Research Statement. Yakun Xi

Transcription

1 Research Statement Yakun Xi 1 Introduction My research interests lie in harmonic and geometric analysis, and in particular, the Kakeya- Nikodym family of problems and eigenfunction estimates on compact manifolds. In Section 2, I will describe my work on the Kakeya-Nikodym problems [36]. I have obtained a new proof of Wolff s L (d+2)/2 bounds [35] without using the induction on scales argument. As an application of my new proof, I generalized Sogge s result for Nikodym maximal function on 3-dimensional manifolds with constant curvature to any dimension d higher than 3. Section 3 is devoted to my work on eigenfunction estimates. In my joint work with Miao, Sogge and Yang [23], we obtained an improvement of the bilinear estimates of Burq, Gérard and Tzvetkov [9] in the spirit of the refined Kakeya-Nikodym estimates of Blair and Sogge [5]. Riemannian surfaces with nonpositive curvature serve as an important model case in the study of eigenfunctions. In my recent work [37], together with Zhang, we proved improved L 4 geodesic restriction estimates for eigenfunctions in this setting. Joint with Sogge and Zhang [30], we studied the geodesic period integral of eigenfunctions, and obtained an improved bound under certain curvature assumptions. Finally, in Section 4, I will discuss my future research plans. 2 Kakeya-Nikodym type maximal operators The original Kakeya problem, proposed by Kakeya [20] in 1917, is to determine the minimal area needed to continuously rotate a unit line segment in the plane by 180 degrees. In 1928, Besicovitch [2] showed that such sets may have arbitrarily small measure. Moreover, Besicovitch s work indicates the existence of measure zero subsets of R d which contain a unit line segment in every direction. Such sets are called Besicovitch sets or Kakeya sets. It was later found that Kakeya sets are closely related to many fundamental problems in harmonic analysis. Fefferman [15] was the first to apply the construction of measure zero Kakeya sets to a problem of Fourier transform, namely the ball multiplier problem. It was also discovered that many problems in analysis require more detailed information about the size of Kakeya sets, and in paticular, their fractal dimension. The Kakeya set conjecture asserts that even though the measure of a Kakeya set can be zero, it still needs to be large in the sense of fractal dimension. Conjecture 1 (Kakeya Set Conjecture). Kakeya sets in R d must have full Hausdorff/ Minkowski dimension. There is also a stronger formulation of the conjecture in terms of maximal functions, which is called the maximal Kakeya conjecture, or the Kakeya maximal function conjecture. Conjecture 2 (Kakeya Maximal Function Conjecture). For any 0 < δ < 1, given ɛ > 0, there exists a constant C ɛ such that f δ L d (S d 1 ) C ɛ δ ɛ f L d (R d ). (2.1) Here fδ : Sd 1 R is the Kakeya maximal function defined by: fδ 1 (ξ) = sup a R d Tξ δ(a) f(y) dy, Tξ δ(a) 1

2 where T δ ξ (a) is an 1 δ δ tube centered at a Rd with direction ξ S d 1. Interpolating between (2.1) and the trivial L 1 L estimate, one sees that natural partial result to Conjecture 2 would be the following estimate: f δ L q (S d 1 ) C ɛ δ 1 d p ɛ f L p (R d ), (2.2) where 1 < p < d, and q = (d 1)p are fixed. Indeed, it is well-known that an estimate like (2.2) for a given p would imply that Kakeya sets have Hausdorff/Minkowski dimension at least p. For the case d = p = 2, Conjecture 2 was fully solved by Córdoba [13]. However, it is still open for any d 3. When p = (d + 1)/2, q = (d 1)p = d + 1, (2.2) follows from Drury s work on X-ray transformations [14] in In 1991, Bourgain [7] improved this result for each d 3 to some p(d) ((d+1)/2, (d+2)/2) by the so-called bush argument. Bourgain studied the bush structure where a large number of tubes intersect at a given point. Four years later, Wolff [35] generalized Bourgain s bush argument to the more refined hairbrush argument, by considering