arXiv:1404.5016v1 [math.CA] 20 Apr 2014 DP120102019. θ nwhich in rhnra ai of basis orthonormal u h tnadshrclhroiscnb aeoie as categorized be can harmonics spherical standard The erc n auae ycranshrclhrois enwintr now We polynomials harmonic use homogeneous harmonics. we the spherical are harmonics certain spherical by The saturated and metric, § stedmnino t iesae dim eigenspace, its of dimension the is eoe∆ denote 0 k · k (iii). 6.I atclr fw s h ogtdnlcoordinate longitudinal the use we if particular, In 16]. (ii). λ < (i). ∈ sa iefnto of eigenfunction an is hnteeigenfrequency the Then nteReana surface, Riemannian the On eerhi atal upre yteAsrla eerhCouncil Research Australian the by supported partially is Research e od n phrases. and words Key 2010 nan In n a oma rhnra basis orthonormal an form can One L [0 PEIA AMNC IHMAXIMAL WITH HARMONICS SPHERICAL p 0 2 m − m ( , M SH ahmtc ujc Classification. Subject Abstract. epoiea xlctlwrbudo h est nti example. this in density Zeldit the and on Sogge bound lower by explicit proposed suc an question provide supporting the we surface answers Riemannian This a of functions. example an giving therefore hnra peia amnc hc civstemaximal the achieves which harmonics spherical thonormal ≤ 2 k < m < k ) n π 0: = = twspoe ySge[o,S2 that So2] [So1, Sogge by proved was It . g C λ dmninlsot n opc imninmnfl ( Riemannian compact and smooth -dimensional othat so ) k h alc-etaioeao soitdwith associated operator Laplace-Beltrami the k,m ± 1 2 odnt h e fsc ucin ihhmgnosdegree homogeneous with functions such of set the denote to ≤ k Z : sthe is λ k Q 2 2 = ± · · · ≤ nti ae,w hwta hr xssapstv est subsequ density positive a exists there that show we paper, this In : k Y S Y = L 0 2 k SH m k 2 ∋ Y σ r aldznlharmonics; zonal called are iefnto siae,shrclhrois maximal harmonics, spherical estimates, omlsto atr and factor, normalisation r aldtsea harmonics; tesseral called are − ± onigmlilcte,adteeigenfunctions the and multiplicities, counting ( k k p x k on ∆ as = ) r aldscoa amnc rGusa beams. Gaussian or harmonics sectoral called are (sin = λ { = Y    Y m S k p m n n k p } 2 2 − ( 2  φ m k ,θ φ, iheigenvalue with 1 k .(.)i hr ntesphere the on sharp is (1.1) 6. = = 2 1 cos (  1. − − k − 1 2 52,3C5 58J50. 33C55, 35P20, IOOGHAN XIAOLONG = ) k { ∆ 1) + − SH : θ, u − Introduction GROWTH p 1 u j ∆  sin } C p 1 k = u j ∞ −  =1 k,m 2 = j ≈ φ k 1 2 = 1 ( in sin P k k k h utpiiyo h eigenvalue the of multiplicity the ; λ k m 2 p 1) + L j 2 ;and 1; + P θ, n (cos k u p < 2 k k u m ( j ≤ cos ( , M j k u. k steascae eedepolynomial. Legendre associated the is p φ p .I hsppr euse we paper, this In ). ≤ 1): + φ φ ) ∞ ≤ . e g ,te n a rt h standard the write can one then ), 2( imθ ∈ Then . L n λ L n − p [0 j σ +1) 2 , . 1 ( ( omgot o 2 for growth norm p π , S ) L 2 = with n aiuia coordinates latitudinal and ] p = ) usqec feigen- of subsequence h − h[Z] Furthermore, [SZ2]. ch p (2 = ∆ n M hog icvr Project Discovery through etitdo h ; the on restricted , dc h terminology: the oduce ⊕ p < g , k ∞ L S { k =0 − p ihu boundary, without ) 2 u o each For . ≤ omgot,density. growth, norm SH ∆ j ⊂ } g 6) j ∞ neo or- of ence =1 k a eigenvalues has R e ..[Z2, e.g. See . p < k · k 3 NORM satisfy: ihround with ≤ p u 6, k omean to ( ∈ k SH (1.1) 1) + k , 2 XIAOLONG HAN

The Gaussian beams concentrate around the equator φ = π/2 and achieve the maximal Lp norms for 2

