YAKUN XI AND CHENG ZHANG

Transcription

1 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES ON RIEMANNIAN SURFACES WITH NONPOSITIVE CURVATURE YAKUN XI AND CHENG ZHANG Abstract. We show that one can obtain improved L 4 geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge s strategy in [3]. We first combine the improved L 2 restriction estimate of Blair and Sogge [3] and the classical improved L estimate of Bérard to obtain an improved weak-type L 4 restriction estimate. We then upgrade this weak estimate to a strong one by using the improved Lorentz space estimate of Bak and Seeger []. This estimate improves the L 4 restriction estimate of Burq, Gérard and Tzvetkov [5] and Hu [9] by a power of (log log λ). Moreover, in the case of compact hyperbolic surfaces, we obtain further improvements in terms of (log λ) by applying the ideas from [7] and [3]. We are able to compute various constants that appeared in [7] explicitly, by proving detailed oscillatory integral estimates and lifting calculations to the universal cover H 2.. Introduction Let (M, g) be a compact n-dimensional Riemannian manifold and let g be the associated Laplace-Beltrami operator. Let e λ denote the L 2 -normalized eigenfunction g e λ = λ 2 e λ, so that λ is the eigenvalue of the first order operator g. Various types of concentrations exhibited by eigenfunctions have been studied. See the recent survey by Sogge [4] for a detailed discussion. A classical result of Sogge [0] states that the L p norm of the eigenfunctions satisfies where 2 p and µ(p) is given by { n µ(p) = max 2 e λ L p (M) Cλ µ(p), ( 2 p ) (, n 2 ) }. p Mathematics Subject Classification. Primary 58J5; Secondary 35A99, 42B37.

2 2 YAKUN XI AND CHENG ZHANG If we let p c = 2(n+) n (.), these bounds can also be written as {Cλ n 2 ( 2 p ), 2 p p c, e λ L p (M) Cλ n( 2 p ) 2, pc p. The estimates (.) are saturated on the round sphere by zonal functions for p p c and for 2 < p p c by the highest weight spherical harmonics. Even though they are sharp on the round sphere S n, it is expected that (.) can be improved for generic Riemannian manifolds. Manifolds with nonpositive sectional curvature have been studied as the model case for such improvements. It is well-known that one can get log improvements for e λ L (M) if M has nonpositive curvature. Indeed, Bérard s results [2] in 977 on improved error term estimates for the pointwise Weyl law imply that e λ L (M) = O(λ n 2 / log λ), which gives log improvements over (.) for p > p c via interpolation. In a recent work of Hassell and Tacy [8], they obtained further improvements for p > p c in the sense that where C p blows up as p p c. λ n( 2 p ) 2 e λ L p (M) C p (log λ) 2 e λ L 2 (M), Recently, Blair and Sogge [3] were able to obtain log improvements over (.) for 2 < p < p c by proving improved Kakeya-Nikodym bounds which measure L 2 - concentration of eigenfunctions on λ 2 tubes about unit length geodesics. Despite the success in improving (.) for the range 2 < p < p c and p c < p <, improvement of the critical case has been elusive. On one hand, the estimate at the critical exponent p c = 2(n+) n (.2) e λ Cλ n L 2(n+) 2(n+) n (M) actually implies (.) for all 2 p via interpolating with the classical L estimate and the trivial L 2 estimate. On the other hand, this bound (.2) is sensitive to both point concentration and concentration along geodesics, in the sense that it is saturated by both zonal functions and spherical harmonics on the round sphere. Recently, Sogge [3] managed to improve over (.2) by a power of (log log λ) under the assumption of nonpositive curvature. Using Bourgain s [4] idea in proving weak-type estimate for the Stein-Tomas restriction theorem, Sogge was able to combine the recent improved L p, 2 < p < p c bounds of Blair and Sogge [3] and the

3 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 3 classical improved sup-norm estimate of Bérard [2], to get improved bounds for the critical case. In the last decade, similar L p estimates have been established for the restriction of eigenfunctions to geodesics. Burq, Gérard and Tzvetkov [5] and Hu [9] showed that for n-dimensional Riemannian manifold (M, g), if Π denotes the space of all unit-length geodesics γ, then ( ) (.3) sup e λ p p ds γ Π γ where Cλ σ(n,p) e λ L 2 (M), (.4) σ(2, p) = {, 4 2 p 4, 2 p 4 p. and (.5) σ(n, p) = n 2, if p 2 and n 3, p here the case n = 3, p = 2 is due to Chen and Sogge [7]. Note that in the 2- dimensional case, the estimates (.3) have a similar flavor compared to Sogge s L p estimates (.). Indeed, when n = 2 the estimates (.3) also have a critical exponent p c = 4. Moreover, on the sphere S 2, (.3) is saturated by zonal functions when p 4, while for p 4, it is saturated by the highest weight spherical harmonics. When n = 3, the critical exponent no longer appears in (.3). However, the estimate for p = 2 is still saturated by both zonal functions and highest weight spherical harmonics. In higher dimensions n > 3, geodesic restriction estimates are too singular to detect concentrations of eigenfunctions near geodesics. In fact, in these dimensions, estimates (.3) are always saturated by zonal functions rather than highest weight spherical harmonics on the round sphere S n. There has been considerable work towards improving (.3) under the assumption of nonpositive curvature in the 2-dimensional case. Bérard s sup-norm estimate [2] provides natural improvements for large p. In [6], Chen managed to improve over (.3) for all p > 4 by a (log λ) 2 factor: ( ) (.6) sup e λ p p ds γ Π γ C λ 2 p e (log λ) λ L 2 (M). 2

4 4 YAKUN XI AND CHENG ZHANG Sogge and Zelditch [5] showed that one can improve (.3) for 2 p < 4, in the sense that ( ) (.7) sup e λ p p ds = o(λ 4 ). γ Π γ A few years later, Chen and Sogge [7] showed that the same conclusion can be drawn for p = 4: ( ) (.8) sup e λ 4 4 ds = o(λ 4 ). γ Π γ (.8) is the first result to improve an estimate that is saturated both by zonal functions and highest weight spherical harmonics. Recently, by using the Toponogov s comparison theorem, Blair and Sogge [3] showed that it is possible to get log improvements for L 2 -restriction: ( ) (.9) sup e λ 2 2 ds γ Π γ C λ 4 e (log λ) λ L 2 (M), 4 Adapting Sogge s idea in proving improved critical L pc estimates [3], we are able to further improve the critical L 4 -restriction estimate in the 2-dimensional case (.3) by a factor of (log log λ) 8. Theorem. Let (M, g) be a 2-dimensional compact Riemannian manifold of nonpositive curvature, let γ M be a fixed unit-length geodesic segment. Then for λ, there is a constant C such that (.0) χ [λ,λ+(log λ) ]f L 4 (γ) Cλ 4 (log log λ) 8 f L 2 (M). Therefore, taking f = e λ, we have (.) e λ L 4 (γ) Cλ 4 (log log λ) 8 eλ L 2 (M). Moreover, if Π denotes the set of unit-length geodesics, there exists a uniform constant C = C(M, g) such that ( ) (.2) sup e λ 4 4 ds Cλ 4 (log log λ) 8 eλ L 2 (M). γ Π γ Furthermore, if we assume further that M has constant negative curvature, we are able to get log improvement for the L 4 -restriction estimate following the ideas in [3] and [7].

