Growth of high $L^p$ norms for eigenfunctions: an application of geodesic beams

This work concerns $L^p$ norms of high energy Laplace eigenfunctions, $(-\Delta_g-\lambda^2)\phi_\lambda=0$, $\|\phi_\lambda\|_{L^2}=1$. In 1988, Sogge gave optimal estimates on the growth of $\|\phi_\lambda\|_{L^p}$ for a general compact Riemannian manifold. The goal of this article is to give general dynamical conditions guaranteeing quantitative improvements in $L^p$ estimates for $p>p_c$, where $p_c$ is the critical exponent. We also apply previous results of the authors to obtain quantitative improvements in concrete geometric settings including all product manifolds. These are the first results improving estimates for the $L^p$ growth of eigenfunctions that only require dynamical assumptions. In contrast with previous improvements, our assumptions are local in the sense that they depend only on the geodesics passing through a shrinking neighborhood of a given set in $M$. Moreover, the article gives a structure theorem for eigenfunctions which saturate the quantitatively improved $L^p$ bound. Modulo an error, the theorem describes these eigenfunctions as finite sums of quasimodes which, roughly, approximate zonal harmonics on the sphere scaled by $1/\sqrt{\log \lambda}$.

This article studies the growth of L p norms of the eigenfunctions, φ λ , as λ → ∞. Since the work of Sogge [Sog88], it has been known that there is a change of behavior in the growth of L p norms for eigenfunctions at the critical exponent p c := 2(n+1) n−1 . In particular, For p ≥ p c , (1.1) is saturated by the zonal harmonics on the round sphere S n . On the other hand, for p ≤ p c , these bounds are saturated by the highest weight spherical harmonics on S n , also known as Gaussian beams. In a very strong sense, the authors showed in [CG19a,page 4] that any eigenfunction saturating (1.1) for p > p c behaves like a zonal harmonic, while Blair-Sogge [BS15a,BS17] showed that for p < p c such eigenfunctions behave like Gaussian beams. In the p ≤ p c , Blair-Sogge have recently made substantial progress on improved L p estimates on manifolds with non-positive curvature [BS19,BS18,BS15b] This article concerns the behavior of L p norms for high p; that is, for p > p c . While there has been a great deal of work on L p norms of eigenfunctions [KTZ07, HR16, Tac19, Tac18, STZ11, SZ02, SZ16, TZ02, TZ03] this article departs from the now standard approaches. We both adapt the geodesic beam methods developed by the authors in [GT17, Gal19, Gal18, CGT18, CG19c, GT18, CG19b, CG19a] and develop a new second microlocal calculus used to understand the number of points at which |u λ | can be large. By doing this we give general dynamical conditions guaranteeing quantitative improvements over (1.1) for p > p c . In order to work in compact subsets of phase space, we semiclassically rescale our problem. Let h = λ −1 and, abusing notation slightly, write φ λ = φ h so that We also work with the semiclassical Sobolev spaces H s scl (M ), s ∈ R, defined by the norm u 2 H s scl (M ) := (−h 2 ∆ g + 1) s u, u L 2 (M ) . We start by stating a consequence of our main theorem. Let Ξ denote the collection of maximal unit speed geodesics for (M, g). For m a positive integer, r > 0, t ∈ R, and x ∈ M define Ξ m,r,t x := γ ∈ Ξ : γ(0) = x, ∃ at least m conjugate points to x in γ(t − r, t + r) , where we count conjugate points with multiplicity. Next, for a set V ⊂ M write Note that if r t → 0 + as |t| → ∞, then saying that y ∈ C n−1,rt,t x for t large indicates that y behaves like a point that is maximally conjugate to x. This is the case for every point x on the sphere when y is either equal to x or its antipodal point. The following result applies under the assumption that this does not happen and obtains quantitative improvements in that setting.
Theorem 1. Let p > p c , U ⊂ M , and assume that there exist t 0 > 0 and a > 0 so that inf with r t = 1 a e −at . Then, there exist C > 0 and h 0 > 0 so that for 0 < h < h 0 and u ∈ D ′ (M ) The assumption in Theorem 1 rules out maximal conjugacy of any two points x, y ∈ U uniformly up to time ∞, and we expect it to hold on a generic manifold M with U = M . Since Theorem 1 includes the case of manifolds without conjugate points, it generalizes the work of [HT15], where it was shown that logarithmic improvements in L p norms for p > p c are possible on manifolds with non-positive curvature.
The proof of Theorem 1 hinges on a much more general theorem that does not require global geometric assumptions on (M, g). As far as the authors are aware, this is the first result giving quantitative estimates for the L p growth of eigenfunctions that only requires dynamical assumptions. We emphasize that, in contrast with previous improvements on Sogge's L p estimates, the assumptions in Theorem 2 below are purely dynamical and, moreover, are local in the sense that they depend only on the geodesics passing through a shrinking neighborhood of a given set in M . Moreover, the techniques do not require long-time wave parametrices.
Theorem 2 below controls u L p (U ) using an assumption on the maximal volume of long geodesics joining any two given points in U . For our proof, it is necessary to control the number points in U where the L ∞ norm of u can be large. This is a very delicate and technical part of the argument, as the points in question may be approaching one another at rates ∼ h δ as h → 0 + , with 0 < δ < 1 2 . We overcome this problem by developing a second microlocal calculus in Section 6.1 which, after a delicate microlocal argument, yields an uncertainty type principle controlling the amount of L 2 mass shared along short geodesics connecting two nearby points. We expect that additional development of these counting techniques will have many other applications, e.g. to estimates on L p norms p ≤ p c .
