DEFECT MEASURES OF EIGENFUNCTIONS WITH MAXIMAL L

ANNALES DE

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Jeffrey Galkowski

Defect measures of eigenfunctions with maximalL^∞growth Tome 69, n^o4 (2019), p. 1757-1798.

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DEFECT MEASURES OF EIGENFUNCTIONS WITH MAXIMAL L

GROWTH

by Jeffrey GALKOWSKI (*)

Abstract. — We characterize the defect measures of sequences of Laplace eigenfunctions with maximalL^∞growth. As a consequence, we obtain new proofs of results on the geometry of manifolds with maximal eigenfunction growth ob- tained by Sogge–Toth–Zelditch, and generalize those of Sogge–Zelditch to the smooth setting. We also obtain explicit geometric dependence on the constant in Hörmander’sL^∞bound for high energy eigenfunctions, improving on estimates of Donnelly.

Résumé. — Nous caractérisons les mesures de défauts de séquences de fonc- tions propres de Laplace avec croissance L^∞ maximale. En conséquence, nous obtenons des nouvelles preuves de résultats sur la géométrie des variétés avec une croissance des fonctions propres maximale obtenus par Sogge–Toth–Zelditch, et nous généralisons ceux de Sogge–Zelditch au cas lisse. Nous obtenons également une dépendance géométrique explicite de la constante de HörmanderL^∞liée aux functions propres de haute énergie, améliorant les estimations de Donnelly.

1. Introduction

Let (M, g) be aC^∞compact manifold of dimensionnwithout boundary.

Consider the solutions to

(1.1) (−∆g−λ²_j)uλ_j = 0, kuλ_jkL² = 1

as λj → ∞. It is well known [1, 11, 14] (see also [25, Chapter 7]) that solutions to (1.1) satisfy

(1.2) kuλ_jk_L^∞_(M)6Cλ

n−1 2

j

Keywords:eigenfunctions, defect measures, sup-norms.

2010Mathematics Subject Classification:35P20, 58J50.

(*) The author would like to thank John Toth for many stimulating discussions and for comments on a previous version. The author is grateful to the referee for careful reading and many helpful comments which improved the exposition. Thanks also to the National Science Foundation for support under the Mathematical Sciences Postdoctoral Research Fellowship DMS-1502661.

and that this bound is saturated e.g. on the sphere. Estimates forL^pnorms of eigenfunctions improving on those given by interpolation between (1.2) andku_λjk_L2= 1 have been available since the seminal work of Sogge [16].

Since there are examples where these estimates are sharp, it is natural to consider situations which produce sharp examples for (1.2). Previous works [2, 12, 19, 20, 21, 22, 23, 24] have studied the connections between growth ofL^∞norms of eigenfunctions and the global geometry of the man- ifoldM. The works of Sogge [18] and Blair–Sogge [4, 5] study similar ques- tions for lowL^p norms.

In this article, we study the relationship betweenL^∞growth andL²con- centration of eigenfunctions (this direction of inquiry was initiated in [9]).

We measure L² concentration of eigenfunctions using defect measures, a sequence{uhj}has defect measure µif for anya∈C_c^∞(T^∗M),

(1.3)

a(x, hjD)uh_j, uh_j

→ Z

T^∗M

a(x, ξ)dµ.

We writea(x, hD) for a semiclassical pseudodifferential operator given by the quantization of the symbol a(x, ξ) (see [25, Chapters 4, 14]) and let hj =λ⁻¹_j when considering the solutions to (1.1).

By an elementary compactness/diagonalization argument it follows that any L² bounded sequence uh possesses a further subsequence that has a defect measure in the sense of (1.3) [25, Theorem 5.2]. Moreover, a standard commutator argument shows that if

p(x, hD)u=o_L2(h), forp∈S^k(T^∗M) real valued with

|p|>chξi^k on|ξ|>R,

thenµis supported on Σ :={p= 0} and is invariant under the bicharac- teristic flow ofp; that is, if G_t= exp(tH_p) : Σ→Σ is the bicharacteristic flow, (Gt)∗µ=µ,∀t∈R[25, Theorems 5.3, 5.4].