5 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 5 Theorem 2. Let (M, g) be a 2-dimensional compact Riemannian manifold of constant negative curvature, let γ M be a fixed unit-length geodesic segment. Then for λ, there is a constant C such that (.3) e λ L 4 (γ) Cλ 4 (log λ) 4 eλ L 2 (M). Moreover, if Π denotes the set of unit-length geodesics, there exists a uniform constant C = C(M, g) such that ( ) (.4) sup e λ 4 4 ds Cλ 4 (log λ) 4 eλ L 2 (M). γ Π γ Our paper is organized as follows. In Section 2, we give the proof of Theorem. We do this by first proving a new local restriction estimate which corresponds to Lemma 2.2 in [3]. Then we use this local estimate together with the improved L 2 -restriction estimate (.9) of Blair and Sogge [3] and the classical improved sup-norm estimate of Bérard [2] to obtain improved L 2 (M) L 4, (γ) estimate. Finally, we prove Theorem by interpolating between the improved L 2 (M) L 4, (γ) estimate and the L 2 (M) L 4,2 (γ) estimate of Bak and Seeger []. In Section 3, we show how to obtain further improvements under the assumption of constant negative curvature. We follow the strategies that were introduced in [7] and [3]. We shall lift all the calculations to the universal cover H 2 and then use the Poincaré half-plane model to compute the dependence of various constants explicitly. Throughout our argument, we shall assume that the injectivity radius of M is sufficiently large, and fix γ to be a unit length geodesic segment. We shall use P to denote the first order operator g. Also, whenever we write A B, it means A CB and C is some unimportant constant. 2. Riemannian surface with nonpositive curvature We start with some standard reductions. Let ρ S(R) such that ρ(0) = and supp ˆρ [ /2, /2], then it is clear that the operator ρ(t (λ P )) reproduces eigenfunctions, in the sense that ρ(t (λ P ))e λ = e λ. Consequently, we would have the estimate (.0) if we could show that (2.) ρ(log λ(λ P )) L 2 (M) L 4 (γ) = O(λ 4 /(log log λ) 8 ). The uniform bound (.2) also follows by a standard compactness argument.

6 6 YAKUN XI AND CHENG ZHANG 2.. A local restriction estimate. To prove (2.), we apply Sogge s strategy in [3]. We shall need the following local restriction estimate. Lemma. Let λ r, γ r be a fixed subsegment of γ with length r. Then we have ρ(λ P )f L 2 (γ r) λ 4 r 4 f L 2 (M). Proof. By a standard T T argument, this is equivalent to showing that (2.2) χ(λ P )h L 2 (γ r) λ 2 r 2 h L 2 (γ r), here χ = ρ 2. Thus χ(λ P )h = 2π χ(t)e iλt e itp h dt. We shall need a preliminary reduction. Let β C0 function, satisfying β(s) =, if s [/2, 2], and β(s) = 0, if s [/4, 4]. Then we claim that it suffices to prove: (2.3) χ(t)e iλt β(p/λ)e itp h dt L 2 (γ) Indeed, we note that the operator (2.4) χ(t)e iλt ( β(p/λ))e itp dt be a Littlewood-Paley bump Cλ 2 r 2 h L 2 (γ). has kernel χ(λ λj )( β)(λ j /λ)e j (γ(s))e j (γ(s )). Since χ C 0 (R) and β is the Littlewood-Paley bump function, we see that χ(λ λ j )( β)(λ j /λ) C( + λ + λ j ) 4. On the other hand, by the Weyl formula, e j (γ(s))e j (γ(s )) C( + λ), λ j [λ,λ+] we conclude that the kernel of the operator given by (2.4) is O(λ ). This means that this operator enjoys better bounds than (2.2), which gives our claim that it suffices to prove (2.3). To prove (2.3), we consider the corresponding kernel K λ (γ(s), γ(s )) = χ(t)e iλt β(p/λ)e itp (γ(s), γ(s )) dt.

7 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 7 We claim that K λ satisfies (2.5) K λ (γ(s), γ(s )) = O(λ 2 s s 2 ). Indeed, one may use a parametrix and the calculus of Fourier integral operators to see that modulo a trivial error term of size O(λ N ) (β(p/λ)e itp )(γ(s), γ(s )) = e i(s s )ξ +it ξ α(t, s, s, ξ ) dξ, R 2 where α is a zero-order symbol. See [, Lemma 5..3] and [2]. Thus, modulo trivial errors, ( ) (2.6) K λ (γ(s), γ(s )) = e it(l λ) α(t, s, s, l) e il(s s ) (0,),ω dω l dldt. 0 S Integrating by parts in t shows that the above expression is majorized by 0 ( + l λ ) 3 l dl = O(λ), thus (2.5) is valid when s s λ. To handle the remaining case, we recall that, by stationary phase, e ix ω dω = O( x 2 ), x. S If we plug this into (2.6) with x = l(s s, 0), and integrate by parts in t, we conclude that if λ s s, we have K λ (γ(s), γ(s )) 0 ( + l λ ) 3 (l s s ) 2 l dl = O(λ 2 s s 2 ), as claimed. By Young s inequality, the left hand side of (2.3) is bounded by ( r r λ 2 h(s ) ds 2 ) 2 ds λ 0 0 s s 2 r 2 h L 2 ([0,r]), 2 completing our proof. Remark. A similar argument gives the same estimate for ρ(t (λ P ))f if T. Indeed, the same argument works for operator with kernel [ (2.7) a(t) e itλ e dt] itp (γ(s), γ(s )), providing a C0 (, ). While the operator ρ(t (λ P )) corresponds to the kernel [ ] a(t/t ) e itλ e itp dt (γ(s), γ(s )), T which can be handled by smoothly partitioning the interval [ T, T ] into subintervals of size. Each piece of the kernel over a subinterval of size enjoys the same bound