To state our theorem, we need to introduce a few geometric objects. First, consider the Hamiltonian function p ∈ C ∞ (T * M \{0}), where · denotes the norm in any metric on T (T * M ). Note that Λ max ∈ [0, ∞), and if Λ max = 0 we may replace it by an arbitrarily small positive constant. We next describe a cover of S * M by geodesic tubes. For each ρ 0 ∈ S * M , the co-sphere bundle to M , let H ρ 0 ⊂ M be a hypersurface so that ρ 0 ∈ SN * H ρ 0 , the unit conormal bundle to H ρ 0 . Then, let be a hypersurface containing SN * H ρ 0 . Next, for q ∈ H ρ 0 , τ > 0, we define the tube through q of radius R(h) > 0 and 'length' τ + R(h) as q ∈ H ρ 0 . Similarly, for A ⊂ S * M , we define Λ τ A (R(h)) in the same way, replacing q with A in (1.3).
Definition 1. Let A ⊂ S * M , r > 0, and {ρ j (r)} Nr j=1 ⊂ A for some N r > 0. We say that the collection of tubes {Λ τ ρ j (r)} Nr j=1 is a (τ, r)-cover of a set A ⊂ S * M provided T j , T j := Λ τ ρ j (r).
We are now ready to state Theorem 2, where we give explicit dynamical conditions guaranteeing quantitative improvements in L p norms.
Suppose that for any pair of points x 1 , x 2 ∈ U , the tubes over x 1 can be partitioned into a disjoint union J Then, there are h 0 > 0 and C > 0 so that for all u ∈ D ′ (M ), and 0 < h < h 0 , In order to interpret (1.4), note that we think of the tubes G x 1 ,x 2 and B x 1 ,x 2 as respectively good (or non-looping) and bad (or looping) tubes. Then, observe that x 1 M ), and j∈B x 1 ,x 2 T j is the set of points over x 1 which may loop through x 2 in time T (h). Therefore, if the volume of points in S * improvements over the standard L p bounds. We expect these non-looping type assumptions to be valid on generic manifolds.
As in [CG19a, Theorem 5 and Section 5], the assumptions of Theorem 2 can be verified in certain integrable situations with T (h) ≫ log h −1 , thus producing o((log h −1 ) − 1 2 ) improvements. Moreover, in [CG19b], we used these types of good and bad tubes to understand averages and L ∞ -norms under various assumptions on the geometry, including that it has Anosov geodesic flow or non-positive curvature. Since our results do not require parametrices for the wave-group, we expect that the arguments leading to Theorem 2 will provide polynomial improvements over Sogge's estimates on manifolds where Egorov type theorems hold for longer than logarithmic times. Remark 1. The proofs below adapt directly to the case of quasimodes for real principal type semiclassical pseudodifferential operators of Laplace type. That is, to operators with principal symbol p satisfying both ∂ ξ p = 0 on {p = 0} and that {p = 0}∩T * x M has positive definite second fundamental form. This is the case, for example, for Schrödinger operators away from the forbidden region. However, for concreteness and simplicity of exposition, we have chosen to consider only the Laplace operator.
1.1. Discussion of the proof of Theorem 2. Our method for proving Theorem 2 differs from the standard approaches for treating L p norms in two major ways. It hinges on adapting the geodesic beam techniques constructed by the authors [CG19a], and on the development of a new second-microlocal calculus.
We start in Section 2 by covering S * M with tubes of radius R(h). Then, in Section 3.2, we decompose the function u, whose L p norm we wish to study, into geodesic beams i.e. into pieces microlocalized along each of these tubes. We then sort these beams into collections which carry ∼ 2 −k u L 2 mass and study the collections for each k separately.
In order to understand the L p norm of u, we next decompose the manifold into balls of radius R(h). By constructing a good cover of M , we are able to think the L p norm of a function on M as the L p norm of a function on a disjoint union of balls of radius R(h). In each ball, B, we are able to apply the methods from [CG19a] to understand the L ∞ norm of u on B in terms of the number of tubes with mass ∼ 2 −k u L 2 passing over that ball.
To bound the L p norm with p < ∞, it then remains to understand the number of balls on which the function u can have a certain L ∞ norm. In Section 3.4 we first observe that when u has relatively low L ∞ norm on a ball, this ball can be neglected by interpolation with Sogge's L pc estimate. It thus remains to understand the number of balls B on which the L ∞ norm of u can be large (i.e. close to extremal). This is done in Section 3.5. It is in this step where a crucial new ingredient is input.
The new method allows us to control the size of the set on which an eigenfunction (or quasimode) can have high L ∞ norm. The method relies on understanding how much L 2 mass can be effectively shared along short geodesics joining two nearby points in such a way as to produce large L ∞ norm at both points. That is, if x α and x β are nearby points on M , and if |u(x α )| and |u(x β )| are near extremal, how much total L 2 mass must the tubes over x α and x β carry?
In order to understand this sharing phenomenon, we develop a new second microlocal calculus associated to a Lagrangian foliation L over a co-isotropic submanifold Γ ⊂ T * M . This calculus allows for simultaneous localization along a leaf of L and along Γ. The calculus, which is developed in Section 5, can be seen as an interpolation between those in [DZ16] and [SZ99]. It is then the incompatibility between the calculi coming from two nearby points which allows us to control this sharing of mass. This incompatibility is demonstrated in Section 6 in the form of an uncertainty principle type of estimate.
Once the number of balls with high L ∞ norm is understood, it remains to employ the non-looping techniques from [CG19a] where the L 2 mass on a collection of tubes is estimated using its non-looping time (see Section 3.5.2).
1.2. Outline of the paper. In section 2, we construct the covers of S * M by tubes and T * M by balls which are necessary in the rest of the article. Section 3 contains the proof of Theorem 2. This proof uses the anisotropic calculus developed in Section 5 and the almost orthogonality results from Section 6. Section 4 contains the necessary dynamical arguments to prove Theorem 1 using Theorem 2.
Acknowledgements. The authors are grateful to the National Science Foundation for support under grant DMS-1900519 (Y.C) and Mathematical Sciences Postdoctoral Research Fellowship DMS-1502661 as well as grant DMS-1900434 (J.G.). Y.C. is grateful to the Alfred P. Sloan Foundation.

Tubes Lemmata
The next few lemmas are aimed at constructing (τ, r)-good covers and partitions of various subsets of T * M (see also [CG19a, Section 3.2]).