{ujm } such that     

kujm kp lim ≥ cM m→∞ σ(p) λjm for some constant cM depending only on (M,g). If pn ≤ p ≤∞, Sogge, Toth, and Zelditch proved some conditions involving “blow-down” points for the manifold to have maximal Lp growth. This in some sense resembles zonal p harmonics with maximal L (pn ≤ p ≤ ∞) norm growth at a point, in which case all the geodesics emitting from this point return back at the same time. We refer to their papers [SZ1, STZ] for details and other related results. If 2 < p ≤ pn, fewer results are known. In the following, we concentrate on the case of σ(p) Riemannian surfaces. Sogge [So3] (see also Bourgain [B2]) proved that kujkp = o λj for 2 0 and 2 λ(ε,p). (1.4) SPHERICAL HARMONICS WITH MAXIMAL Lp (2

Duistermaat-Guillemin’s small-o improvement on the remainder term in Weyl law expan- sion [DG] provides a natural condition of the periodic geodesics. Sogge and Zelditch [SZ2, Theorem 2.1] proved that Theorem 1 (Sogge and Zelditch [SZ2]). If the measure of periodic geodesics is zero, then

for any orthonormal eigenfunction basis {uj}, one can find a full density σ(p) subsequence {ujm } such that kujm kp = o λjm for 2 0 or = 1), we call such subsequence a zero (positive or full) density subsequence. On S2, Gaussian beams maximize the Lp (2

Theorem 3. There exists a constant 0

σ(p) p Recall that Cpk is the L norm of a Gaussian beam Qk ∈ SHk. Moreover, one can choose D such that 1 (7 + 1296D)c−1/D ≤ , (1.7) 0 25 1/72 where c0 = e > 1. For example, D =1/400 = 0.25% will do. We remark that (1.7) in the theorem is not optimal, but achieving this non-optimal density condition requires careful work in the proof. m SH If we complete the orthonormal set {ui}i=1 ⊂ k to an orthonormal basis, then evidently m SH the density of ∪k{ui}i=1 is D in this basis since dim k =2k + 1 and D(2k +1) − 1 < m ≤ D(2k + 1). Thus when k is large enough, 1 2k+1 1 m ⌊D(2k + 1)⌋ 1 D ku k ≥ ku k ≥ · C kσ(p) ≥ · C kσ(p). 2k +1 i p 2k +1 i p 2k +1 2 p 3 p i=1 i=1 X X We deduce the following result. Corollary 4. Using the same notations and constants as in Theorem 3, there exists an 2k+1 SH orthonormal basis {ui}i=1 in k such that 1 2k+1 D ku k ≥ · C kσ(p), k →∞, 2k +1 i p 3 p i=1 X for 2

6, one can ask a similar question as Problem 2: Problem 5. For which Riemannian surface (if any) does there exist an orthonormal basis of eigenfunctions {uj} for which there exists a positive density subsequence {ujm } so that it achieves the maximal Lp norm growth for some p> 6? To my knowledge, this is unknown now. Following the argument in this paper, we are able to prove a weaker version: Corollary 6. Using the same notations and constants as in Theorem 3, there exists an m SH orthonormal spherical harmonics set {ui}i=1 ⊂ k with positive density D satisfying (1.7) such that 1 1 4 − 2 kuikp ≈ k p , 1/4 for p> 6. In particular, kuik∞ ≈ k . 1/4−1/(2p) Here, recall that for Gaussian beams, kQkkp ≈ k in (1.2), comparing to the max- p 1/2−2/p imal L norm growth for p> 6 of zonal harmonics kZkkp ≈ k .

Further remarks and some background. Since kukp & kuk2 for p > 2 on compact manifolds, it is also interesting to ask whether one can achieve the minimal Lp growth of orthonormal eigenfunctions on (M,g). Toth and Zelditch [TZ] proved that under certain condition, if all the eigenfunctions have minimal L∞ norm growth O(1), then the manifold must be a flat torus. See [Z1, Z2] for more information. On S2, VanderKam [V, Theorem 1.2] proved that for almost all orthonormal eigenfunction bases, the L∞ norm of eigenfunctions is bounded by log2 λ. But it is unknown whether there exists a uniformly bounded orthonormal eigenfunction basis or a uniformly bounded subsequence of eigenfunctions with some density in an orthonormal basis. On S3, regarded as the boundary of the unit in C2, Bourgain [B1] constructed a uniformly bounded orthonormal basis for the of holomorphic polynomials. It is still open whether a bounded basis exists in higher dimensions, i.e. on S2n−1 ⊂ Cn for n ≥ 3, or more generally on the boundary of a relatively compact strictly pseudoconvex domain in a complex manifold. Very recently, Shiffman [Sh] proved a partial result that one can select SPHERICAL HARMONICS WITH MAXIMAL Lp (2

2, which is a much stronger condition than (M,g) having maximal Lp growth, then in this case what can we say about the dynamical properties of its geodesic flow? One can weaken the question by assuming the density is of order O(1/N) (as the density of Gaussian beams (1.6) in the standard spherical harmonics basis). In dimension two, Theorem 1 from [SZ2] says that if the measure of periodic geodesic is zero, then there does not exist a posi- tive density subsequence of eigenfunctions achieving maximal Lp norm growth for 2 6 can be found in [SZ1, STZ, TZ] and references therein.