tubes with lots of bushes on them. Combining the hairbrush argument and an induction on scales argument, Wolff showed that (2.2) holds for all d 3, p = (d + 2)/2. Wolff s result is still the best result for Conjecture 2 when d 8. Imporved bounds have been proven in the higher dimensional cases, and for the weaker Conjecture 1 in lower dimensional cases, see e.g. [8], [21], [22]. The induction on scales argument introduced by Wolff has been an essential technique for proving such maximal inequalities. To be more specific, one can reformulate Conjecture 2 by looking at the corresponding restricted weak type bound and discretization. Conjecture 3 (Maximal Kakeya Conjecture, discrete version). Let 0 < δ, λ < 1, 1 p d, and {T 1,..., T M } be a collection of 1 δ... δ tubes oriented in a δ-separated set of directions. For each 1 i M, let E i T i be a set with E i λ T i. Then M E i C ɛ (Mδ d 1 )λ d δ d p+ɛ. i=1 Remark: The Minkowski dimension version of Conjecture 1 corresponds to the case where λ = 1, and the Hausdorff dimension version essentially corresponds to the case where λ 1/(log 2 (1/δ)). Thus, while the Kakeya set conjecture is concerned with how small one can make unions of tubes T i, Conjecture 2 is concerned with how small one can make unions of density λ (possibly very small) portions E i of tubes T i. Wolff s induction on scales argument is also often called the two-ends reduction (see [33] for a detailed discussion), because it allows one to avoid the situations where each E i is concentrated only in some small portion of the tube. That is, by two-ends reduction, it suffices to only consider portions E i which occupy both ends of the tube in some sense. This reduction exploits the approximate scale-invariance of the Euclidean Kakeya problem, and has become a standard technique in similar problems. It is tempting to remove such a technical argument. In 1999, Sogge [27] managed to avoid the two-ends reduction in his work on the closely related Nikodym maximal functions in 3-dimensional manifolds with constant curvature. Sogge s idea was to use a modified hairbrush argument and an optimal bound for an auxiliary maximal function. Following Sogge s idea, in 2014, Miao, Yang and Zheng [24] were able to recover Wolff s result for Kakeya maximal functions in R 3 without the use of the two-ends reduction. In fact, Miao, Yang and Zheng also tried to recover Wolff s results for all dimension d 3, but it seemed impossible to extend the same argument to higher dimension d > 3, due to the fact that their auxiliary maximal function bound involves a δ (d 3)/2 loss. 2

3 The first part of my recent work [36] addresses this problem. By using a more natural auxiliary maximal function and taking into account certain geometric observations, I obtained an optimal auxiliary maximal function bound. This leads to a new proof of Wolff s Kakeya maximal function bounds for all dimension d 3, without the use of the induction on scales argument. Theorem 1 (Xi [36]). It can be shown without the induction on scales argument that the Kakeya maximal function in R d satisfies fδ 2d 1 L (d 1)(d 2) C ɛ δ d+2 ɛ f d (S d 1 d+2. (2.3) ) L 2 (R d ) This new proof shows that Wolff s L (d+2)/2 bounds of the Kakeya maximal function follows directly from some geometric combinatorics and Córdoba s optimal bounds for the 2-dimensional case. On one hand, it opens up a new route to get Wolff s bounds where different values of λ and different dimensions can be handled in the same way. On the other hand, since we now know how to avoid the rescaling argument, it is easier to apply similar ideas to the non-euclidean case following arguments in [27]. As mentioned above, Nikodym problems are close cousins to the Kakeya problems. The Nikodym set problem is concerned with the fractal dimension of the so-called Nikodym sets. Similar to the Kakeya problems, the conjectured dimension bound for the Nikodym sets follows from a L d L d bound for the corresponding Nikodym