8 8 YAKUN XI AND CHENG ZHANG as in Lemma, thanks to the fact that e itp is unitary. Now if we sum up the T pieces resulting from the partition, we obtain the desired estimate for ρ(t (λ P )) An improved weak-type estimate. In this section, we prove the following improved weak-type estimate. Proposition. Let (M, g) be a 2-dimensional compact Riemannian manifold of nonpositive curvature. Then for λ (2.8) ρ(log λ(λ P )) L 2 (M) L 4, (γ) = O(λ 4 /(log log λ) 4 ). As discussed before, the L 4 restriction bound is saturated by both zonal functions and highest weight spherical harmonics. Thus as in [3], to get improved L 4 bounds, we shall need the following two improved results which corresponds to the range 2 p < 4 and the range 4 < p respectively. Lemma 2 ([3]). Let (M, g) be as above. Then for λ we have (2.9) ρ(log λ(λ P )) L 2 (M) L 2 (γ) = O(λ 4 /(log λ) 4 ). Lemma 3 ([2]). If (M, g) is as above then there is a constant C = C(M, g) so that for T and large λ we have the following bounds for the kernel of η(t (λ P )), η = ρ 2, ( (2.0) η(t (λ P ))(x, y) C [T λ d g (x, y) ) ] 2 + λ 2 e CT, The first Lemma is a recent result of Blair and Sogge [3]. The other bound (2.0) is well-known and follows from the arguments in the paper of Bérard [2]. Now we are ready to prove Proposition. It suffices to show that (2.) {x γ : ρ(log λ(λ P ))f(x) > α} Cα 4 λ(log log λ). assuming f is L 2 normalized. By Chebyshev inequality and (2.9), we have {x γ : ρ(log λ(λ P ))f(x) > α} α 2 ρ(log λ(λ P ))f 2 ds Note that for large λ we have Thus it remains to show when α λ 4 (log λ) 8. γ α 2 λ 2 (log λ) 2. α 2 λ 2 (log λ) 2 α 4 λ(log log λ), if α λ 4 (log λ) 8. {x γ : ρ(log λ(λ P ))f(x) > α} Cα 4 λ(log log λ),

9 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 9 We notice that [ ρ(c 0 log log λ(λ τ)) ] ρ(log λ(λ τ)) log log λ log λ ( + λ τ ) N, together with the estimate χ λ L 2 (M) L 4 (γ) = O(λ 4 ), we see that [ ρ(c 0 log log λ(λ P )) I ] ρ(log λ(λ P ))f L 4 (γ) Therefore we would be done if we could show that log log λ log λ λ 4 f L 2 (M). {x γ : ρ(c 0 log log λ(λ P ))h(x) > α} Cα 4 λ(log log λ), if α λ 4 (log λ) 8, and h L 2 (M) =. Let A = {x γ : ρ(c 0 log log λ(λ P ))h(x) > α}. Take r = λα 4 (log log λ) 2. We decompose A into r-separated subsets j A j = A with length r. By replacing A by a set of proportional measure, we may assume that if j k, we have dist(a j, A k ) > C 0 r, where C 0 will be specified momentarily. Let T λ = ρ(c 0 log log λ(λ P )), which has dual operator Tλ mapping L2 (γ) L 2 (M). Let ψ λ (x) = T λ f(x)/ T λ f(x), if T λ f(x) 0, otherwise let ψ λ (x) =. Let S λ = T λ Tλ and a j = ψ λ Aj. Then by Chebyshev s inequality and Cauchy-Schwarz inequality, we have α A T λ fψ λ A ds T λ fa j ds = γ M ( Tλa j f dv g squaring both sides, we see that α 2 A 2 Tλa j 2 dv g + j M j k j γ j M j γ T λa j 2 dvg ) 2, S λ a j a k ds = I + II. By the dual version of Lemma (see Remark ), we see that I r 2 λ 2 a j 2 ds = r 2 λ 2 A = λα 2 (log log λ) A. j γ

10 0 YAKUN XI AND CHENG ZHANG By making c 0 sufficiently small, we see from (2.0) that we can control the kernel, K λ (s, s ), of S λ by ( K λ (s, s ) C [(log log λ) λ ) ] 2 + λ s s 2 (log λ) 40, thus [ ( λ ) ] II (log log λ) 2 + λ 2 (log λ) 40 a j L a k L C 0 r C 2 0 α 2 A 2 + λ 2 (log λ) 40 A 2. Since we are assuming α λ 4 (log λ) 8, for sufficiently large C 0, we have thus II 2 α2 A 2, j k α 2 A 2 Cλα 2 (log log λ) A + 2 α2 A 2, which gives A Cλα 4 (log log λ), if α λ 4 (log λ) 8, completing the proof of Proposition Proof of Theorem. We shall combine the improved L 2 (M) L 4, (γ) estimate (2.8) we obtained in the last section with the following L 2 (M) L 4,2 (γ) estimate established by Bak and Seeger [] to prove our main theorem. This estimate of Bak and Seeger holds for general Riemannian manifold without any curvature condition. Lemma 4 ([]). Let (M, g) be a 2-dimensional Riemannian manifold. Fix γ M to be a geodesic segment. Then we have the following estimate for the unit band spectral projection operator χ [λ,λ+] (2.2) χ [λ,λ+] f L 4,2 (γ) C( + λ) 4 f L 2 (M). We remark that Lemma 4 is a special case of the results in [] regarding the restriction of eigenfunctions to hypersurfaces for manifolds with dimension n 2. Let us recall some basic facts about the Lorentz space L p,q (γ). First, for a function u on M, we define the corresponding distribution function ω(α) with respect to γ as ω(α) = {x γ : u(x) > α}, α > 0. u is the nonincreasing rearrangement of u on γ, given by u (t) = inf{α : ω(α) t}, t > 0.