After this paper was completed, I was informed about Bourgain’s unpublished note: In [B3], Bourgain proved the existence of a positive density subsequence of spherical harmonics on S2 with maximal L4 norm growth in Theorem 3, therefore answers Problem 2. His result can be generalized to Lp for all 2

Throughout this paper, A . B (A & B) means A ≤ cB (A ≥ cB) for some constant c depending only on the sphere; A ≈ B means A . B and B . A; the constants c and C may vary from line to line. Organization of the paper. In Section 2, we outline the proof of the main theorem, Theorem 3, and recall the crucial tools used after: diagonally dominant matrices and Wigner D-matrix; in Section 3, we give the proofs of Theorem 3 and Corollary 6; in Section 4, we give some characterization on the L2 localization of the spherical harmonics in Theorem 3. 2. Outline of the proof and preliminaries

Outline of the proof. In SHk with dimension 2k + 1, to construct an orthonormal set of spherical harmonics with large Lp (2

iBourgain’s argument [B3], however, uses Gram-Schmidt process together with some “Hilbertian system” tools to construct a new orthonormal system from a set of Gaussian beams. 6 XIAOLONG HAN necessary to recall some standard facts about this type of matrix. For more information, see [HJ, Chapter 6]. Let A = (aij) be a n × n complex matrix, i.e. A ∈ Mn(C), A is said to be diagonally dominant if n ′ |aii| ≥ |aij| = Ri(A) j=1X,j6=i for all i =1, ..., n. And it is said to be strictly diagonally dominant if all the above inequalities ′ are strict for i =1, ..., n. Here, Ri(A) is called the i-th deleted absolute row sums of A. The following theorem states that all the eigenvalues of A fall into the “Gerˇsgorin discs” involving the diagonal entries and the deleted absolute row sums. Lemma 7 (Gerˇsgorin disc theorem). All the eigenvalues of A are located in the union of n Gerˇsgorin discs: n C ′ z ∈ : |z − aii| ≤ Ri(A) . i=1 [  One can regard A as a perturbation of the diagonal matrix diag(a11, ..., ann) by adding ′ Ri(A) to the i-th row, then Lemma 7 concludes that the eigenvalues of A are in the neigh- borhoods of a11, ..., ann. See [HJ, Theorem 6.1.1] for a detailed proof.

Inner product of two Gaussian beams. Recall that the two Gaussian beams Q±k defined in the introduction are k ±ikθ Q±k(φ, θ)= CkPk (cos φ)e . Both of them concentrate around the equator G = {φ = π/2}, the great circle perpendic- ular to the north pole φ = 0, and decrease in φ exponentially away from G. However, they propagate in opposite directions on G; by the right-hand rule, we say that Qk has pole as the north pole and propagates in the positive direction on G. So in our terminology, Q−k has pole φ = π as the south pole. Any other Gaussian beam with pole as the north pole differs with Qk only by a phase shift, and has the form α k ik(θ+α) Qk (φ, θ)= CkPk (cos φ)e , where α ∈ [−π/k,π/k), and ikα is the initial phase. This is equivalent to the statement that α ikα Qk = e Qk on S2. Given two Gaussian beams q1 and q2 with respect to poles x1 and x2 and concentrating around great circles G1 and G2, we need to compute their inner product

hq1, q2i = q1q2. S2 Z Example. Using the above notation, it is easy to see that for two Gaussian beams Qk and α Qk with the same pole as the north pole, α ikα ikα hQk , Qki = e hQk, Qki = e , which is exactly the phase shift. Generally, we have the following lemma.