maximal function. Recall that the Nikodym maximal function fδ in R d is defined by: fδ (x) = sup Tγ δ f(y) dy, (2.4) x x γ x 1 where γ x denotes any unit line segment that contains the point x. Correspondingly, we have the Nikodym maximal function conjecture. Conjecture 4 (Nikodym Maximal Function Conjecture). For any 0 < δ < 1, given ɛ > 0 then there exists a constant C ɛ such that T δ γx f δ L d (R d ) C ɛ δ ɛ f L d (R d ). (2.5) Wolff s hairbrush argument [35] applies equally well to the Nikodym maximal function, so we have a similar bound: fδ 2d 1 L (d 1)(d 2) C ɛ δ d+2 ɛ f d (R d d+2. (2.6) ) L 2 (R d ) Indeed, Tao [32] showed that Kakeya maximal function conjecture is equivalent to Nikodym maximal function conjecture in Euclidean space, and furthermore, any bound like (2.2) is equivalent to the corresponding bound for the Nikodym problem. Even though Kakeya problems and Nikodym problems are equivalent in Euclidean spaces, Kakeya problems are not natural on general manifolds since there is no unique way to identify directions at different points on a general manifold. However, we can naturally extend the definition of the Nikodym maximal function (3.3) to any Riemannian manifold (M, g), by replacing γ x by any geodesic segment that contains x with length α < min{1, 1 2Inj(M)} fixed. In 1997, Minicozzi and Sogge [25] were the first to study the Nikodym maximal functions on general manifolds. By using a modified bush argument, they showed that for a general manifold Drury s bounds for p = (d + 1)/2 still hold. On the other hand, they noticed that Bourgain and Wolff s arguments relied heavily on reducing to lower dimensional subspaces. So, to extend these 3

4 arguments to a manifold, one would need the existence of many totally geodesic submanifolds. Unfortunately, for generic manifolds, this is rarely the case. Minicozzi and Sogge were able to build counter-examples by exploiting this fact. They showed that for each d there exists a manifold (M d, g), such that this estimate breaks down if p > (d + 1)/2. In other words, Drury s result is the best possible at least for odd dimensional manifolds. Clearly, if one wants to generalize Wolff s Hairbrush argument to manifolds, some additional assumptions are needed. In the later work of Sogge on Nikodym sets in 3-dimensional manifolds [27], Sogge noticed that if M has constant curvature, all 2-planes in the geodesic normal coordinates about a point are totally geodesic. So it seemed possible to generalize Wolff s hairbrush argument to manifolds with constant curvature. However, there was one obstacle on the way. The induction on scales argument Wolff had used seemed hard to generalize to the non-euclidean setting. By introducing a weighted auxiliary maximal function and a more precise multiplicity argument, Sogge was able to avoid the induction on scales argument and was able to prove the L 5/2 -bounds for the Nikodym maximal function in the 3-dimensional constant curvature case. As an application of the proof for Theorem 1, it is shown in the second part of my work [36] that if one works with a more natural auxiliary maximal function, Sogge s idea for 3-dimensional manifolds with constant curvature actually works for all dimensions d 3. Theorem 2 (Xi [36]). For any d 3, assume that (M d, g) has constant curvature. Then for f supported in a compact subset K of a coordinate patch and all ɛ > 0, fδ 2d 1 L (d 1)(d+2) C ɛ δ d+2 ɛ f d (M d d+2. (2.7) ) L 2 (M d ) We remark that just like the Kakeya problem in Euclidean space, the Nikodym problem here is a local problem, so Theorem 2 implies the more general case (without support assumption on f). Thus it is easy to see that Theorem 2 implies Wolff s result for Nikodym maximal function in R d [35] as a special 0 curvature case. Also, Theorem 2 generalizes Sogge s [27] result for 3-dimensional manifolds to any dimension higher than 3. 