11 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D Then the Lorentz space L p,q (γ) for p < and q < is defined as all u so that ( q [ (2.3) u L p,q (γ) = t p u (t) ] q dt ) q <, p t 0 It s well known that for the special case p = q, the Lorentz norm L p,p (γ) agrees with the standard L p norm L p (γ). Moreover, we also have sup t>0 t p u (t) = sup α[ω(α)] p. α>0 If we take u = χ [λ,λ+(log λ) ]f, and assume f L 2 (M) =, then by (2.8) we have (2.4) sup t 4 u (t) Cλ 4 (log log λ) 4. t>0 On the other hand, since χ [λ,λ+] u = u, by Lemma 4 we have (2.5) u L 4,2 (γ) Cλ 4 u L 2 (M) Cλ 4. Interpolating between (2.4) and (2.5), we then get ( [ u L 4 (γ) = t 4 u (t) ] ) 4 dt 4 0 t ( ) sup t 4 u 2 (t) u t>0 which completes the proof of Theorem. ( λ 4 (log log λ) 4 = λ 4 (log log λ) 8, ) 2 λ 8 2 L 4,2 (γ) 3. Riemannian surfaces with constant negative curvature We shall apply the strategies in [7] and [3] to prove Theorem 2. Recall in [7], Chen and Sogge showed that for Riemannian surfaces with nonpositive curvature, ( ) 4 4 (3.) χ(t (λ P ))(γ(t), γ(s))h(s)ds dt 0 0 CT 2 λ 2 h L 4 3 ([0,]) + C T λ 3 8 h L 4 3 ([0,]),

12 2 YAKUN XI AND CHENG ZHANG here χ(t (λ P ))(x, y) denotes the kernel of the multiplier operator χ(t (λ P )). Clearly, this would imply (.8) if one takes T to be sufficiently large. We shall show that under the assumption of constant negative curvature, the constant C T in (3.) can be taken to be e CT where C > 0 is some constant independent of T. Then Theorem 2 would be proved if we set T = c log λ, for c > 0 to be sufficiently small. From now on, we shall use C to denote various positive constants that are independent of T. 3.. Some reductions. Choose a bump function β C 0 (R) satisfying β(τ) = for τ 3/2, and β(τ) = 0, τ 2. Then we may write χ(t (λ P ))(x, y) = β(τ)ˆχ(τ/t )e iλτ (e iτp )(x, y)dτ 2πT + ( β(τ))ˆχ(τ/t )e iλτ (e iτp )(x, y)dτ = K 0 (x, y) + K (x, y). 2πT As (2.5) in the proof of Lemma, one may use a parametrix to see that (3.2) ( K 0 (γ(t), γ(s))h(s)ds dt ) 4 CT λ 2 h L 4 3 ([0,]), which is better than the bounds in (3.). (See [7, p.8].) Since the kernel of χ(t (λ+p )) is O(λ N ) with constants independent of T, we are left to consider the integral operator S λ : (3.3) S λ h(t) = πt 0 ( β(τ))ˆχ(τ/t )e iλτ (cos τp )(γ(t), γ(s))h(s)dsdτ. As in [7] and [3], we now use the Hadamard parametrix and the Cartan-Hadamard theorem to lift the calculations up to the universal cover (R 2, g) of (M, g). Let Γ denote the group of deck transformations preserving the associated covering map κ : R 2 M coming from the exponential map from γ(0) associated with the metric g on M. The metric g is its pullback via κ. Choose also a Dirchlet fundamental domain, D M, for M centered at the lift γ(0) of γ(0). We shall let γ(t), t R denote the lift of the geodesic γ(t), t R, containing the unit geodesic segment γ(t), t [0, ]. We measure the distances in (R 2, g) using its Riemannian distance function d g (, ).

13 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 3 Following [7], we recall that if x denotes the lift of x M to D, then we have the following formula (cos t g )(x, y) = α Γ (cos t g )( x, α(ỹ)). Consequently, we have, for t [0, ], S λ h(t) = ( β(τ))ˆχ(τ/t )e iλτ (cos τ g )( γ(t), α( γ(s)))h(s) dsdτ. πt Let α Γ 0 (3.4) T R ( γ) = {(x, y) R 2 : d g ((x, y), γ) R} and Γ TR ( γ) = {α Γ : α(d) T R ( γ) }. From now on we fix R InjM. We write S λ h(t) = Sλ tube h(t) + Sλ osc h(t) = Sλ,α tube h(t) + α Γ TR ( γ) By the Huygens principle, α/ Γ TR ( γ) S osc λ,αh(t), t [0, ]. (cos τ g )( γ(t), α( γ(s))) = 0, if d g ( γ(t), α( γ(s))) > τ. Recall that ˆχ(τ) = 0 if τ. Hence d g ( γ(t), α( γ(s))) T, s, t [0, ]. Since there are only O() translates of D, α(d), that intersect any geodesic ball with arbitrary center of radius R, it follows that (3.5) #{α Γ TR ( γ) : d g (0, α(0)) [2 k, 2 k+ ]} C2 k. Thus the number of nonzero summands in Sλ tube h(t) is O(T ) and in Sλ osch(t) is O(eCT ). Given α Γ set with s, t [0, ] K α (t, s) = πt T T ( β(τ))ˆχ(τ/t )e iλτ (cos τ g )( γ(t), α( γ(s)))dτ. When α = Identity, by using the Hadamard parametrix (see e.g. [6], p. 9), we get Thus, If α Identity, we set K Id (t, s) CT λ 2 t s 2. 0 K Id (t, s) ds CT λ 2. φ(t, s) = d g ( γ(t), α( γ(s))), s, t [0, ].

14 4 YAKUN XI AND CHENG ZHANG Then by the Huygens principle and α Identity, we have (3.6) 2 φ(t, s) T, if s, t [0, ]. Following Lemma 3. in [7], we can write K α (t, s) = w( γ(t), α( γ(s))) ± a ± (T, λ; φ(t, s))e ±iλφ(t,s) + R(t, s), where w(x, y) C, and for each j = 0,, 2,..., there is a constant C j independent of T, λ so that (3.7) j r a ± (T, λ; r) Cj T λ 2 r 2 j, r. Using the Hadamard parametix with an estimate on the remainder term (see [2]), we see that R(t, s) e CT. Therefore we are able to estimate Sλ tube (t, s) of S tube satisfies K tube λ Consequently, (3.8) λ K tube λ (t, s) CT λ 2 2 k T h by Young s inequality. Indeed, the kernel 2 k 2 k/2 + e CT = CT 2 λ 2 + e CT. S tube λ h L 4 [0,] (CT 2 λ 2 + e CT ) h L 4 3 [0,] A stationary phase argument. To deal with the remaining part Sλ osch(t), we need the following detailed version of the oscillatory integral estimates. (See e.g. Chapter of []). Proposition 2. Let a C 0 (R 2 ), let φ C (R 2 ) be real valued and λ > 0, set T λ f(t) = If φ st 0 on supp a, then where (3.9) C a,φ = Cdiam(supp a) 2 e iλφ(t,s) a(t, s)f(s)ds, f C 0 (R). T λ f L 2 (R) C a,φλ 2 f L 2 (R), { a + 0 i,j 2 i ta j t φ inf φ st 2 st }.