Remark 1.1. — We will usually writeGt(q) for the bicharacteristic flow applied to a pointq ∈T^∗M. However, it will sometimes be useful distin- guish between the position and momentum ofqand in these cases we will writeq= (x, ξ) and write G_t(x, ξ) for the bicharacteristic flow applied to (x, ξ)∈T^∗M.

Rather than studying only eigenfunctions of the Laplacian, we replace

−∆g−λ²_jby a general semiclassical pseudodifferential operator and replace

eigenfunctions with quasimodes. To this end, we say that uis compactly microlocalized if there existsχ∈C_c^∞(R) with

Oph(1−χ(|ξ|))u=O_S(h^∞kuk_L2(M))

where Oph is a quantization procedure giving pseuodifferential operators onM (see e.g. [8, Appendix E], see also Appendix A). ForP∈Ψ^m(M) i.e.

anh-pseudodifferential operator of orderm, we say thatuis aquasimode forP if

P u=o_L2(h), kuk_L2 = 1.

Remark 1.2. — Althoughuimplicitly depends onh, we suppress this in our notation to avoid overburdening the writing.

Forx₀∈M, let Σx0 := Σ∩T_x^∗

0M and define respectively theflow out of Σx₀ andtimeT flowout ofΣx₀ by

Λx₀ :=

∞

[

T=0

Λx₀,T, Λx₀,T :=

T

[

t=−T

Gt(Σx₀).

Remark 1.3. — Note that in the case that P =−h²∆g−1, Σ =S^∗M and Σx₀ =S_x^∗

0M.

LetH^rdenote the Hausdorff-rmeasure with respect to the Sasaki metric onT^∗M or more precisely the metric induced onT^∗M by pulling back the Sasaki metric onT M (see for example [3, Chapter 9] for a treatment of the Sasaki metric). Note that we choose to use the Sasaki metric onT^∗M induced by the metric onM for concreteness, but any other metric onT^∗M will work equally well for our purposes. For a Borel measureρ on T^∗M, letρx₀ :=ρ|Λ_x0 i.e. ρx₀(A) :=ρ(A∩Λx₀). Recall that two Borel measures on a set Ω,µ and ρ, aremutually singular (written µ⊥ρ) if there exist disjoint setsN, P ⊂Ω so that Ω =N∪P andµ(N) =ρ(P) = 0.

The main theorem characterizes the defect measures of quasimodes with maximal growth.

Theorem 1.4. — LetP ∈Ψ^m(M)be anh-pseudodifferential operator with real principal symbolpsatisfying

(1.4) ∂ξp6= 0on{p= 0}.

Supposeuis a compactly microlocalized quasimode forP with

(1.5) lim sup

h→0

hⁿ⁻¹² kukL^∞ >0

and defect measureµ. Then there existsx₀∈M andx(h)→x₀ so that (1.6) lim sup

h→0

hⁿ⁻¹² |u(x(h))|>0, µx₀=ρx₀+fdH_xⁿ₀,

where 0 6= f ∈ L¹(Λx0,H_xⁿ₀), ρx0 ⊥ Hⁿ_x0, and both fdHⁿ_x0, and ρx0 are invariant underGt.

One way of interpreting Theorem 1.4 is that a quasimode with maximal L^∞growth nearx0must have energy on a positive measure set of directions entering T_x^∗₀M. That is, it must have concentration comparable to that of the zonal harmonic. (See [9, Section 4] for a description of the defect measure of the zonal harmonic.)

Theorem 1.4 is an easy consequence of the following theorem (see sec- tion 2 for the proof that Theorem 1.5 implies Theorem 1.4).

Theorem 1.5. — Letx0 ∈M and P ∈Ψ^m(M)be anh-pseudodiffer- ential operator with real principal symbolpsatisfying

∂ξp6= 0on{p= 0}.

There exists a constantC_ndepending only onnwith the following property:

Suppose thatuis compactly microlocalized quasimode forP and has defect measureµ. Defineρ_x0⊥ Hⁿ_x

0 andf ∈L¹(Λ_x0;Hⁿ_x

0)by µx₀ =:ρx₀+fdHⁿ_x0. Then for allr(h) =o(1),

lim sup

h→0

hⁿ⁻¹² kukL^∞(B(x₀,r(h)))6C_n Z

Σ_x0

pf

s|ν(H_p)|

|∂ξp|g

d vol_Σx

0

where ν is a unit (with respect to the Sasaki metric) conormal to Σx₀

in Λ_x0, vol_Σx

0 is the measure induced by the Sasaki metric on T^∗M, and

|∂ξp|g=|∂ξp·∂x|g. Furthermore,fdHⁿ_x0 isGtinvariant.