Lemma 8. Admitting the above notations about q1 and q2, β 2k hq , q i = eikα cos , 1 2 2   SPHERICAL HARMONICS WITH MAXIMAL Lp (2

ikα in which 0 ≤ β ≤ π is the angle between the poles x1 and x2, and e is the phase shift at the intersecting points of G1 and G2. That is, q (y ) q (y ) eikα = 1 1 = 1 2 q2(y1) q2(y2) where α ∈ [−π/k,π/k), and y1 and y2 are the two intersecting points of G1 and G2. If the 2 poles x1 and x2 coincide, we can choose any non-vanishing point on S to get the phase shift. Let us fix the cartesian coordinates (x(1), x(2), x(3)) and the corresponding polar coordinates (φ, θ) for x ∈ S2: x(1) = sin φ cos θ, (2) x = sin φ sin θ, x(3) = cos φ. For each orthogonal matrix R in SO(3), the special orthogonal in R3, it can be  specified by its Euler angles (α,β,γ), written as Rαβγ . Then Rαβγ is associated with the operator Pαβγ of (a). rotating about x(3)-axis by an angle α, (b). rotating about x(2)-axis by an angle β, (c). rotating about x(1)-axis by an angle γ: 2 Pαβγf(y)= f(Rαβγ y), y ∈ S , for a function f on the sphere. The axes remain fixed in these rotations. Here, we follow [W, Appendix A] for the convention of orientation by the right-hand rule. Recall that from the introduction, the standard orthonormal basis in SHk consists of k m imθ Ym(φ, θ)= Ck,mPk (cos φ)e , m = −k, ..., k. (3) For example, Rα00 is the rotation about x -axis through an angle α, then in polar coor- dinates (φ, θ), φ is unchanged, and θ goes into θ + α. Thus k imα k Pα00Ym(φ, θ)= e Ym(φ, θ), (2.1) N k k and if α = 2lπ/m, l ∈ , then Pα00 keeps Ym unchanged. This is indeed the fact that Ym is a joint eigenfunction of the Laplace-Beltrami operator ∆ and L3 = i∂θ, the generator of rotations about x(3)-axis. See [Z2, §16] for a detailed discussion. k S2 Generally, Pαβγ Ym, as a new function on , is still an eigenfunction of −∆ since ∆ is invariant under rotations, (and may not be an eigenfunction of L3 since it is not rotational k SH invariant). Therefore, Pαβγ Ym ∈ k, and can be expanded by the standard spherical har- monics basis: k k k k PαβγYm = Dm′m(α,β,γ)Ym′ , ′ mX=−k where k k k Dm′m(α,β,γ)= hPαβγYm,Ym′ i k k since Ym′ ’s are orthonormal. The (2k + 1) × (2k + 1) matrix with entries Dm′m(α,β,γ) is called a Wigner D-matrix. See [W, Chapter 15] for a complete treatment on this subject. In particular, Equation (15.27a) in [W] is β 2k hP Y k,Y ki = Dk (α,β,γ)= eimα cos eimγ . (2.2) αβγ m m mm 2   Now we proceed to prove Lemma 8 using the above tools. 8 XIAOLONG HAN

Proof of Lemma 8. Because we only care about two Gaussian beams q1 and q2 with poles (1) (3) x1 and x2, we can assume x1 is the north pole, and x2 is on the x x -plane with polar coordinate (β, 0), 0 < β ≤ π. When β = 0, the two poles coincide, and this case has been discussed in the beginning of this subsection. We aim to find a rotation R such that the operator P associated with R satisfies

P q1 = q2.

Denote the two great circles perpendicular to x1 and x2 as G1 and G2. Then they intersect at two points on x(2)-axis: π π π 3π y = , and y = , . 1 2 2 2 2 2     k We know that |q1| achieves its maximal value CkPk (0) on G1, and |q2| achieves the same maximal value on G2. Therefore, at y1 and y2, k |q1(y1)| = |q1(y2)| = |q2(y1)| = |q1(y2)| = CkPk (0). Then there is a complex constant eikα for α ∈ [−π/k,π/k) such that q (y ) 1 1 = eikα. (2.3) q2(y1) We can also conclude that ikπ q1(y2) e q1(y1) ikα = ikπ = e , q2(y2) e q2(y1) which just states that the phase changing from y1 =(π/2,π/2) to y2 =(π/2, 3π/2) is kπ for both q1 and q2. Now observe that from (2.1), −ikα P(−α)(−β)0q1 = P0(−β)0P(−α)00q1 = e P0(−β)0q1 is also a Gaussian beam with pole x2. Furthermore, because y1 is invariant under the rotation (2) R0(−β)0 as it is on x -axis: R0(−β)0y1 = y1, −ikα −ikα −ikα P(−α)(−β)0q1(y1)= e P0(−β)0q1(y1)= e q1 R0(−β)0y1 = e q1(y1)= q2(y1) from (2.3). Then  P(−α)(−β)0q1 = q2 must hold from the fact that P(−α)(−β)0q1 and q2 are two Gaussian beams with the same pole and have the same phase at one point y1. This implies β 2k hq , q i = hq , q i = P q , q = e−ikαDk (0, β, 0) = eikα cos 1 2 2 1 (−α)(−β)0 1 1 kk 2   by (2.2).  m Remark 9. In the next section, we shall select a family of Gaussian beams {qi}i=1 on the sphere with arbitrary phase shifts at the intersecting points. It is crucial to observe that |hqi, qji| depends only on the angle between the corresponding poles. SPHERICAL HARMONICS WITH MAXIMAL Lp (2