15 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 5 Assume supp a is contained in some compact set F R 2. Denote the range of t in F by F t R. If there is one t 0 F t such that φ st(t 0, s) = 0, φ stt(t 0, s) 0 on {s : (t 0, s) F }, and φ st(t, s) 0 on {(t, s) F : t t 0 }, then where (3.0) C a,φ = Cdiam(supp a) 4 T λ f L 2 (R) C a,φλ 4 f L 2 (R), { a + 0 i,j 2 i ta j t φ inf φ st/(t t 0 ) 2 st The norms and the infima are taken on supp a. The constant C > 0 is independent of λ, a, φ and F. Proof. By a standard T T argument and Young s inequality, it suffices to estimate the kernel of Tλ T λ K(s, s ) = e iλ(φ(t,s) φ(t,s )) a(t, s)a(t, s )dt. Let and let ϕ(t, s, s ) = φ(t, s) φ(t, s ), s s, and ϕ(t, s, s) = φ s s s(t, s), Then the kernel becomes (3.) K(s, s ) = ã(t, s, s ) = a(t, s)a(t, s ). e iλ(s s )ϕ(t,s,s )ã(t, s, s )dt. If φ st 0 on supp a, then by the mean value theorem, ϕ t(t, s, s ) = φ st(t, s ) inf φ st, where s is some number between s and s. If λ(s s ), it is easy to see that K(s, s ) a(t, s) a(t, s ) dt diam(supp a) a 2. For λ(s s ), we integrate by parts twice to see that ( ( )) K(s, s ) (λ s s ) 2 ã dt t ϕ t t ϕ t ( ) 2 ta i j t φ st C(λ s s ) 2 0 i,j 2 diam(supp a), inf φ st 4 }.

16 6 YAKUN XI AND CHENG ZHANG where C is some constant independent of λ, a and φ. Hence K(s, s ) Cdiam(supp a) { a 2 + ( 0 i,j 2 ) 2 ta i j t φ st } inf φ st 4 again C is some constant independent of λ, a and φ. Consequently, K(s, s ) ds C 2 a,φλ, which finishes the proof of the first case. ( + λ s s ) 2, Now we prove the second part of our proposition. Let δ > 0. Choose ρ C0 (R) satisfying ρ(t) =, t, and ρ(t) = 0, t 2. Then e iλ(s s )ϕãρ((t t 0 )/δ)dt 4δ a 2. For the remainder term with factor ρ, we integrate by parts twice to see that if s s, e iλ(s s )ϕã( ρ((t t 0 )/δ))dt ( (ã( )) (λ s s ) 2 ρ((t t0 )/δ)) dt t t t t 0 >δ ( ) 2 ta i tφ i st C(λ s s ) 2 0 i,j 2 inf ( φ st / t t 0 ) 4 ( ) 2 ta i tφ i st Cδ 3 (λ s s ) 2 0 i,j 2 inf ( φ st / t t 0 ) 4, ϕ t ϕ t t t 0 >δ where C is a constant independent of λ, a, φ and F. ( t t δ 2 t t 0 2 )dt By setting δ = (λ s s ) 2, we get ( ) 2 { ta i j t φ st } K(s, s ) C a 2 0 i,j 2 + (λ s s ) inf( φ st / t t 0 ) 4 2, if s s.

17 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 7 Therefore, which completes the proof. K(s, s ) ds C 2 a,φ λ 2, 3.3. Proof of Theorem 2. Noting that diam(supp a ± ) 2 and we have good control on the size of a ± and its derivatives by (3.7), it remains to estimate the size of φ st and its derivatives. On general manifolds with nonpositive curvature, it seems difficult to get desirable bounds. However, under our assumption of constant curvature, we can compute φ st and its derivatives explicitly. Without loss of generality, we may assume that (M, g) is a compact 2-dimensional Riemannian manifold with constant curvature equal to. It is well known that the universal cover of any 2-dimensional manifolds with constant negative curvature is the hyperbolic plane H 2. We consider the Poincaré half-plane model H 2 = {(x, y) R 2 : y > 0}, with the metric given by ds 2 = y 2 (dx 2 + dy 2 ). Recall that the distance function for the Poincaré half-plane model is given by ( ) dist((x, y ), (x 2, y 2 )) = arcosh + (x 2 x ) 2 + (y 2 y ) 2, 2y y 2 where arcosh is the inverse hyperbolic cosine function arcosh(x) = ln(x + x 2 ), x. Moreover, the geodesics are the straight vertical rays orthogonal to the x-axis and the half-circles whose origins are on the x-axis. Any pair of geodesics can intersect at at most one point. Without loss of generality, we may assume that γ is the y-axis. There are three possibilities for the image α( γ). It can be a straight line parallel to γ, a half-circle parallel to γ, or a half-circle intersecting γ at one point. We need to treat these cases separately. Let γ(t) = (0, e t ), t R, be the infinite geodesic parameterized by arclength. Our unit geodesic segment is given by γ(t), t [0, ]. Then its image α( γ(s)), s [0, ], is a unit geodesic segment of α( γ). Lemma 5. If α / Γ TR ( γ) and α( γ) γ =, then we have and inf φ st e CT, φ st + φ stt + φ sttt e CT,

18 8 YAKUN XI AND CHENG ZHANG where C > 0 is independent of T. The infimum and the norm are taken on the unit square {(t, s) R 2 : t, s [0, ]}. Lemma 6. Let α / Γ TR ( γ) and α( γ) is a half-circle intersecting γ at the point (0, e t 0 ), t 0 R. If t 0 / [, 2], then the intersection point (0, e t 0 ) is outside some neighbourhood of the unit geodesic segment { γ(t) : t [, 2]}. We have inf φ st e CT, and φ st + φ stt + φ sttt e CT, where C > 0 is independent of T. On the other hand, if t 0 [, 2], we have inf φ st/(t t 0 ) e CT, and φ st + φ stt + φ sttt e CT, where C > 0 is independent of T. The infima and the norms are taken on the unit square {(t, s) R 2 : t, s [0, ]}. We shall postpone the proof of Lemma 5 and Lemma 6 to the last section. Now we see first how to finish the proof of Theorem 2 using Lemma 5 and Lemma 6. Proof of Theorem 2. By (3.2) and (3.8), we only need to show that (3.2) S osc λ h L 4 ([0,]) e CT λ 3 8 h L 4 3 ([0,]), where C is independent of T. Let α / Γ TR ( γ). If α( γ) γ =, by Proposition 2, Lemma 5 and the condition on the amplitude (3.7), we have S osc λ,αh L 2 ([0,]) e CT h L 2 ([0,]). Assume that α( γ) intersects γ at the point γ(t 0 ). Since α / Γ TR ( γ), the intersection point cannot lie on the unit geodesic segment α( γ(s)), s [0, ]. Thus, by Proposition 2, Lemma 6 and (3.7) we obtain S osc λ,αh L 2 ([0,]) e CT h L 2 ([0,]), if t 0 / [, 2], and Sλ,αh osc L 2 ([0,]) e CT λ 4 h L 2 ([0,]), if t 0 [, 2], where we set F = [, 2] [0, ] in Proposition 2 for the case t 0 [, 2].