In particular, ifµx₀ ⊥ Hⁿ_x0, then

kuk_L^∞_(B(x0,r(h)))=o(h¹⁻ⁿ² ).

Remark 1.6.

(1) We may assume without loss of generality that Σ is compact. This follows from the fact thatuis compactly microlocalized. In partic- ular, letχ ∈C_c^∞(R) have Oph(1−χ(|ξ|))u=OS(h^∞). Thenu is a quasimode forPe =P+N Op_h(hξi^m)Op_h(1−χ(|ξ|)) and for N large enough,{pe= 0} is compact. Therefore, we may work withpe rather thanp. This furthermore implies that we may assume Σ_x0 is a manifold since∂ξp6= 0 on Σ.

(2) Note that ∂_ξp·∂_x = dπH_p where π : T^∗M → M is the natural projection map. Therefore,∂ξp·∂xis a well defined invariant vector field. The appearance of this factor in Theorem 1.5 quantifies the fact that bicharacteristics ofHp are not tangent to vertical fibers.

It is precisely the tangency of these bicharacteristics which causes a change of behavior when∂ξp= 0.

(3) Finally, observe that if one fixes geodesic normal coordinates atx0, then the Sasaki metric onT_x^∗

0M is equal to the Euclidean metric and hence, in these coordinates, d volΣ_x0 is the volume induced by the Euclidean metric.

To see that Theorem 1.5 applies to solutions of (1.1), let hj = λ⁻¹_j . Writingu=u_λj andh=h_j,

(−h²∆g−1)u= 0.

Then, (−h²∆g−1) =p(x, hD) withp=|ξ|²_g−1 +hr and therefore, the elliptic parametrix construction shows thatuis compactly microlocalized.

Since∂ξ_jp= 2g^ijξi, ∂ξp6= 0 onp= 0 and Theorem 1.5 applies. In Section 2, we use Theorem 1.5 withP =−h²∆_g−1 to give explicit bounds on the constantCin (1.2) in terms of theinjectivity radius ofM, inj(M), thereby improving on the bounds of [7] at high energies.

Corollary 1.7. — There exists Cen >0 depending only onnso that for all(M, g) compact, boundaryless Riemannian manifolds of dimension nand allε >0, there exists λ0 =λ0(ε, M, g)>0 so that forλj > λ0 and u_λj solving (1.1)

kuλ_jkL^∞6 Cen

inj(M)^1/2 +ε

! λ

n−1 2

j .

Theorem 1.5 is sharp in the following sense. Let P=−h²∆_g−1 andG_t as above.

Theorem 1.8. — Suppose there exists z₀ ∈ M, T > 0 so that GT(z0, ξ) = (z0, ξ) for all (z0, ξ) ∈ S_z^∗₀M. Let ρz₀ ⊥ H_zⁿ₀ be a Radon measure onΛ_z0 invariant under G_t and06f ∈L¹(Λ_z0,H_zⁿ

0)be invariant underGtso that

kfk_L1(Λz0,Hⁿ_z

0)+ρz₀(Λz₀) = 1.

Then there existhj→0 and{uh_j}^∞_j=1 solving

(−h²_j∆g−1)uh_j =o(hj), kuh_jk_L2 = 1, lim sup

j→∞

h

n−1 2

j kuhjkL^∞ >(2π)¹⁻ⁿ² Z

Σ_z0

pfd Vol_Σz

0

and having defect measureµ=ρ_z0+fd volΛz0.

Notice that we do not claim the existence of exact eigenfunctions hav- ing prescribed defect measures in Theorem 1.8, instead constructing only quasimodes.

1.1. Relation with previous results

As far as the author is aware, the only previous work giving conditions on the defect measures of eigenfunctions with maximalL^∞ growth is [9].