3. Proofs of Theorem 3 and Corollary 6 Now we begin the proof of Theorem 3. Follow the notations in the previous section: For 0

For each point xi, let qi bep a Gaussian beam withp the pole xi. (Recall the terminology of Gaussian beams and their corresponding poles in Section 2.) From Lemma 8, β 2k |hq , q i| = cos ij , i j 2   in which βij is the angle between the poles xi and xj (0 ≤ βij ≤ π). Denote the n × n Hermitian matrix E = hqi, qji . We show that if the density D is chosen small enough, then E is strictly diagonally domi-  nant. In order to evaluate the deleted absolute row sums m ′ Ri(E)= |hqi, qji|, j=1X,j6=i we assume i = 1 and x1 is the north pole without loss of generality. Using the coordinates (φ, θ), we divide φ ∈ [0, π] into ⌈π/d⌉ intervals with equal length π δ = ⌈π/d⌉ where ⌈π/d⌉ is the ceiling function, that is, the smallest integer not less than ⌈π/d⌉: π/d ≤ ⌈π/d⌉ < π/d +1. Then d/2 ≤ δ ≤ d, and by (3.1) 1 6 ≤ δ ≤ . (3.2) 2 ⌊D(2k + 1)⌋ ⌊D(2k + 1)⌋ Write the partition p p P :0 <δ< 2δ < ··· < ⌈π/d⌉δ = π, then P provides ⌈π/d⌉ strips Sl with l = 1, ..., ⌈π/d⌉ on the sphere, which can be divided into three groups:

(I). l = 1: The number of poles in S1 is bounded by a constant since δ ≤ d. They correspond to the Gaussian beams which have the largest inner product with q1. (II). l is small: The strips are close to the north pole x1, and they contain less poles comparing the ones with large l. However, the poles have relatively small angles with x1, and the corresponding Gaussian beams have relatively large inner product with q1. The summation in this group can be carefully examined, and then controlled if we increase the length of intervals δ, which subsequently results to decreasing the density D. 10 XIAOLONG HAN

(III). l is large: The strips are away from the north pole x1. Even they contain more poles with number of them growing polynomially, the inner product with q1 decrease exponentially when k → ∞. Therefore, the summation in this group can always be well controlled, and in fact is independent of δ and D if the strips are away from the north pole by a constant angle. Since cos2k decreases exponentially away from 1, the summations in Groups (I) and (II) ′ above essentially contributes the majority of R1(E). We divide P into three groups: l = 1, lδ ≤ 1/2, and lδ > 1/2 according to the relation β2 β2 cos(β)=1 − + O(β4) ≤ 1 − , (3.3) 2 3 if 0 ≤ β ≤ 1/2. On the l-th strip Sl, the poles xj falling into the strip Sl have angles with the north pole between (l − 1)δ and lδ. To estimate the number of poles in Sl, we have

• The number of poles in S1 is bounded by 2π sin δ ≤ 7. d • The surface area of Sl (l ≥ 2) is lδ δ (2l − 1)δ 2π sin φdφ =4π sin sin , 2 2 Z(l−1)δ and the number of poles in Sl is bounded by 4π sin δ sin (2l−1)δ 18 sin (2l−1)δ 36 sin(l − 1)δ 2 2 ≤ 2 ≤ . π(d/3)2 δ δ Compute that ′ R1(E) m ⌈π/d⌉

= |hq1, qji| = |hq1, qji| j=2 x ∈S X Xl=1 Xj l ⌈π/d⌉ 36 sin(l − 1)δ ≤ 7 max |hq1, qji| + max |hq1, qji| xj ∈S1 δ xj ∈Sl Xl=2 δ 2k (l − 1)δ 2k (l − 1)δ 2k ≤ 7 cos + 36 (l − 1) cos + 36 (l − 1) cos . 2 2 2   l=2X,lδ≤1/2   l=2X,lδ>1/2   Note that the angle between xj and x1, βij ≥ d ≥ δ, and we use this estimate for the poles in S1. Then

(I). l = 1: The number of poles in S1 is bounded by 7. Using (3.2) and (3.3), δ 2k δ2 7 cos ≤ 7 exp 2k log 1 − 2 12      kδ2 ≤ 7 exp − 6   k ≤ 7 exp − 24⌊D(2k + 1)⌋   1 ≤ 7 exp − 72D   −1/D = 7c0 → 0, as D → 0, SPHERICAL HARMONICS WITH MAXIMAL Lp (2