Definition 2 (good covers and partitions). Let A ⊂ T * M , r > 0, and {ρ j (r)} Nr j=1 ⊂ A be a collection of points, for some N r > 0. Let D be a positive integer. We say that the collection of tubes In addition, for 0 ≤ δ ≤ 1 2 and R(h) ≥ 8h δ , we say that a collection {χ j } N h j=1 ⊂ S δ (T * M ; [0, 1]) is a δ-good partition for A associated to a (D, τ, R(h))-good cover if . Remark 2. We show below that for any compact Riemannian manifold M , there are D M , R 0 , τ 0 > 0, depending only on (M, g), such that for 0 < τ < τ 0 , 0 < r < R 0 , there exists a (D M , τ, r) good cover for S * M .
We start by constructing a useful cover of any Riemannian manifold with bounded curvature.
Lemma 2.1. LetM be a compact Riemannian manifold. There exist D n > 0, depending only on n, and R 0 > 0 depending only on n and a lower bound for the sectional curvature ofM , so that the following holds. For 0 < r < R 0 , there exist a finite collection of points {x α } α∈I ⊂M , I = {1, . . . , N r }, and a partition {I i } Dn i=1 of I so that • {x α } α∈I is an r 2 maximal separated set inM .
Proof. Let {x α } α∈I be a maximal r 2 separated set inM . Fix α 0 ∈ I and suppose that B(x α 0 , 3r) ∩ B(x α , 3r) = ∅ for all α ∈ K α 0 ⊂ I. Then for all α ∈ K α 0 , B(x α , r 2 ) ⊂ B(x α 0 , 8r). In particular, Now, there exist R 0 > 0 depending on n and a lower bound on the sectional curvature ofM , and D n > 0 depending only on n, so that for all 0 < r < R 0 , Hence, it follows from (2.1) that In particular, |K α 0 | ≤ D n . At this point we have proved that each of the balls B(x α , 3r) intersects at most D n − 1 other balls. We now construct the sets I 1 , . . . , I Dn using a greedy algorithm. We will say that the index α 1 intersects the index α 2 if First place the index 1 ∈ I 1 . Then suppose we have placed the indices {1, . . . , α} in I 1 , . . . , I Dn so that each of the I i 's consists of disjoint indices. Then, since α + 1 intersects at most D n − 1 indices, it is disjoint from I i for some i. We add the index α to I i . By induction we obtain the partition I 1 , . . . , I Dn . Now, suppose that there exists x ∈M so that x / ∈ α∈I B(x α , r). Then, a contradiction of the r/2 maximality of x α .
There exist C 1 > 0 so that for all 0 < h ≤ h 1 and every (τ, R(h))-cover of Σ H there exists a partition of unity Proof. The proof is identical to that of [CG19a,Proposition 3.4]. Although the claim that j χ j ≡ 1 on Λ τ Σ H ( 1 2 R(h)) does not appear in the statement of [CG19a, Proposition 3.4], it is included in its proof.

Proof of Theorem 2
For each q ∈ S * M , choose a hypersurface H q ⊂ M with q ∈ SN * H q and τ injHq > inj(M ) 2 , where τ injHq is defined in (2.4) and inj(M ) is the injectivity radius of M . We next use Lemma 2.2 to generate a cover of Σ Hq . Lemma 2.2 yields the existence of D n > 0 depending only on n and R 0 = R 0 (n, H q ) > 0, such that the following holds. Since by assumption R(h) ≤ h δ 1 , there is h 0 > 0 such that h δ 2 ≤ R(h) ≤ R 0 for all 0 < h < h 0 . Also, set r 1 := R(h) and r 0 := 1 2 R(h). Then, by Lemma 2.2 there exist By (3.1) there is an h-independent open neighborhood of q, V q ⊂ S * M , covered by tubes as in Lemma 2.2. Since S * M is compact, we may choose {q ℓ } L ℓ=1 with L independent of h, so that S * M ⊂ ∪ L ℓ=1 V q ℓ . In particular, if 0 < τ ≤ min 1≤ℓ≤L τ Hq ℓ , and for each ℓ ∈ {1, . . . , L} we let We split the analysis of u in two parts: near and away from the characteristic variety {p = 0} = S * M . In what follows we use C to denote a positive constant that may change from line to line.
3.1. It suffices to study u near the characteristic variety. In this section we reduce the study of u L p (U ) to an h-dependent neighborhood of the characteristic variety {p = 0} = S * M . We will use repeatedly the following result.
Lemma 3.1. For all ε > 0 and all p ≥ 2, there exists C > 0 such that (3.4) Proof. By [Gal19, Lemma 6.1] (or more precisely its proof), for any ε > 0, there exists C ε ≥ 1 so that Id By complex interpolation of the operators Id : L 2 → L 2 and Id : H and this yields (3.4) as claimed.
Observe that First, note that since 1 − L ℓ=1 ψ q ℓ =0 in an h-independent neighborhood of S * M = {p = 0}, by the standard elliptic parametrix construction (see e.g. (3.5) Next, combining (3.5) with Lemma 3.1, and using that h It remains to understand the terms Op h (ψ q ℓ )u. Since there are finitely many such terms, we write and consider each term Op h (ψ q ℓ )u L p individually. By Proposition 2.3 for each ℓ = 1, . . . , L there exist τ 1 (q ℓ ) > 0, ε 1 (q ℓ ) > 0, and a family of cut-offs {χ T q ℓ ,j } j∈J q ℓ , withχ T q ℓ ,j supported in Λ τ +ε 1 (q ℓ ) ρ j (R(h)) and such that for all 0 < τ < τ 1 (q ℓ ) ( 1 2 R(h)). (3.8) Let τ 0 (q ℓ ) from [CG19a, Theorem 8]. Then, set From now on we work with tubes Next, we localize u near and away from Λ τ Σ Hq ℓ (h δ ): In particular, by (3.8), (3.10) Combining (3.6), (3.7) and (3.10) we have proved that for U ⊂ M 3.2. Filtering tubes by L 2 -mass. By (3.11) it only remains to control the terms where u is localized to the V q ℓ patch within the characteristic variety S * M and, more importantly, to the tubes T q ℓ ,j . Therefore, we fix ℓ and, abusing notation slightly, write )∩S δ be a smooth cutoff function with supp χ T j ⊂ T j , χ T j ≡ 1 on suppχ T j , and such that {χ j } j is bounded in S δ . We shall work with the modified norm Note that this norm is the natural norm for obtaining T − 1 2 improved estimates in L p bounds since the fact that u is an o(T −1 h) quasimode implies, roughly, that u is an accurate solution to (hD t + P )u = 0 for times t ≤ T . For each integer k ≥ −1 we consider the set (3.13) It follows that A k consists of those tubes T j with L 2 mass comparable to 2 −k .