3 Eigenfunction estimates The investigation of the eigenfunctions of the Laplace-Beltrami operator on Riemannian manifolds has been an ongoing endeavor for over one hundred years, and remains a central area in both mathematics and physics. Studying various types of concentration exhibited by eigenfunctions is essential in the development of this mathematical theory. Let e λ denote the L 2 -normalized eigenfunction on a compact boundaryless manifold, g e λ = λ 2 e λ, so that λ is the eigenvalue of the first order operator g. It is a classical result of Sogge [26] that the L p norms of the eigenfunctions satisfy e λ L p (M) λ σ(p) e λ L 2 (M), (3.1) where 2 p and σ(p) is given by { ( n 1 1 σ(p) = max ) ( 1, n p 2 1 ) 1 }. p 2 4

5 The L p bounds can also be written as {λ n 1 2 ( p ) e λ e λ L p (M) L 2 (M), 2 p 2(n+1) n 1, λ n( p ) 1 2(n+1) 2 e λ L 2 (M), n 1 p. Although the above estimates are sharp for the round sphere S n, it is expected that one should be able to improve it for generic Riemannian manifolds. 3.1 Kakeya-Nikodym norms The Kakeya-Nikodym norm introduced by Sogge [28] serves as the crucial tool to get improved L p bounds for small p. Given λ > 0, the Kakeya-Nikodym norm of a function f is given by f KN(λ) = sup f L 2 (T γ Π λ 1/2 (γ)), where Π denotes the space of unit length geodesics and T λ 1/2(γ)) is the λ 1/2 -tube about γ. Remark: One can see that here f KN(λ) equals a multiple of the L norm of the Nikodym maximal function, ( f 2 ) δ L, when δ = λ 1/2. However, L norm is the natural one that is relevant in this case, so one cannot get any improvement using bounds like (2.7). Various connections between the L p norm and the Kakeya-Nikodym norm of the eigenfunctions have been established by Sogge in [28], and Blair and Sogge in [5], [4]. For instance, in the twodimensional case, Sogge [28] showed that (3.2) e λ L 4 (M) λ 1 8 eλ 1 4 KN(λ) e λ 3 4 L 2 (M), (3.3) while in [4], Blair and Sogge proved an optimal version in terms of the power of e λ KN(λ), but with a λ ɛ loss e λ L 4 (M) C ɛ λ 1 8 +ɛ e λ 1 2 KN(λ) e λ 1 2 L, ɛ > 0. (3.4) 2 (M) Therefore, any improvement for the Kakeya-Nikodym norm would automatically yields improvement for L 4 norm, which in turn gives improvement for all L p norms for 2 < p < 6 via interpolation. Some recent work of Sogge and Blair gives logarithmic improvement for L p norms with small p following this route. On the other hand, a result of Burq, Gérard and Tzvetkov [9] generalized Sogge s L p estimates for the special case p = 4 in the two-dimensional case. They proved the following bilinear estimate. e λ e µ L 2 (M) λ 1 4 eλ L 2 (M) e µ L 2 (M), (3.5) where λ µ. One easily sees that Sogge s L 4 -estimate can be recovered from (3.5) by taking λ = µ. It then is a natural question to ask whether one can combine (3.3) and (3.5) to get a bilinear Kakeya-Nikodym bound. In a recent joint work [23] with Miao, Sogge and Yang, we were able to address this question with a positive answer. Theorem 3 (Miao, Sogge, Xi and Yang [23]). Assume 0 < λ µ and e λ, e µ are two eigenfunctions of g associated to the frequencies λ and µ respectively. Then for every 0 < ɛ 1 2, there is a C ɛ > 0 such that e λ e µ L 2 (M) C ɛ λ 1 4 +ɛ e µ L 2 (M) e λ KN(λ), (3.6) and e λ e µ L 2 (M) C ɛ λ 1 4 +ɛ e λ L 2 (M) e µ KN(λ), (3.7) 5

6 Note that it has to be the geodesic tubes corresponding to the lower frequency that accounts for the above bound of e λ e µ 2. That is, one cannot take the KN(µ)-norm of e µ on the right hand side of (3.7). To see this, consider the Torus T d = ( π, π] d, and take the eigenfunctions e λ = e ij x, j = λ, and e µ = e ik x, k = µ, then the analog of (3.7) involving e µ KN(µ) is obviously false if µ λ. 3.2 Geodesic restriction estimates Another active area of eigenfunction study is for manifolds with nonpositive curvature. In this setting, the eigenfunctions are conjectured to be distributed more and more evenly as the frequency λ. The L p norms of eigenfunctions are thus expected to satisfy much better bounds than those in (3.1). It is a classical result of Bérard that