19 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 9 Recall that the number of nonzero summands in Sλ osc λ > we always have S osc λ h L 2 ([0,]) e CT λ 4 h L 2 ([0,]). is O(e CT ). Consequently, for By interpolating with the trivial L L bound, we obtain (3.2), finishing the proof Proof of Lemmas. Before proving the lemmas, we remark that in the Poincaré half-plane model T R ( γ) = {(x, y) R 2 : y > 0 and y x / (coshr) 2 }. Indeed, the distance between (0, e t ) and (x, y), y > 0, is f(t) = arcosh ( + x2 + (y e t ) 2 ) ( x 2 + y 2 + e 2t ) = arcosh. 2ye t 2ye t Setting f (t) = 0 gives t = ln x 2 + y 2, which must be the only minimum point. Thus the distance between (x, y) and the infinite geodesic γ is dist((x, y), γ) = arcosh( + (x/y) 2 ). Since dist((x, y), γ) R in T R ( γ), it follows that y x / (coshr) 2. From now on, we shall always parametrize γ and α( γ) by arc-length, denoted by γ (t) and γ 2 (s) respectively. The explicit expressions for the corresponding segments that we concern will be given in the proof case by case. Proof of Lemma 5. Note that in this case, the image α( γ) = γ 2 can be either a straight line or a half-circle parallel to γ = γ. We treat these two cases separately. Let γ (t) = (0, e t ), t [0, ], γ 2 (s) = (a, e s ) be the two unit geodesic segments, where a R and s is in some unit closed interval of R. See Figure. The distance function is φ(t, s) = dist(γ (t), γ 2 (s)) = arcosh ( + a2 + (e s e t ) 2 ) ( a 2 + e 2t + e 2s ) = arcosh. 2e s+t 2e t+s Then we have 8e 2s+2t a 2 φ st(t, s) = ((a 2 + e 2t + e 2s ) 2 4e 2s+2t ). 3/2 One can obtain this expression by direct computations or refer to the proof of (3.6). By (3.6), we have φ T. Thus a 2 e t s + e t s + e s t 2coshT, which gives s [ T, T + ] and a Ce T. Here C is independent of T.

20 20 YAKUN XI AND CHENG ZHANG Figure. α( γ) is a line parallel to γ. To get the lower bound of φ st, we need to use the condition that α / Γ TR ( γ). We claim that (3.3) α / Γ TR ( γ) a Ce T, where C is independent of T. Note that if the segment γ 2 (s), s [ T, T + ] is completely included in T R ( γ), then we must have α Γ TR ( γ), meaning that which implies our claim. Consequently, e T a (coshr) 2 α Γ TR ( γ), φ st C e 2T e 2T e 6T = Ce 0T. This gives the lower bound of φ st. Moreover, direct computations give φ stt = 6a2 e 2s+2t ((a 2 + e 2s ) 2 + e 2s+2t a 2 e 2t 2e 4t ) ((a 2 + e 2t + e 2s ) 2 4e 2s+2t ) 5/2, φ sttt = 32a2 e 2s+2t ((a 2 + e 2s ) 4 + lower order terms) ((a 2 + e 2t + e 2s ) 2 4e 2s+2t ) 7/2. Here the lower order terms in the bracket are lower order in the sense of the multivariate polynomial degree in terms of a and e s.

21 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 2 The upper bounds can be estimated similarly. By (3.6), we have φ 2, namely a 2 + e 2t + e 2s 2(cosh2)e t e s. Thus So we have Thus and a 2 + e 2t + e 2s 2e s+t (2cosh2 2)e t e s Ce T. (a 2 + e 2t + e 2s ) 2 4e 2s+2t Ce 2T. φ st C e2t e 2T e 3T Ce 7T. φ stt C e2t e 2T e 4T e 5T Ce 3T, φ sttt C e2t e 2T e 8T e 7T Ce 9T. This completes the proof of the first case. Now we turn to the case when γ 2 is a half-circle centered at (a, 0) with radius r > 0. See Figure 2. Let γ (t) = (0, e t ), t [0, ], γ 2 (s) = (a + r e2s, 2res ) be two +e 2s +e 2s unit geodesic segments, where a r > 0 and s is in some unit closed interval of R. Without loss of generality, we may only consider the case a r > 0. Then the distance function is ( A ) (3.4) φ(t, s) = dist(γ (t), γ 2 (s)) = arcosh, 4re s+t where (3.5) A = e 2s+2t + (a r) 2 e 2s + e 2t + (a + r) 2. Thus we have (3.6) φ st = 6re2s+2t (a + r + (a r)e 2s )(a 2 r 2 + e 2t ) (A 2 6r 2 e 2s+2t ) 3/2. To see this, we write e s+t coshφ = A 4r. Taking derivatives on both sides, we obtain (3.7) (φ t + φ s + φ ts)sinhφ + ( + φ tφ s)coshφ = e s+t /r. Denote X = e s+t, Y = (a r) 2 e s t, Z = e t s, and W = (a + r) 2 e s t. Since taking derivatives gives 4rcoshφ = X + Y + Z + W, 4rφ tsinhφ = X Y + Z W, 4rφ ssinhφ = X + Y Z W.

22 22 YAKUN XI AND CHENG ZHANG Figure 2. α( γ) is a half-circle parallel to γ. Then we multiply both sides of (3.7) by 4r 2 (sinhφ) 2 and use the hyperbolic trigonometric identity (sinhφ) 2 = (coshφ) 2 to obtain 4r 2 (sinhφ) 3 φ st = (a r)(x+w )+(a+r)(y +Z) = e s t (a+r+(a r)e 2s )(a 2 r 2 +e 2t ). This gives our desired expression (3.6). Again by (3.6), we get φ T. Namely, (3.8) (e 2t + (a r) 2 )e 2s 4r(coshT )e t e s + e 2t + (a + r) 2 0, which implies r (3.9) 4coshT es 4rcoshT. Moreover, note that if we view the left hand side of (3.8) as a quadratic polynomial in terms of e s, then the discriminant has to be nonnegative: we obtain that (3.20) 6r 2 (cosht ) 2 e 2t 4(e 2t + (a r) 2 )(e 2t + (a + r) 2 ) 0, a r 2ecoshT, (3.2) a r 2ecoshT,