Observe that since |χ T j | ≤ 1, for h small enough depending on finitely many semi- (3.14) Lemma 3.2. There exists C n > 0 so that for all k ≥ −1 Proof. According to (3.2), the collection {T j } j∈J can be partitioned into D n sets of disjoint tubes. Thus, we have j∈J |χ T j | 2 ≤ D n and there is C n > 0 depending only on n such that In particular, Therefore, Next, let Then, by (3.12) and (3.14) we have The goal is therefore to control w k L p (U ) for each k since the triangle inequality yields 3.3. Filtering tubes by L ∞ weight on shrinking balls. By Lemma 2.1, there are Next, for each positive integer k and α ∈ I consider the sets where π M : T * M → M is the standard projection. The indices in A k are those that correspond to tubes with mass comparable to 1 2 k u P,T , while indices in A k (α) correspond to tubes of mass 1 2 k u P,T that run over the ball B(x α , 2R(h)). In particular, Lemma 3.2 and [CG19a, Lemma 3.7] yield the existence of C n , c M > 0 such that Indeed, for α ∈ I k,m , (3.20) In addition, (3.13) and Lemma [CG19a, Lemma 3.7] imply that there exist c M > 0, τ M > 0, and C n > 0, depending on M and n respectively, such that for all N > 0 there exists C N > 0 with which, combined with (3.20), proves the lower bound in (3.19). Furthermore, to simplify notation, let Note that for each α ∈ I k,m there isx α ∈ B(x α , R(h)) such that We finish this section with a result that controls the size of I k,m in terms of that of A k,m . Let 1)), and define the operator χ h,xα by and Ψ −∞ Γx α ,Lx α ,ρ is a class of smoothing pseudodifferential operators that allows for localization to h ρ neighborhoods of Γx α and is compatible with localization to h ρ neighborhoods of the foliation Lx α of Γx α generated by Ωx α .
In Theorem 3 for ε > 0 we explain how to build a cut-off operator where inj M denotes the injectivity radius of M .
Our next goal is to produce a lower bound for | A k,m | in terms of | I k,m | by using the lower bound (3.22) on χ h,xα w k,m L ∞ for indices α ∈ I k,m .
By (3.24), we have for α ∈ I k,m . In particular, by (3.22) and (3.26), (3.27) Therefore, applying the standard L ∞ bound for quasimodes of the Laplacian (see e.g. [Zwo12, Theorem 7.12]) and using that by (3.24) we have that Xx α nearly com- (3.28) Note that we have canceled the factor h 1−n 2 which appears both in (3.27) and the standard L ∞ bounds for quasimodes. Using The last inequality follows from the definition of w k,m together with the definition (3.13) of A k .
In particular, we have proved that there is C > 0 such that for allĨ ⊂ I k,m ≤ 1 and so by (3.29) Note that for w k,m defined as in (3.26), Finally, we split the study of w k L p (U ) into two regimes: tubes with low or high L ∞ mass. Fix N > 0 large, to be determined later. (Indeed, we will see that it suffices to take N > 1 2 (1 − pc p ) −1 .) Then, we claim that for each k ≥ −1, (3.32) where m 1,k and m 2,k are defined by where c 0 , c n are described in what follows. Indeed, note that the bound (3.19) yields that 2 m is bounded by | A k (α)| for all α ∈ I k,m and the latter is controlled by c 0 R(h) n−1 for some c 0 > 0, depending only on (M, g). In addition, note that by (3.19) the w k,m are only defined for m satisfying 2 m ≤ c n 2 2k . These two observations justify that the second sum in (3.32) runs only up to m 2,k .
3.4. Control of the low L ∞ mass term, m ≤ m 1,k . We first estimate the small m term in (3.32). The estimates here essentially amount to interpolation between L pc and L ∞ . From the definition (3.18) of I km , together with 1−n 2 (p − p c )−1 = −pδ(p) and w k,m L pc (U k,m ) ≤ h − 1 pc u P,T , (3.33) Finally, define k 1 , k 2 such that If k ≤ k 1 , then 2 m 1,k = c n 2 2k , so there exists C n,p > 0 such that .
Putting these three bounds together with (3.33), we obtain (3.35) 3.5. Control of the high L ∞ mass term, m ≥ m 1,k . In this section we estimate the large m term in (3.32). To do this we split where the set of 'good' tubes j∈G k,m T j is [t 0 , T ] non-self looping and the number of 'bad' tubes |B k,m | is small in the following sense. We ask that for some c > 0, where That is, |B U | is the maximum number of loops of length in [t 0 , T ] joining any two points in U . Then, define Next, consider (3.39) 3.5.1. Bound on the looping piece. We start by estimating the 'bad' piece Observe that if 2 m 1,k = min(c 0 R(h) 1−n , c n 2 2k ), then m 1,k = m 2,k and we need not consider this part of the sum. Therefore, the high L ∞ mass term has and In particular, as a consequence of Lemma 3.3 we have the existence of h 0 > 0 and C > 0 such that for all 0 < h ≤ h 0 , where we have used again Lemma 3.2 to bound | A k,m |. Next, note that for each point in I k,m there are at most c| I k,m ||B U | tubes in B k,m touching it. Therefore, we may apply [CG19a, Lemma 3.7] to obtain C > 0 such that Using (3.43) and interpolating between L ∞ and L pc we obtain (3.44) In addition, since combining (3.13) with (3.36) yields the bounds in (3.44) and (3.42), together with the definition of m 1,k (3.40) yield Then, with k 1 , k 2 defined as in (3.34), we have that Finally, since we only need to consider 3.5.2. Bound on the non self-looping piece. In this section we aim to control the 'good' piece We first estimate the number of non-self looping tubes T j with j ∈ A k . That is, tubes on which the L 2 mass of u is comparable 2 −k u P,T .
