3). By ∗ p′ the similar argument, we show the adjoint operator Tk is also bounded in L (w) provided n − 1 λ − µ < < λ + ν + 2, p′ which holds for 1 < p < ∞. Now we show the vector valued inequality (4.2). When 2 ≤ p ≤ ∞ and then p ≥ q := 2 1, we have ∑ ∑ ( ) 1 2 2 2 2 |Tkfk| = |Tkfk| Lp(w) Lq (w) k k ∫ ∞ ∑ 2 n−2 = sup |Tkfk| g(ρ)ρ dρ { ≥ ∈ q′ } g 0:g L (w) 0 k ∑ ∫ ∞ 2 n−2 = sup |Tkfk| g(ρ)ρ dρ ′ {g≥0:g∈Lq (w)} 0 ∑k ∥ ∥2 = Tkfk Lp(w). k LINEAR RESTRICTION ESTIMATES FOR WAVE WITH INVERSE SQUARE POTENTIAL 17

By (4.3), we can see that ∑ ∑ ( ) 1 2 | |2 2 ≤ ∥ ∥2 Tkfk C fk Lp(w) Lp(w) k k ∑ ∫ ∞ 2 n−2 = C sup |fk| g(ρ)ρ dρ { ≥ ∈ q′ } k g 0:g L (w) 0 ∫ ∞ ∑ (4.7) 2 n−2 ≤ C sup |fk| g(ρ)ρ dρ { ≥ ∈ q′ } g 0:g L (w) 0 k ∑ ∑ 2 ( ) 1 2 2 2 2 ≤ C |fk| ≤ C |fk| . Lq (w) Lp(w) k k ≤ ∗ H H Finally, if 1 < p 2, since the adjoint operator Tk = ν(k) µ(k) is bounded in ′ Lp (w), thus we get (4.2) in this case by duality. □

References

[1] B. Barcelo. On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc. 292 (1985), 321-333. [2] N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh. Strichartz estimates for the wave and Schrödingerequations with the inverse-square potential. J. Funct. Anal. 203 (2003), 519- 549. [3] N. Burq, F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh. Strichartz estimates for the wave and Schrödingerequations with potentials of critical decay. Indiana Univ. Math. J. 53(2004), 1665-1680. [4] P. Chen, E. M. Ouhabaz, A. Sikora and L. Yan, Restriction estimates, sharp spectral multi- pliers and endpoint estimates for Bochner-Riesz means. arXiv:1202.4052v1. [5] J. Cheeger, M. Taylor, On the diffraction of waves by conical singularities. I. Comm. Pure Appl. Math., 35(1982), 275-331. [6] H. Kalf, U. W. Schmincke, J. Walter and R. Wüst,On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials. In Spectral theory and differential equations, 182-226. Lect. Notes in Math., 448 (1975) Springer, Berlin. [7] F. Nicola, Slicing surfaces and Fourier restriction conjecture, Proceedings of the Edinburgh Mathematical Society 52 (2009), 515-527. [8] J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173(2000) 103-153. [9] C. Miao, J. Zhang, and J. Zheng. A Note on the Cone Restriction Conjecture. To appear Proceedings of the AMS. [10] C. Miao, J. Zhang and J. Zheng. Linear Adjoint Restriction Estimates for Paraboloid, Preprint. [11] C. Miao, J. Zhang, and J. Zheng. Strichartz estimates for wave equation with inverse square potential, Preprint. [12] F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh. Lp estimates for the wave equation with the inverse-square potential. Discrete Contin. Dynam. Systems, 9(2003), 427-442. [13] F. Planchon, J. Stalker and A. S. Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete Contin. Dynam. Systems, 9(2003), 1387-1400. [14] S. Shao, Sharp linear and bilinear restriction estimates for paraboloids in the cylindrically symmetric case, Revista Matemática Iberoamericana. 25(2009), 1127-1168. [15] S. Shao, A note on the cone restriction conjecture in the cylindrically symmetric case, Proc. Amer. Math. Soc. 137(2009) 135-143. [16] E.M. Stein, Some problems in harmonic analysis, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll. Williamstown, Mass., 1978), Part1, pp. 3-20. [17] E.M. Stein, Harmonic analysis: real variable methods, orthogonality and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. 18 JUNYONG ZHANG AND JIQIANG ZHENG

[18] E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N. J., 1971, Princeton Mathematical Series, No. 32. MR0304972. [19] R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke. Math. J., 44 (1977), 705-714. [20] J. Sterbenz and Appendix by I. Rodnianski, Angular Regularity and Strichartz Estimates for the Wave Equation. Int Math Res Notices 4 (2005), 187-231. [21] K. Stempak, A Weighted uniform Lp estimate of Bessel functions: A note on a paper of Guo. Proceedings of the AMS. 128 (2000) 2943-2945. [22] T.Tao, Recent progress on the restriction conjecture. ArXiv:math/0311181. [23] T.Tao, The Bochner-Riesz conjecture implies the restriction conjecture, Duke Math. J. 96 (1999), 363-375. [24] T.Tao, Endpoint bilinear restriction theorems for the cone and some sharp null form estimates, Math. Z. 238 (2001), 215-268. [25] T.Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), 1359-1384. [26] T.Tao, A. Vargas and L. Vega, A bilinear approach to the restrction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), 967-1000. [27] G. N. Watson, A Treatise on the Theory of Bessel Functions. Second Edition Cambridge University Press, (1944). [28] T. Wolﬀ, A sharp bilinear cone restriction estimate, Ann. of Math. 153 (2001), 661-698.

Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China and Beijing Computational Science Research Center, Beijing 100084 E-mail address: zhang [email protected] The Graduate School of China Academy of Engineering Physics, P. O. Box 2101, Beijing, China, 100088 E-mail address: [email protected]