23 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 23 and (3.22) r 2coshT. To get the lower bound of φ st, we need to use the condition that α / Γ TR ( γ). We claim that there exists some constant C independent of T such that (3.23) α / Γ TR ( γ) r CcoshT or a r CcoshT. Indeed, we shall prove the contrapositive: (3.24) r CcoshT and a r CcoshT α Γ T R ( γ). We obtain this by showing that under the above assumptions on r and a r, the segment γ 2 (s), s [ ln(4r cosht ), ln(4rcosht )] is completely included in T R ( γ), which implies α Γ TR ( γ). By solving the polynomial system { y = x / (coshr) 2 (x a) 2 + y 2 = r 2 and recalling that x = a + r e2s + e, 2s we can see that {γ 2 (s) : s R} T R ( γ) (3.25) = {γ 2 (s) : (a r) 2 e 4s + 2(a 2 (2(coshR) 2 )r 2 )e 2s + (a + r) 2 0}. Note that our assumptions imply a/r coshr, namely { r CcoshT a r (CcoshT ) a/r + (CcoshT ) 2 coshr. Thus in the case when a r, the RHS of (3.25) becomes (3.26) {γ 2 (s) : u e 2s u + }, where (3.27) u ± = (2(coshR)2 ) (a/r) 2 ± ((coshr) 2 (a/r) 2 )((coshr) 2 ) (a/r ) 2. It is easy to see that (3.28) u ((a/r) 2 ) 2 (a/r ) 2 (2(coshR) 2 (a/r) 2 ) (a/r + )2 (coshr) 2 coshr + coshr,

24 24 YAKUN XI AND CHENG ZHANG (3.29) u + (2(coshR)2 ) (a/r) 2 (a/r ) 2 (coshr)2 (a/r ) 2. So under our assumptions r CcoshT and a r, if we choose C = CcoshT 4 coshr + / coshr, we see that (3.30) { { r CcoshT u r 2 (4coshT ) 2 a r and α Γ a r (CcoshT ) u+ (4rcoshT ) 2 TR ( γ). In the easier case a = r, we have u + = +. Consequently, we obtain { r CcoshT a r (CcoshT ) α Γ TR ( γ). This finishes the proof of our claim. We note that by φ T, ( ) 2 A (3.3) φ st φ st a + r + (a r)e2s a 2 r 2 + e 2t. 4re s+t cosht (cosht ) 2 ra Remark 2. Since we have not used the assumption that a r so far, (3.4)-(3.3) are applicable to the case a < r in the proof of Lemma 6. Therefore, by (3.23) we have to consider two cases and estimate φ st respectively. By (3.3), we only need to estimate the upper bound of A and the lower bound of the numerator in (3.3). (I) Assume r CcoshT. If a r, then by (3.9)-(3.22), A is bounded by Cr 2 (a r) 2 (cosht ) 2. And the numerator in (3.3) is bounded below by (a r)e 2s (a 2 r 2 ). So we obtain φ st C (a + r)(a r) 2 r 2 (cosht ) 2 (cosht ) 2 r(r 2 (a r) 2 (cosht ) 2 ) If 0 a r, then similarly we have C a + r φ st (cosht ) 2 r(r 2 (cosht ) 2 ) Ce 6T. (II) Assume a r first case. CcoshT If a r, then using (3.9)-(3.22) we get φ st Ce 6T.. We may assume r, otherwise it is reduced to the C (a + r)(a r) 2 r 2 (cosht ) 2 (cosht ) 2 r((a r) 2 r 2 (cosht ) 2 ) Ce 6T.

25 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 25 If 0 a r then similarly we obtain φ st C (a + r)(a r) 2 r 2 (cosht ) 2 (cosht ) 2 r(r 2 (cosht ) 2 ) Ce 8T. Note that the constant C is independent of T. Hence we finish the proof of the lower bound of φ st. The upper bounds can be obtained in a similar fashion. By the expression (3.6), direct computations give (3.32) φ stt = 32re2s+2t (a + r + (a r)e 2s )((a + r)(a r) 5 e 4s + lower order terms), (A 2 6r 2 e 2s+2t ) 5/2 (3.33) φ sttt = 64re2s+2t (a + r + (a r)e 2s )((a + r)(a r) 9 e 8s + lower order terms) (A 2 6r 2 e 2s+2t ) 7/2. Here again the lower order terms in the bracket are lower order in the sense of the multivariate polynomial degree in terms of a, r and e s. By (3.32)-(3.33), we only need to estimate the lower bound of A 2 6r 2 e 2s+2t and the upper bounds of the absolute values of the numerators. By (3.6), we have φ 2, namely A 4(cosh2)re s+t. Thus (3.34) A 4re s+t (4cosh2 4)re t e s. (I) Assume r CcoshT. Using (3.9)-(3.22) and (3.34), we get which implies Then by (3.9)-(3.2), A 4re s+t (4cosh2 4)re t e s C(coshT ) 3, A 2 6r 2 e 2s+2t C(coshT ) 6. φ st C(coshT )(cosht )4 (cosht ) 5 (cosht ) 2 (cosht ) 9 Ce 2T, φ stt C(coshT )(cosht )4 (cosht ) 5 (cosht ) 4 (cosht ) 5 Ce 39T, φ stt C(coshT )(cosht )4 (cosht ) 5 (cosht ) 26 (cosht ) 2 Ce 57T. (II) Assume r CcoshT. By (3.9) and (3.34), we have A 4re s+t (4cosh2 4)re t e s Cr 2 (cosht ),