Then, there exists a constant C n > 0, depending only on n, such that Proof. Using that G ⊂ A k , we have Combining (3.47), (3.48), and (3.49) yields We may now proceed to estimate the L p -norm of the non-looping piece (3.46). The first step is to notice that we only need to sum up to m ≤ m 3,k , where m 3,k is defined by and c M > 0 is as defined in (3.19) and C n > 0 is the constant in Lemma 3.4. To see this, first observe that, using (3.18), (3.41) and (3.43), for each α ∈ I k,m Next, since m 1,k ≤ m ≤ m 2,k , we may apply Lemma 3.3 to bound | I k,m | as in (3.41) to obtain that for some C > 0 In addition, provided we have, that for m ≥ m 1,k and k 1 ≤ k ≤ k 2 where we used that by (3.19), |A k,m | is controlled by 2 2k to get the first inequality, that m ≥ m 1,k to get the second, and that k ≥ k 1 to get the third. Combining (3.51) and the bound in (3.53) we obtain | A k,m | ≤ C n t 0 2 2k T , and so, by (3.19), As claimed, this shows that to deal with (3.46) we only need to sum up to m ≤ m 3,k . The next step is to use interpolation to control the first sum in (3.46) by (3.54) We claim that (3.50) yields Indeed, using the bound (3.52) on |B U |, that | A k,m | is controlled by 2 2k , that m ≥ m 1,k as in (3.40), and that k 1 ≤ k ≤ k 2 , we have Using (3.55), the standard bound on w G k,m pc L pc (U k,m ) , and that w G k,m 2 Then, summing in k, and again using that only Note that the sum over k in (3.56) is controlled by the value of k for which since the sum is geometrically increasing before this value of k and geometrically decreasing afterward.
for some C > 0, as requested in (3.52). Since this estimate holds only when |B U |R(h) n−1 ≤ CT −6N , we will replace T above by T 0 := min{ 1 where the constant C is adjusted from line to line.
Finally, to finish the proof of Theorem 2, we need to show that for any (τ, R(h)) cover {T j } j of S * M , up to a constant depending only on M , |B U | can be bounded by |B U | whereB U is defined as in (3.37) using a (D, τ, R(h)) good cover {T k } k of S * M .
Lemma 3.5. There exists C M > 0 depending only on M so that if {T j } j∈J and {T k } k∈K are respectively a (τ, R(h)) cover S * M and a (D, τ, R(h)) good cover of S * M , and |B U |, |B U | are defined as in (3.37) for respectively the covers {T j } j∈J ,{T k } k , then Proof. Fix α, β such that x α , x β ∈ U . Suppose that j ∈ B U (α, β) where B U (α, β) is as in (3.38). Then, there is k ∈B U (α, β) such thatT k ∩ T j = ∅. Now, fix j ∈ J and let We claim that there is c M > 0 such that for each k ∈ C j (3.58) Assuming (3.58) for now, there exists C M > 0 such that Thus, for each j ∈ B U (α, β), there are at most C MD elements inB U (α, β) and hence |B U (α, β)| ≥ |B U (α, β)|/(C MD ) as claimed.
We now prove (3.58). Let q ∈T k . Then, there are ρ ′ k , ρ ′ j , q ′ ∈ S * M and t k , t j , s ∈ . Applying ϕ −t j , and adjusting c M in a way depending only on M , and the claim follows.

Proof of theorem 1
In order to finish the proof of Theorem 1, we need to verify that the hypotheses of Theorem 2 hold with T (h) = b log h −1 for some b > 0, and such that for all x 1 , x 2 ∈ U there is some splitting J and ε 0 > 0. Fix x 1 , x 2 ∈ U and let F 1 , F 2 : T * M → R n+1 be two smooth functions such that for i = 1, 2, Define also ψ i : To find B x 1 ,x 2 , we apply the arguments from [CG19b, Sections 2, 4]. In particular, fix a > 0 and let r t := a −1 e −a|t| . Suppose that Next, let {Λ τ ρ j (r 1 ) be a (D M , τ, r 1 )-good cover for S * M . We apply [CG19b, Proposition 2.2] to construct B x 1 ,x 2 and G x 1 ,x 2 . Remark 3. We must point out that we are applying the proof of that proposition rather than the proposition as stated. The only difference here is that the loops we are interested in go from a point x 1 to a point x 2 where x 1 and x 2 are not necessarily equal. This does not affect the proof.

Anisotropic Pseudodifferential calculus
In this section, we develop the second microlocal calculi necessary to understand 'effective sharing' of L 2 mass between two nearby points. That is, to answer the question: how much L 2 mass is necessary to produce high L ∞ growth at two nearby points? To that end, we develop a calculus associated to the co-isotropic which allows for localization to a Lagrangian leaves ϕ t (Ω x ). In Section 6.2 we will see, using a type of uncertainty principle, that the calculi associated to two distinct points, x α , x β ∈ M , are incompatible in the sense that, despite the fact that Γ xα and Γ x β intersect in a dimension 2 submanifold, for operators X xα and X x β localizing to Γ xα and Γ x β respectively, Let Γ ⊂ T * M be a co-isotropic submanifold and L = {L q } q∈Γ be a family of Lagrangian subspaces L q ⊂ T q Γ that is integrable in the sense that if U is a neighborhood of Γ, and V, W are smooth vector fields on T * M such that V q , W q ∈ L q for all q ∈ Γ, then [V, W ] q ∈ L q for all q ∈ Γ. The aim of this section is to introduce a calculus of pseudodifferential operators associated to (L, Γ) that allows for localization to h ρ neighborhoods of Γ with 0 ≤ ρ < 1 and is compatible with localization to h ρ neighborhoods of the foliation of Γ generated by L. This calculus is close in spirit to those developed in [SZ99] and [DZ16]. To see the relationships between these calculi, note that the calculus in [DZ16] allows for localization to any leaf of a Lagrangian foliation defined over an open subset of T * M and that in [SZ99] allows for localization to a single hypersurface. The calculus developed in this paper is designed to allow localization along leaves of a Lagrangian foliation defined only over a co-isotropic submanifold of T * M . In the case that the co-istropic is a whole open set, this calculus is the same as the one developed in [DZ16]. Similarly, in the case that the co-isotropic is a hypersurface and no Lagrangian foliation is prescribed, the calculus becomes that developed in [SZ99].