one can get log improvements over (3.1) assuming nonpositive curvature, that is e λ L (M) = O(λ n 1 2 / log λ), which gives an improvement over (3.1) for p > p c via interpolation. Recently, log improvements over (3.1) for 2 < p < p c and p = p c were obtained by Blair-Sogge [6] and Sogge [29] respectively, essentially by proving log improved Kakeya-Nikodym bounds using global harmonic analysis. In the last decade, similar L p estimates have been established for the restriction of eigenfunctions to geodesics. Burq, Gérard and Tzvetkov [9] and Hu [19] showed that for n-dimensional Riemannian manifold (M, g), if Π denotes the space of all unit-length geodesics γ, then where and ( ) 1 sup e λ p p ds γ Π γ Cλ σ(n,p) e λ L 2 (M), (3.8) { 1 4 σ(2, p) =, 2 p 4, p, 4 p. (3.9) σ(n, p) = n 1 2 1, if p 2 and n 3, (3.10) p here the case n = 3, p = 2 is due to Chen and Sogge [11]. Note that in the 2-dimensional case, the estimates (3.8) have a similar flavor to Sogge s L p estimates (3.1). Indeed, when n = 2, (3.8) also has a critical exponent p c = 4. Moreover, on the sphere S 2, (3.8) 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 (3.8). 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 (3.8) 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 (3.8) under the assumption of nonpositive curvature in the 2-dimensional case. Bérard s sup-norm estimate [1] provides natural improvements for large p. In [10], Chen managed to improve over (3.8) for all p > 4 by a factor of (log λ) 1 2 : ( ) 1 sup e λ p p ds γ Π γ C λ p e (log λ) 1 λ L 2 (M). (3.11) 2 6

7 Sogge and Zelditch [31] showed that one can improve (3.8) for 2 p < 4, in the sense that ( ) 1 sup e λ p p ds γ Π γ = o(λ 1 4 ). (3.12) A few years later, Chen and Sogge [11] showed that the same conclusion can be drawn for p = 4: ( ) 1 sup e λ 4 4 ds γ Π γ = o(λ 1 4 ). (3.13) (3.13) is the first result to improve an estimate that is saturated by both zonal functions and highest weight spherical harmonics. Recently, by using Toponogov s comparison theorem, Blair and Sogge [6] showed that it is possible to get log improvements for L 2 -restriction: ( ) 1 sup e λ 2 2 ds γ Π γ C λ 1 4 e (log λ) 1 λ L 2 (M), (3.14) 4 In a joint work with Zhang, we obtained further improvements for the L 4 -restriction estimate. Theorem 4 (Xi and Zhang [37]). Let (M, g) be a 2-dimensional compact Riemannian manifold of nonpositive curvature, and let γ M be a fixed unit-length geodesic segment. Then for λ 1, there is a constant C such that In particular, taking f = e λ χ [λ,λ+(log λ) 1 ]f L 4 (γ) Cλ 1 4 (log log λ) 1 8 f L 2 (M). (3.15) Furthermore, if M has constant negative curvature e λ L 4 (γ) Cλ 1 4 (log log λ) 1 8 eλ L 2 (M). (3.16) e λ L 4 (γ) Cλ 1 4 (log λ) 1 2 eλ L 2 (M). (3.17) Remark: Indeed, (3.17) is slightly better than the estimate stated in [37], however, as pointed out to us by Professor Sogge, this can be easily seen by a more careful analysis of the leading coefficient of the Hadamard parametrix. Also, I would like to mention that in a recent posted work of Blair [3], estimates like (3.17) has been generalized to all surfaces with nonpositive curvature. 3.3 Geodesic period integrals The study of geodesic period integrals of eigenfunctions on compact hyperbolic surfaces goes back to Good [16] and Hejhal [18]. They showed that the period integrals of eigenfunctions over closed geodesics is bounded by using Kuznecov formula, that is, e λ ds C γper. (3.18) γ per Indeed, as shown by Chen and Sogge [12], the above bound is valid for any compact Riemannian surfaces without any curvature assumption, and is sharp on the round sphere S 2 and the flat torus T 2. Despite this, it is also shown in [12] that the period integral (3.18) are o(1) as λ, if (M 2, g) has strictly negative curvature. Their proof exploited the fact that, in this case, quadrilaterals 7