1/72 with an absolute constant c0 = e > 1. (II). l ≥ 2 and lδ ≤ 1/2: Similarly as in (I), (l − 1)δ 2k 36 (l − 1) cos 2 l=2X,lδ≤1/2   (l − 1)2 ≤ 36 (l − 1) exp − 72D l=2X,lδ≤1/2   ∞ l2 ≤ 36 l · exp − 72D l=1   X 2 ∞ 1 −t ≤ 36 t · exp dt 72D Z1    −1/D = 1296Dc0 → 0, as D → 0. (III). l ≥ 2 and lδ > 1/2: We have (l − 1)δ > 1/4, then (l − 1)δ 2k 36 (l − 1) cos 2 l=2X,lδ>1/2   1 2k ≤ 36 cos (l − 1) 8   l=2X,lδ>1/2 1 2k ≤ 36 cos (2k + 1)2 → 0, as k →∞, 8   which is independent of δ and D. Remark 10. Comparing (I) and (II), −1/D −1/D 7c0 ≥ 1296Dc0 if and only if 7 D ≤ , 1296 and this is the case when the number of the poles m = ⌊D(2k + 1)⌋ is small on the sphere.

The majority of the contribution to j |hq1, qji| just comes from the poles around x1 in the first strip S1. Away from S1, the contribution to the summation is essentially negligible. P Adding the above quantities in (I), (II), and (III) together, we arrive at ′ −1/D −1/D −1/D R1(E) ≤ 7c0 + 1296Dc0 + o(k) = (7+1296D)c0 + o(k) := r. (3.4) ′ Thus we deduce that Ri(E) ≤ r < aii = 1 if r < 1, and E is strictly diagonally dominant. From Gerˇsgorin disc theorem in Lemma 7, all the eigenvalues of E fall into the interval [1 − r, 1+ r]. (Here, we use the fact that E is Hermitian, and its eigenvalues are real.) In particular, E is positive definite. Therefore, there is a unique positive definite Hermitian matrix F = E−1/2. See e.g. [HJ, Theorem 7.2.6]. We next show that F is also strictly diagonally dominant by letting r be smaller. Define the matrix norm |||A||| of A by its l∞ → l∞ mapping kAyk ∞ A : Rn → Rn : |||A||| = sup l . y∈Rn kykl∞ Then m

|||A||| = max |aij|. i j=1 X 12 XIAOLONG HAN

Notice that E = I + B, where I is the identity matrix and B is a matrix with diagonal ′ −1/2 entries all zero. Then |||B||| = maxi Ri(E)= r< 1, and therefore we can expand F = E by power series: ∞ 1 1 − F =(I + B)− 2 = I + 2 Bi = I + H, i i=1   in which X ∞ − 1 ∞ |||H||| ≤ 2 |||B|||i ≤ 3 ri ≤ 6r. i i=1   i=1 Thus, X X ′ ′ 1 − 6r ≤ Fii =1+ Hii ≤ 1+6r, and Ri(F )= Ri(H) ≤ |||H||| ≤ 6r. (3.5) This shows that F = I + H is strictly diagonally dominant if we choose 6r ≤ 1/4, that is, r ≤ 1/24. Then 3 5 1 ≤ F ≤ , and R′(F ) ≤ . (3.6) 4 ii 4 i 4 This requires that by (3.4), 1 r = (7+1296D)c−1/D + o(k) ≤ , 0 24 and thus, 1 (7 + 1296D)c−1/D ≤ , 0 25 if k is large enough. t t m Let u = F q with u =(u1, ..., un) and q =(q1, ..., qn) , then it is easy to check that {ui}i=1 is orthogonal:

hui,uji = Fisqs, Fjtqt = FisFjtEst = F EF ij = Iij. * s t + s,t X X X  Furthermore, by (3.6), m m

kuikp = Fijqj ≥ kFiiqikp − kFijqjkp

j=1 p j=1,j6=i X X ′ 1 σ(p) ≥ |Fii|kqikp − R (F )kqjkp ≥ Cpk (3.7) i 2 for 2

= |hq1, qji| Xl=1 xXj ∈Sl ⌈π/d⌉ 36 sin(l − 1)δ ≤ 7 max |hq1, qji| + max |hq1, qji| xj∈S1 δ xj ∈Sl Xl=2 ⌈π/d⌉ 36 δ 2k (l − 1)δ 2k ≤ sin δ cos · δ + sin(l − 1)δ cos · δ δ2  2 2    Xl=2     SPHERICAL HARMONICS WITH MAXIMAL Lp (2