Definition 3. Let Γ be a co-isotropic submanifold and L a Lagrangian foliation on Γ. Fix 0 ≤ ρ < 1 and let k be a positive integer. We say that a ∈ S k Γ,L,ρ if a ∈ C ∞ (T * M ), a is supported in an h-independent compact set, and where W 1 , . . . W ℓ 2 are any vector fields on T * M , V 1 , . . . V ℓ 1 are vector fields on T * M with (V 1 ) q , . . . (V ℓ 1 ) q ∈ L q for q ∈ Γ, and q → d(Γ, q) denotes the distance from q to Γ induced by the Sakai metric on T * M .
We also define symbol classes associated to only the co-isotropic Definition 4. Let Γ be a co-isotropic submanifold. We say that a ∈ S k Γ,ρ if a ∈ C ∞ (T * M ), a is supported in an h-independent compact set, and where V 1 , . . . V ℓ 1 are tangent vector fields to Γ, and W 1 , . . . W ℓ 2 are any vector fields. 5.1. Model case. The goal of this section is to define the quantization of symbols in S k Γ 0 ,L 0 ,ρ , where Γ 0 , L 0 are a model pair of co-isotropic and Lagrangian foliation defined below. The model co-isotropic submanifolds of dimension 2n − r is with Lagrangian foliation Note that in this model case the distance from a point (x, ξ) to Γ 0 is controlled by |x ′ |. Therefore, a ∈ S k Γ 0 ,L 0 ,ρ if and only if a is supported in an h-independent compact set and for all (α, β) ∈ N n × N n there exists C α,β > 0 such that In the model case, it will be convenient to defineã ∈ C ∞ (R n x × R n ξ × R r λ ) such that a(x, ξ) =ã(x, ξ, h −ρ x ′ ), and for all (α ′ , α ′′ , β, γ) ∈ N r × N n−r × N n × N r there exists C α,β,γ > 0 such that Similarly, if a ∈ S k Γ 0 ,ρ , then for all (α ′ , α ′′ , β, γ) ∈ N r × N n−r × N n × N r there exists C α,β,γ > 0 such that Definition 5. The symbols associated with this submanifold are as follows. We say a ∈ S k Γ 0 ,L 0 ,ρ if a ∈ C ∞ (R n x × R n ξ × R r λ ) satisfies (5.2) and a is supported in an hindependent set in (x, ξ). If we have the improved estimates (5.3) then we say that a ∈ S k Γ 0 ,ρ .
Remark 4. Note that although there is no ρ in the definition of S k Γ 0 ,ρ , we keep it in the notation for consistency Let a ∈ S k Γ 0 ,L 0 ,ρ . We then define Since a ∈ S k Γ 0 ,L 0 ,ρ is compactly supported in x, there exists C > 0 such that on the support of the integrand λ ≤ Ch −ρ and hence h ≤ Ch 1−ρ λ −1 .
This will be important when computing certain asymptotic expansions.
Lemma 5.1. Let k ∈ R and a ∈ S k Γ 0 ,L 0 ,ρ . Then, . Then, for all α, β ∈ N n there exists C α,β such that , by [Zwo12, Theorem 4.23] there exists a universal constant M > 0 such that In particular, If instead, a ∈ S k 1 Γ 0 ,ρ and b ∈ S k 2 Γ 0 ,ρ , then the remainder in (5.5) lies in h 1−ρ S k 1 +k 2 −1 Proof. With T δ as in (5.4), we have In particular, using that a and b are compactly supported, a h ∈ h − max(ρk 1 ,0) S ρ/2 and b h ∈ h − max(ρk 2 ,0) S ρ/2 and hence [Zwo12, Theorems 4.14,4.17] apply. In particular, if we let M > 0 andk := max(k 1 , 0) + max(k 2 , 0), we obtain where, for any N > 0, the remainder is O(h M ) S ρ/2 . Moreover, since a and b were compactly supported, we may assume introducing an h ∞ error, that the remainder is supported in {(x, ξ) : we thus have If instead, a ∈ S m 1 Γ 0 ,ρ and b ∈ S m 2 Γ 0 ,ρ , then the remainder lies in h 1−ρ S m 1 +m 2 −2 Similarly, the same conclusion holds if b ∈ S m 2 Γ 0 ,ρ with the error term in c being Proof. In each case, we need only apply the formula (5.6).

5.2.
Reduction to normal form. In order to define the quantization of symbols in S Γ,L,ρ for general (Γ, L), we first explain how one can reduce the problem to working in the model case (Γ 0 , L 0 ).
In order to create a well-defined global calculus of psuedodifferential operators associated to (Γ, L), we will need to show invariance under conjugation by FIOs preserving the pair (L 0 , Γ 0 ).
Indeed, this follows from the fact that in this case an FIO quantizing κ is T u(x) = u(x 1 , . . . x j−1 , −x j , x j+1 , . . . , x n ) and so the conclusion of the proposition follows from a direct computation together with the identity case. Thus, we may assume that Lemma 5.6. Let κ be a symplectomorphism satisfying (5.8) and (5.9). Then, there is a piecewise smooth family of symplectomorphisms [0, 1] ∋ t → κ t such that κ t satisfies (5.8), (5.9), κ 0 = Id, and κ 1 = κ.