26 26 YAKUN XI AND CHENG ZHANG which implies Thus by (3.9)-(3.2), A 2 6r 2 e 2s+2t Cr 4 (cosht ) 2. φ st Cr3 (cosht ) 2 (r 2 (cosht ) 3 )(rcosht ) ((cosht ) 2 r 4 ) 3/2 Ce 9T, φ stt Cr3 (cosht ) 2 (r 2 (cosht ) 3 )(r 5 (cosht ) 9 ) ((cosht ) 2 r 4 ) 5/2 Ce 9T, φ sttt Cr3 (cosht ) 2 (r 2 (cosht ) 3 )(r 9 (cosht ) 7 ) ((cosht ) 2 r 4 ) 7/2 Ce 29T. Since the constant C is independent of T, the proof is complete. Remark 3. Since we did not use the assumption that a r in the proof of the upper bounds of various derivatives, these upper bounds are also valid for the case a < r in Lemma 6. Remark 4. Indeed, the upper bounds for the derivatives hold for not only α / Γ TR ( γ) but all α Id, as we only use the condition that 2 φ T. Proof of Lemma 6. Let γ (t) = (0, e t ), t [0, ], γ 2 (s) = (a + r e2s +e 2s, 2res +e 2s ) be two unit geodesic segments, where r > a 0 and s is in some unit closed interval of R. Without loss of generality, we may only consider the case r > a 0. The expressions of the distance function φ and its derivatives are the same as in (3.4)- (3.6), (3.32) and (3.33). Moreover, from Remark 2 we can see that (3.9)-(3.3) are also applicable. So the zero set of φ st(t, s) is { (t, s) R 2 : t = t 0 or s = s 0, where e 2t 0 = r 2 a 2 and e 2s 0 = r + a r a It is not difficult to check that the point p = γ (t 0 ) = γ 2 (s 0 ) is the intersection point of the two infinite geodesics γ and γ 2. If the unit geodesic segment γ 2 (s) passes through the intersection point, then this segment must be contained in T R ( γ), thus our geodesic segment γ 2 (s) cannot pass through the point p. We need to consider the following two cases: (I) t 0 [, 2], (II) t 0 / [, 2]. By Remark 3, we only need to consider the lower bounds of φ st or φ st/(t t 0 ). (I) Assume t 0 [, 2]. See Figure 3. By Remark 2, we also have the claim (3.23). Since t 0 [, 2], we have (3.35) e 2t 0 = (r + a)(r a) [e 2, e 4 ], }.

27 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 27 Figure 3. t 0 [, 2] which implies that the two inequalities in (3.23), r CcoshT and r a CcoshT, are equivalent. By (3.26), α / Γ TR ( γ) implies e 2s > u + or e 2s < u. If e 2s > u +, by (3.29) we have (r a)e 2s (r + a) (coshr)2 ( a/r) 2 (r a) (r + a) = (coshr)2 2 + (a/r) 2 r ((coshr) 2 2)r. a/r If e 2s < u, similarly by (3.28), we have (r + a) (r a)e 2s (a/r + )2 (r + a) (r a) (coshr) 2 = (coshr)2 2 + (a/r) 2 ( + a/r)r (coshr)2 2 (coshr) 2 (coshr) 2 r. Hence for some constant C independent of T, we always have (3.36) a + r + (a r)e 2s Cr. By (3.35) and (3.23), we see that e r CcoshT and r a e 2. We get A Cr 2 (cosht ) 2.

28 28 YAKUN XI AND CHENG ZHANG Figure 4. t 0 [, 2] Thus by (3.3), for t [, 2] \ {t 0 } φ st t t 0 Cr e 2t r 2 + a 2 (cosht ) 2 r(r 2 (cosht ) 2 ) t t 0 Ce 6T, where we used the fact e 2t 0 = r 2 a 2 and the mean value theorem. This lower bound also implies φ stt(t 0, s) 0, and φ ts 0 for t [, 2] \ {t 0 }. (II) Assume t 0 / [, 2]. See Figure 4. By Remark 2, the claim (3.23) is also applicable here. By (3.2), we have A Cr 2 (cosht ) 4. Since t 0 / [, 2], e 2t r 2 + a 2 = e 2t e 2t 0 e 2. If r C(coshT ) 4, by (3.3) and (3.36), we have If r a CcoshT φ st Cr (cosht ) 2 r(r 2 (cosht ) 4 ) Ce 4T. and r (cosht )4, then (r a)e 2s (a + r) Cr 2 (cosht ) 3 2r Cr 2 (cosht ) 3, r 2 a 2 e 2t Cr(coshT ) e 2 Cr(coshT ).

29 IMPROVED CRITICAL EIGENFUNCTION RESTRICTION ESTIMATES IN 2-D 29 Thus by (3.3) we get which completes our proof. φ st C(r2 (cosht ) 3 )(r(cosht ) ) (cosht ) 2 r(r 2 (cosht ) 4 ) Ce 0T, Remark 5. As pointed out in [3], the various upper bounds of pure derivatives D α t φ + D α s φ C α e CT follow from Proposition 3 and Lemma 4 in [2]. But it seems that the upper bounds for mixed derivatives are unknown. So we are including the proofs for these upper bounds for the sake of completeness. 4. Acknowledgement We would like to thank Professor C. Sogge for his guidance and patient discussions during this study. This paper would not have been possible without his generous support. It s our pleasure to thank our colleagues C. Antelope, D. Ginsberg, and X. Wang for going through an early draft of this paper. References [] J. G. Bak and A. Seeger. Extensions of the Stein-Tomas theorem. Math. Res. Lett., 8(4):767 78, 20. [2] P. H. Bérard. On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z., 55(3): , 977. [3] M. D. Blair and C. D. Sogge. Concerning Toponogov s Theorem and logarithmic improvement of estimates of eigenfunctions. Preprint. [4] J. Bourgain. Besicovitch type maximal operators and applications to Fourier analysis. Geom. Funct. Anal., 2:45 87, 99. [5] N. Burq, P. Gérard, and N. Tzvetkov. Restriction of the Laplace-Beltrami eigenfunctions to submanifolds. Duke Math. J., 38: , [6] X. Chen. An improvement on eigenfunction restriction estimates for compact boundaryless Riemannian manifolds with nonpositive sectional curvature. Trans. Amer. Math. Soc., 367: , 205. [7] X. Chen and C. D. Sogge. A few endpoint geodesic restriction estimate for eigenfunctions. Comm. Math. Phys., 329(3): , 204. [8] A. Hassell and M. Tacy. Improvement of eigenfunction estimates on manifolds of non-positive curvature. Forum Math., 27:435 45, 205. [9] R. Hu. L p norm estimates of eigenfunctions restricted to submanifolds. Forum Math., 6:02 052, [0] C. D. Sogge. Concerning the L p norm of spectral cluster of second-order elliptic operators on compact manifolds. J. Funct. Anal, 77:23 38, 988. [] C. D. Sogge. Fourier integrals in classical analysis, volume 05 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 993. [2] C. D. Sogge. Hangzhou lectures on eigenfunctions of the Laplacian, volume 88 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 204.

30 30 YAKUN XI AND CHENG ZHANG [3] C. D. Sogge. Improved critical eigenfunction estimates on manifolds of nonpositive curvature. Preprint. [4] C. D. Sogge. Problems related to the concentration of eigenfunctions. Preprint. [5] C. D. Sogge and S. Zelditch. On eigenfunction restriction estimates and L 4 -bounds for compact surfaces with nonpositive curvature. In Advances in Anlysis: The Legacy of Elias M. Stein, Priceton Mathematical Series, pages Princeton University Press, 204. Department of Mathematics, Johns Hopkins University, Baltimore, MD 228, USA address: ykxi@math.jhu.edu, czhang67@math.jhu.edu