⌈π/d⌉ δ 2k (l − 1)δ 2k ≤ 144⌊D(2k + 1)⌋· sin δ cos · δ + sin(l − 1)δ cos · δ .  2 2    Xl=2   One observes that in the bracket,  ⌈π/d⌉ δ 2k (l − 1)δ 2k sin δ cos · δ + sin(l − 1)δ cos · δ (3.8) 2 2   Xl=2   is indeed a Riemann sum of the function φ 2k sin φ cos 2   with respect to the partition P in [0, π], where in the first interval [0, δ], we choose the value of the function at the right endpoint φ = δ; and in other intervals, we choose the value at the left endpoint φ =(l − 1)δ. Thus one can expect that when k →∞, the summation tends to the integral π φ 2k 2 sin φ cos dφ = . (3.9) 2 k +1 Z0   Therefore, back to the summation, 288⌊D(2k + 1)⌋ R′ (E) ≤ ≤ 576D → 0 as D → 0, 1 k +1 which is a weaker statement than (3.4) since it decreases polynomially with D → 0, while (3.4) goes to zero exponentially. However, to rigorously justify that the Riemann sum (3.8) converges to the integral (3.9) as k →∞ is not trivial, since both the partition P and the integrating function involve k. Next we give a simple proof of Corollary 6. Proof of Corollary 6. From [Z2, §16], 1 1 k 2 4 k ikθ 4 − 2 w |Qk(φ, θ)| = |Ck (sin φ) e | ≈ k e , (3.10) in which w = π/2 − φ and is small. Therefore, the mass of Qk concentrates in a tube w −1/2 G1 around the equator with width w ∼ k and have Gaussian decay in the transverse −1/2 1/4 direction. Furthermore, one can also see that if w ≈ k , then q1 & k . Thus for p> 6, 1 1 4 − 2 p 2 p w p kQkkL (S ) ≥kq1kL (G1 ) & k justifying (1.2). Repeat the computation in (3.7), we have 1 1 4 − 2 kuikp & k p 1/4 for p> 6. In particular, kuik∞ & k .  p Remark 12. Recall that the L norm maximizers for p> 6, the zonal harmonics Zk, have 1 2 2 − kZkkp = Ω k p , from (1.3). Therefore, the result we have for p> 6 in Corollary 6 is weaker than Problem 5, that is, this positive density orthonormal set of spherical harmonics {ui} does not maximize the Lp norm for p> 6. Following the same spirit as in the cases of 2 < p ≤ 6, one can of course try to solve Problem 5 using zonal harmonics: We first choose a set of (non-orthogonal) zonal harmonics on the sphere with well separated poles, then modify them such that they are orthonormal and still close to zonal harmonics in Lp norms for p> 6. However, the main obstacle is that the zonal harmonics decrease away from the concentration points rather slowly comparing 14 XIAOLONG HAN to the exponential decreasing of Gaussian beams, then prevents one to evaluate the deleted absolute row sums as we did in this section.

4. Localization of qi and ui m m The construction of the orthonormal set {ui}i=1 from the Gaussian beams {qi}i=1 in Section 3 raises a natural question: What do ui’s look like? Particularly, do they inherit the localization properties of Gaussian beams around great circles? In this section, we characterize the localization properties of ui’s. Roughly speaking, they are still very close to the Gaussian beams in terms of mass localization. In the following discussion, we focus on u1 and q1, and assume that the pole of q1 is the north pole without loss of generality.

Localization of the Gaussian beam q1. We know that the Gaussian beams q1 concentrates around the equator G1. Moreover, (3.10) implies that ∀ε > 0, there exists c > 0 such that w −1/2 mass outside of the tube G1 for w = ck is smaller than ε:

2 2 w 2 w kq1kL (S \G1 ) =1 −kq1kL (G1 ) ≤ ε. (4.1) w Localization of u1 in G1 with fixed width w and density D. In particular, one deduces that from (4.1) 1 kq k 2 w ≥ , 1 L (G1 ) 2 where w = ck−1/2 for some fixed constant c. (Strictly speaking, out notation of “fixed width w” really means that w has fixed dependence on k.) Recall (3.5) and (3.4) that ′ 1 − 6r ≤ Fii ≤ 1+6r, and Ri(F ) ≤ 6r, where −1/D r = (7+1296D)c0 + o(k) → 0 as D → 0. If we let 6r ≤ 1/4, then similar to (3.7), m

2 ε ku1kL (G1) = F1jqj j=1 2 ε L (G1) X m 2 ε 2 2 ≥ kF11q1kL (G1) − kF1jqjkL (S ) j=1,j6=i X′ 2 ε 2 2 ≥ |F11|kq1kL (G1) − R1(F )kqjkL (S ) 3 1 1 1 ≥ · − = . 4 2 4 8 2 w This means that the L mass of u1 in the tube G1 has a constant lower bound. To achieve a stronger localization statement of u1 as in (4.1) for q1, we again have to further maneuver the density D. w Localization of u1 outside G1 depending on w and D. Resume the notation in (4.1), then, m m