Finally, we proceed with the proof of Proposition 5.5.
Proof of Proposition 5.5. Let κ t be as in Lemma 5.6. That is, a piecewise smooth deformation from κ 0 = Id to κ 1 = κ such that κ t preserves Γ 0 and (κ t ) * | Γ 0 perserves L 0 . Let T t be piecewise smooth family of elliptic FIOs defined microlocally near (0, 0), quantizing κ t , and satisfying Here, Q t is a smooth family of pseudodifferential operator with symbol q t satisfying (Such an FIO exists, for example, by [Zwo12, Chapter 10] and q t exists by [Zwo12, Thoerems 11.3, 11.4]) Next, define and hence since the Proposition follows by direct calculation when κ = Id, we may assume that T = T 1 .
In that case, our goal is to find a symbol b such that A 1 = Op h (b). First, observe that (5.17) implies that hD t T −1 t − Q t T −1 t = 0 and so This would yield that B t −A t = O(h ∞ ) L 2 →L 2 and the argument would then be finished by setting hence, since T 0 = Id and B 0 = Op h (a), Combining this with the fact that both T t and T −1 t are bounded on H k h completes the proof.
Iterating this procedure and solving away successive errors finishes the proof of Proposition 5.5. If a ∈ S k Γ 0 ,ρ , then we need only use that ∂ ξ ′ q t = r t x ′ and we obtain the remaining results.
Our next lemma follows [SZ99, Lemma 4.1] and gives a characterization of our second microlocal calculus in terms of the action of an operator. In what follows, given operators A and B, we define the operator ad A by Lemma 5.7 (Beal's criteria). Let A h : S(R n ) → S ′ (R n ) and k ∈ Z. Then, A h = Op h (a) for some a ∈ S k Γ 0 ,L 0 ,ρ if and only if for any α, β ∈ R n there exists C > 0 with Similarly, A h = Op h (a) for some a ∈ S k Γ 0 ,ρ if and only if Proof. The fact that A h = Op h (a) for some a ∈ S k Γ 0 ,L 0 ,ρ implies the estimates above follows directly from the model calculus. Let U h be the unitary (on L 2 ) operator, h . For fixed h, we can use Beal's criteria (see e.g. [Zwo12,Theorem 8.3]) to see that there is a h such thatÃ and hence, Note that for φ, ψ ∈ S(R n ), Our goal is then to understand the behavior of b h (x, ξ) in terms of h and h 1−ρ x ′ .
Let τ x 0 andτ ξ 0 be the physical and frequency shift operators with Fτ ξ 0 = τ ξ 0 F and Fτ x 0 =τ −x 0 . In addition, write u (−r) := h 1−ρ x ′ −r u L 2 for the dual norm to u (r) := U −1 h u r . Assume that k ≥ 0. Then, the definition of B h combined with the assumptions of the lemma yield In addition, note that for fixed ψ, φ ∈ S, Therefore, (5.2) leads to On the other hand, we have by (5.20) that Combining (5.22) with (5.21) we have Next, note that χ can be replaced by any fixed function in C ∞ c by taking ψ, φ witĥ ψ(ξ)φ(x) = 0 on supp χ.
Putting ζ = η 0 − ξ 0 and z = x 0 − y 0 , we obtain that for everyα,β ∈ N n and, as a consequence, we obtain . This gives the first claim of the lemma for k ≥ 0. For k ≤ 0, we consider h −ρ x ′ −k A and use the composition formulae.
A nearly identical argument yields the second claim.
5.3. Definition of the second microlocal class. With Proposition 5.5 in place, we are now in a position to define the class of operators with symbols in S k Γ,L,ρ . Definition 6. Let Γ ⊂ U ⊂ T * M be a co-isotropic submanifold, U an open set, and L a Lagrangian folation on Γ. A chart for (Γ, L) is a symplectomorphism for q ∈ Γ ∩ U .
We now define the pseudodifferential operators associated to (Γ, L).
Definition 7. Let M be a smooth, compact manifold and U ⊂ T * M open, Γ ⊂ U a co-isotropic submanifold, L a Lagrangian foliation on Γ and ρ ∈ [0, 1). We say that is a semiclassical pseudodifferential operator with symbol class S k Γ,L,ρ (U ) (and write A ∈ Ψ k Γ,L,ρ (U )) if there are charts {κ ℓ } N ℓ=1 for (Γ, L) and symbols {a ℓ } N ℓ=1 ⊂ S k Γ,L,ρ (U ) such that A can be written in the form where T ℓ and T ′ ℓ are FIOs quantizing κ ℓ and κ −1 ℓ for ℓ = 1, . . . , N . We say that A is a semiclassical pseudodifferential operator with symbol class S k Γ,ρ (U ), and write A ∈ Ψ k Γ,ρ (U ), if there are symbols {a ℓ } N ℓ=1 ⊂ S k Γ,ρ (U ) such that A can be written in the form (5.23).