8 always have a defect, which provides favorable lower bounds for the derivatives of the corresponding phase function. It is conjectured that the period integral (3.18) would decay like a negative power of λ in the case of strictly negative curvature. In a recent joint work with Sogge and Zhang, some progress has been made in this direction. By using the Gauss-Bonnet formula, we were able to prove more precise estimates for the derivatives of the corresponding phase function, and as a result, we obtained a (log λ) 1 2 decay rate for a much broader class of surfaces. Our main result is the following. Theorem 5 (Sogge, Xi and Zhang [30]). Let (M 2, g) be a closed compact Riemannian surface. Assume that there exist constants δ, N > 0, such that the curvature K of M satisfies B r K dv g δr N, r 1. (3.19) Then if γ(t) is a geodesic in M parametrized by arc length and if b C0 b(t)e λ (γ(t) dt) < C M,b (log λ) 1/2. (( 1/2, 1/2)), we have Moreover, if γ per is a periodic geodesic, we have e λ ds C M γ per (log λ) 1 2. (3.20) γ per Clearly, our assumption (3.19) includes the case of strictly negative curvature, while it also allows the curvature to vanish at an averaged rate of finite type. Our result is the first one to give an explicit decay rate of such period integrals. 4 Research plan in the future In the next few years, I intend to work on the following problems. I) The Kakeya family of problems. Analysis of the structures of the Kakeya sets in R d, d 3 contributes to the basic understanding of Fourier transform in higher dimensions. I am interested in continuing my work on this broad subject. In particular, I would like to study the existence of more detailed geometrical structures of Kakeya sets to get improvements in dimension 3 and 4. As suggested by Tao [34], the stickiness, graininess and planiness of Kakeya sets play important roles here. Also, the polynomial method developped by Guth (see, e.g. [17]) serves as a main new idea in this subject. II) Eigenfunction estimates on manifolds with nonpositive curvature. I would like to continue my study on the concentration of eigenfunctions in the nonpositively curved case. The related problems include but are not limited to: L p estimates, QUE conjecture, and restriction estimates. Sogge s argument [29] shows us how to obtain improvement for the critical L pc norm of eigenfunctions by using existing bounds for the non-critical cases. Meanwhile, there is still room for further improvements, and a more direct approach might yield better result. I would like to investigate the possibility of building a bridge between the critical L 4 restriction estimates and the critical L 6 estimates in 2-D. Indeed, an estimate analogous to (3.3) for the critical exponents would lead to a log improvement for the critical L 6 norm of eigenfunctions. III) Relationship between spectral projection operator estimates and Nikodym maximal function estimates. It is well-known that in Euclidean space, the Fourier restriction problems are closely related to the Kakeya maximal function estimates. Since eigenfunctions of the 8

9 Laplacian provide us with the analogue of the Fourier transform on compact manifolds, eigenfunction estimates and the more general spectral projection operator estimates are in some sense analogous to the Fourier restriction problems. As shown by Sogge [28], L p -estimates for eigenfunction on compact manifolds are related to the L -norm of the Nikodym maximal functions. I am interested in exploring the connection between the mapping properties of the spectral projection operators and the Nikodym maximal function estimates. References [1] P. H. Bérard. On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z., 155(3): , [2] A. S. Besicovitch. On Kakeya s problem and a similar one. Math. Zeitschrift, 27: , [3] M. D. Blair. On logarithmic improvements of critical geodesic restriction bounds in the presence of nonpositive curvature. Preprint. [4] M. D. Blair and C. D. Sogge. Kakeya-Nikodym averages, L p -norms and lower bounds for nodal sets of eigenfunctions in higher dimensions. J. European Math. Soc., 17: , [5] M. D. Blair and C. D. Sogge. Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions. Anal. PDE, 8: , [6] M. D. Blair and C. D. Sogge. Concerning Toponogov s Theorem and logarithmic improvement of estimates of eigenfunctions. Preprint. [7] J. Bourgain. Besicovitch type maximal operators and applications to Fourier analysis. Geom. Funct. Anal., 2: , [8] J. Bourgain. On the dimension of Kakeya sets and related maximal inequalities. Geom. Funct. Anal., 2: , [9] N. Burq, P. Gérard, and N. Tzvetkov. Bilinear eigenfunction estimates and the nonlinear schrödinger equation on surfaces. Inventiones mathematicae, 159(1): , [10] X. Chen. An improvement on eigenfunction restriction estimates for compact boundaryless Riemannian manifolds with nonpositive sectional curvature. Trans. Amer. Math. Soc., 367: , [11] X. Chen and C. D. Sogge. A few endpoint geodesic restriction estimate for eigenfunctions. Comm. Math. Phys., 329(3): , [12] X. Chen and C. D. Sogge. On integrals of eigenfunctions over geodesics. Proc. Amer. Math. Socs, 143(1): , [13] A. Córdoba. The Kakeya maximal function and spherical summation multipliers. Amer. J. Math., 99:1 22, [14] S. Drury. L p estimates for the x-ray transformation. Illinois. J. Math., 27: , [15] C. Fefferman. Inequalities for strongly singular convolution operators. Acta Math., 124:9 36, [16] A. Good. Local analysis of Selbergs trace formula. volume 1040 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, [17] L. Guth. A restriction estimate using polynomial partitioning. J. Amer. Math. Soc., 29: , [18] D. A. Hejhal. Sur certaines séries de Dirichlet associées aux géodésiques fermées dúne surface de Riemann compacte. C. R. Acad. Sci. Paris Sér. I Math., 29(8), [19] R. Hu. L p norm estimates of eigenfunctions restricted to submanifolds. Forum Math., 6:1021 9

10 1052, [20] S. Kakeya. Some problems on maximum and minimum regarding ovals. Tôhoku Science Reprots, 6:71 88, [21] N. H. Katz, I. Laba, and T. Tao. An improved bound on the Minkowski dimension of Besicovitch sets in R 3. Ann. of Math, 152(2): , [22] N. H. Katz and T. Tao. New bounds for Kakeya problem. J. Anal. of Math, 87: , [23] C. Miao, C. D. Sogge, Y. Xi, and J. Yang. Bilinear KakeyaNikodym averages of eigenfunctions on compact Riemannian surfaces. J. Funct. Anal, In press. [24] C. Miao, J. Yang, and J. Zheng. On Wolff s L 5/2 -Kakeya maximal inequality in R 3. Forum Mathematicum, [25] W. Minicozzi and C. D. Sogge. Negative results for Nikodym maximal functions and related oscillatory integrals in curved space. Math. Research Letters, 4: , [26] C. D. Sogge. Concerning the L p norm of spectral cluster of second-order elliptic operators on compact manifolds. J. Funct. Anal, 77: , [27] C. D. Sogge. Concerning Nykodym-type sets in 3-dimensional curved spaces. J. Amer. Math. Soc., 12:1 31, [28] C. D. Sogge. Kakeya-Nikodygm averages and L p -norms of eigenfunctions. Tohoku Math. J., 63: , [29] C. D. Sogge. Improved critical eigenfunction estimates on manifolds of nonpositive curvature. Preprint. [30] C. D. Sogge, Y. Xi, and C. Zhang. Geodesic period integrals of eigenfunctions on Riemann surfaces and the Gauss-Bonnet theorem. preprint. [31] 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, [32] T. Tao. The bochner-riesz conjecture implies the restriction conjecture. Duke Math. J., 96: , [33] T. Tao. The two-ends reduction for the Kakeya maximal conjecture. wordpress.com/tag/two-ends-reduction/, [34] T. Tao. Stickiness, graininess, planiness, and a sum-product approach to the Kakeya problem. stickiness-graininess-planiness-and-a-sum-product-approach-to-the-kakeya-problem/, [35] T. Wolff. An improved bound for Kakeya type maximal functions. Rev. Math. Iberoam, 11: , [36] Y. Xi. On Kakeya-Nikodym type maximal inequalities. Trans. Amer. Math. Soc., in press. [37] Y. Xi and C. Zhang. Improved critical eigenfunction restriction estimates on Riemannian surfaces with nonpositive curvature. Comm. Math. Phys., in press. ykxi@math.jhu.edu 10