2 2 w 2 2 w 2 2 ku1kL (S \G1 ) = F1jqj ≤ kF11q1kL (S \G1 ) + kF1jqjkL (S ) j=1 2 S2 w j=1,j6=i X L ( \G1 ) X ′ ≤ |F |kq k 2 S2 w + R (F )kq k 2 S2 11 1 L ( \G1 ) 1 j L ( ) ≤ (1+6r)ε +6r SPHERICAL HARMONICS WITH MAXIMAL Lp (2

≤ ε. for r small by choosing the density D small, and w = ck−1/2 for some c depending on ε. Acknowledgements Steve Zelditch told me Problem 2 in [SZ2] when I visited Northwestern University in May 2013, I thank him and the hospitality of Northwestern University. Alex Barnett explained to me the relation between Wigner D-matrix and spherical harmonics; Alan McIntosh suggested me to use ||| · ||| norm of matrices in the computation; Andrew Hassell encouraged me to calculate an explicit density bound in Theorem 3. I thank them for the help throughout the preparation of this paper. I also want to thank Jean Bourgain for sending his unpublished note [B3] to me. References [B1] J. Bourgain, Applications of the spaces of homogeneous polynomials to some problems on the ball algebra. Proc. Amer. Math. Soc. 93 (1985), 277–283. [B2] J. Bourgain, Geodesic restrictions and Lp-estimates for eigenfunctions of Riemannian surfaces. Linear and complex analysis, 27–35, Amer. Math. Soc. Transl. Ser. 2, 226, Amer. Math. Soc., Providence, RI, 2009. [B3] J. Bourgain, Spherical harmonics on S2 and Gaussian beams. Unpublished note. [BS] M. Blair and C. Sogge, On Kakeya-Nikodym averages, Lp-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions. arXiv:1301.7468 (2013). To appear in J. Eur. Math. Soc.. [DG] J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bichar- acteristics. Invent. Math. 29 (1975), 39–79. [HJ] R. A. Horn and C. R. Johnson, Matrix analysis. Cambridge University Press, Cambridge, 1985. [Sh] B. Shiffman, Uniformly bounded orthonormal sections of positive line bundles on complex manifolds. arXiv:1404.1508 (2014). [So1] C. Sogge, Oscillatory integrals and spherical harmonics. Duke Math. J. 53 (1986), no. 1, 43–65. [So2] C. Sogge, Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds. J. Funct. Anal. 77 (1988), no. 1, 123–138. [So3] C. Sogge, Kakeya-Nikodym averages and Lp-norms of eigenfunctions. Tohoku Math. J. (2) 63 (2011), no. 4, 519–538. [STZ] C. Sogge, J. Toth, and S. Zelditch, About the blowup of quasimodes on Riemannian manifolds.J. Geom. Anal. 21 (2011), no. 1, 150–173. [SZ1] C. Sogge and S. Zelditch, Riemannian manifolds with maximal eigenfunction growth. Duke Math. J. 114 (2002), no. 3, 387–437. [SZ2] C. Sogge and S. Zelditch, Concerning the L4 norms of typical eigenfunctions on compact surfaces. Recent developments in geometry and analysis, 407–423, Adv. Lect. Math. (ALM), 23, Int. Press, Somerville, MA, 2012. [SZ3] C. Sogge and S. Zelditch, On eigenfunction restriction estimates and L4-bounds for compact surfaces with nonpositive curvature. Advances in Analysis: The Legacy of Elias M. Stein, 437–447. Princeton University Press, Princeton, NJ, 2014. [TZ] J. Toth and S. Zelditch, Riemannian manifolds with uniformly bounded eigenfunctions, Duke Math. J. 111(1), 97–132 (2002). [V] J. VanderKam, L∞ norms and quantum ergodicity on the sphere. Internat. Math. Res. Notices 1997, no. 7, 329–347. Correction. 1998, no. 1, 65. [W] E. P. Wigner, : And its application to the of atomic spectra. Academic Press, New York-London, 1959. [Z1] S. Zelditch, Local and global analysis of eigenfunctions on Riemannian manifolds. Handbook of geometric analysis. No. 1, 545–658, Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA, 2008. [Z2] S. Zelditch, Park City lectures on Eigenfunctions. arXiv:1310.7888 (2013).

Department of Mathematics, Australian National University, Canberra, ACT 0200, Aus- tralia E-mail address: [email protected]