Furthermore, if A ∈ Ψ k Γ,ρ , then A ∈ Ψ k Γ,L,ρ and σ Γ (A) = σ Γ,L (A). Lemma 5.10. Let Γ ⊂ U ⊂ T * M be a co-isotropic submanifold, U an open set, and L a Lagrangian foliation on Γ. There is a non-canonical quantization procedure Proof. Let {(κ ℓ , U ℓ )} N ℓ=1 be charts for (Γ, L) such that {U ℓ } N ℓ=1 is a locally finite cover for U , T ℓ and T ′ ℓ quantize respectively κ ℓ and κ −1 ℓ , and σ(T ′ ℓ T ℓ ) ∈ C ∞ c (U ℓ ) is a partition of unity on U . Let a ∈ S k Γ,L,ρ (U ). Then, define a ℓ ∈ S k Γ 0 ,L 0 ,ρ such that where χ ℓ ≡ 1 on supp σ(T ′ ℓ T ℓ ). We then define the quantization map The fact that σ Γ,L • Op Γ,L h is the natural projection follows immediately. Now, fix A ∈ Ψ k Γ,L,ρ (U ). Put a 0 = σ Γ,L (A). Then, Then, letting a ∼ k h k(1−ρ) a k , we have Remark 5. Note that E := N ℓ=1 T ℓ T ′ ℓ is an elliptic pseudodifferential operator with symbol 1. Therefore, there is E ′ ∈ Ψ 0 with σ(E ′ ) = 1 such that E ′ EE ′ = Id. Replacing T ℓ by E ′ T ℓ and T ′ ℓ by T ′ ℓ E ′ , we may (and will) ask for N ℓ=1 T ℓ T ′ ℓ = Id, and so Op Γ,L h (1) = Id. Lemma 5.11. Let Γ ⊂ U ⊂ T * M be a co-isotropic submanifold. If A ∈ Ψ k Γ,ρ (U ) and P ∈ Ψ m (U ) with symbol p such that for every q ∈ Γ we have H p (q) ∈ T q Γ. Then, Proof. Suppose that W F ′ h (A) ⊂ U ℓ for U ℓ ⊂ U open, and suppose that κ : U ℓ → T * R n is a chart for (Γ, L). Note that we may assume that WF h (A) ′ ⊂ U ℓ and then use a partition of unity to cover U with a family {U ℓ } ℓ . Therefore, there exist a ∈ S k Γ,ρ and a Fourier integral operator T that is microlocally elliptic on U ℓ and quantizes κ, such that Then, on WF h ′ (A), Hence, a direct computation using Lemma 5.3 gives . In particular, 6. An Uncertainty principle for co-isotropic localizers The first goal of this section is to build a family of cut-off operators X y with y ∈ M that act as the identity on the shrinking ball B(y, h ρ ) and such that they commute with P in a fixed size neighborhood of y. This is the content of section 6.1. The second goal is to control X y 1 X y 2 L 2 →L 2 in terms of the distance d(y 1 , y 2 ), as this distance shrinks to 0. We do this in Section 6.2. Finally, in Section 6.3, we study the consequences of these estimates for the almost orthogonality of X y i .
In order to localize to the ball B(y, h ρ ) in a way compatible with microlocalization we need to make sense of 1, 1)), as an operator in some anisotropic pseudodifferential calculus. As a function, χ y is in the symbol class S −∞ Γy,Ly , where Γ y , L y are the co-isotropic submanifold and Lagrangian foliation defined as follows: Fix δ > 0, to be chosen small later, and for each x ∈ M let Γ y := |t|< 1 2 inj(M ) ϕ t (Ω y ), Ω y := ξ ∈ T * y M : 1 − |ξ| g < δ . (6.1) In this section, we construct localizers to Γ y adapted to the Laplacian and study the incompatibility between localization to Γ y 1 and Γ y 2 as a function of the distance between y 1 , y 2 ∈ M . Let y ∈ M . In what follows we work with the Lagrangian foliation L y of Γ y given by L y = {L y,q }q ∈Γy , L y,q = (ϕ t ) * (T q T * y M ), whereq = ϕ t (q) for some |t| < 1 2 inj(M ) and q ∈ Ω y .
Remark 6. In fact, it will be enough for us to show that χ y (x)χ(δ −1 (|hD| g − 1)) ∈ Ψ Γy,Ly,ρ since we will be working near the characteristic variety for the Laplacian.
First, performing stationary phase in (w n , η n ) yields Next, note that since φ(0, x, η) = x, η , withφ such that for every multi-index α there exists C α > 0 with |∂ α t,x,ηφ | ≤ C α . Next, we claim that there exists C > 0 such that x, η) = 0. (6.10) We postpone the proof of (6.10) and proceed to finish the proof of the lemma.
For each x ∈ M let Γ x be as in (6.1). (See Figure 6.2 for a schematic representation of these two co-isotropic submanifolds.) Then we have the following result. Figure 1. A pictorial representation of the co-isotropics involved in Corollary 6.3 where γ x i ,x j is the geodesic from x i to x j . Localization to both Γ x i and Γ x j implies localization in the non-symplectically orthogonal directions, x ′ and ξ ′ . The uncertainty principle then rules this behavior out.
Note that for all t H p (H q i (t) q j (t)) = 0 and {q i (t), q j (t)} | H t = ∂ ξn q i (t)∂ xn q j (t) − ∂ ξn q j (t)∂ xn q i (t) +H q i (t) q j (t), 6.3. Almost orthogonality for coisotropic cutoffs. In this section, we finally prove an estimate which shows that co-isotropic cutoffs associated with Γ x i for many x i are almost orthogonal. This, together with the fact that these cutoffs respect pointwise values near x i , is what allows us to control the number of points at which a quasimode may be large. Proof. To prove this bound we will decompose the sum in (6.15) as First, we note that by Corollary 6.3, there exists C > 0 such that for i = j Therefore, by the the Cotlar-Stein lemma, j∈J X j ≤ sup j∈J X j + i∈J \{j} X * j X i To estimate the sum, observe that there exists C > 0 such that for any j ∈ J and any positive integer k 1 C 2 kn ≤ #{i : 2 k R ≤ d(x i , x j ) ≤ 2 k+1 R} ≤ C2 (k+1)n . In particular, there is C > 0 such that for any j ∈ J We next proceed to control the second term in (6.16). LetX j ∈ Ψ −∞ Γx j ,ρ such that X j X j = X j + O(h ∞ ) L 2 →L 2 . By the Cotlar-Stein Lemma, (6.19) By Corollary 6.3 there exists C > 0 such that for k = ℓ, i = j, X * kXℓ X ℓ X * jX * j X i ≤ Ch (n−1)(2ρ−1) min{1, h (n−1) 2 (2ρ−1) d(x j , x ℓ ) − n−1 2 } (d(x k , x ℓ )d(x j , x i ))