Theorem 1.5 improves on the conditions given in [9, Theorem 3]; replacing Hⁿ_x0(suppµx₀) = 0 with the sharp conditionµx₀ ⊥ Hⁿ_x0. To see an example of how these conditions differ, fixx₀∈M such that every geodesic through x0is closed and let{ξk}^∞_k=1⊂S_x^∗₀M be a countable dense subset. Suppose that the defect measure of{uλ_j}is given by

µ=X

k

a_kδ_γk, a_k>0

where γ_k is the geodesic emanating from (x₀, ξ_k). Then suppµ_x0 = Λ_x0, but µx₀ ⊥ Hⁿ_x0, so Theorem 1.5 applies to this sequence but the results of [9] do not. Furthermore, Theorem 1.5 gives quantitative estimates on the growth rates of quasimodes in terms of their defect measures.

We are able to draw substantial conclusions about the global geometry of a manifoldM having quasimodes with maximalL^∞ growth from The- orem 1.5. The results of Sogge–Toth–Zelditch [19, Theorems 1 (1), 2] and hence also Sogge–Zelditch [20, Theorem 1.1] are corollaries of Thoerem 1.5.

Forx0∈M, define the mapTx₀: Σx₀ →Rt {∞}by (1.7) Tx₀(ξ) := inf{t >0|Gt(x0, ξ)∈Σx₀}.

Then, define theloop setby

Lx0 :={ξ∈Σ_x0|T_x0(ξ)<∞}, and thefirst return map ηx₀ :Lx₀ →Σx₀ by

G_Tx

0(ξ)(x0, ξ) = (x0, ηx₀(ξ)).

Finally, define the set ofrecurrent points by (1.8) Rx0 :=





 ξ∈Σ_x0

ξ∈ T

T >0

S

t>TGt(x0, ξ)∩Σx₀

∩ T

T >0

S

t>TG_−t(x0, ξ)∩Σx₀





 ,

where the closure is with respect to the subspace topology on Σx₀.

Corollary 1.9. — Let(M, g)be a compact boundaryless Riemannian manifold andP satisfy (1.4). Suppose thatvolΣ_x0(Rx₀) = 0. Then for any r(h) =o(1)andua compactly microlocalized quasimode forP,

kuk_L∞(B(x₀,r(h)))=o(h¹⁻ⁿ² ).

Moreover, the forward direction of [21, Theorem 1.1] with the analyticity assumption removed is an easy corollary of Theorem 1.5. To state the theorem recall that d vol_Σx

0 denote the measure induced on Σ_x0 from the Sasaki metric onT^∗M. We define the unitary Perron–Frobenius operator U_x0:L²(R_x0,d vol_Σx

0)→L²(R_x0,d vol_Σx

0) by (1.9) Ux₀(f)(ξ) :=p

Jx₀(ξ)f(ηx₀(ξ)), where, writing

Gt(x0, ξ) = (xt(x0, ξ), ηt(x0, ξ)), we have that

(1.10) J_x0(ξ) =

detD_ξη_t|t=T_x0(ξ)

is the Jacobian factor so that forf ∈L¹(Σ_x0) supported onL_x0, Z

η_x^∗₀f Jx₀(ξ)d volΣx0 = Z

f(ξ)d volΣx0.

See [15, Section 4] for a more detailed discussion ofUx₀. We say thatx0 is dissipativeif

(1.11) n

f ∈L²(Rx₀,d volΣ_x0)

Ux₀(f) =fo

={0}.

Corollary 1.10. — Let(M, g)be a compact boundaryless Riemann- ian manifold andP satisfy (1.4). Suppose thatx₀ is dissipative. Then for r(h) =o(1)andua compactly microlocalized quasimode forP,

kukL^∞(B(x₀,r(h)))=o(h¹⁻ⁿ² ).

The dynamical arguments in [22] show that if (M, g) is a real analytic surface and P = −h²∆_g−1, then x₀ being non-dissipative implies that x0 is a periodic point for the geodesic flow, i.e. a point so that there is a T >0 so that every geodesic starting from (x₀, ξ)∈S_x^∗

0M smoothly closes at timeT.

1.2. Comments on the proof

While the assumption P u = oL²(h) implies a global assumption on u, similar to that in [9], the analysis here is entirely local. The global conse- quences in Corollaries 1.9 and 1.10 follow from dynamical arguments using invariance of defect measures.

We take a different approach from that in [9] choosing to base our method on the Koch–Tataru–Zworski method [13] rather than explicit knowledge of the spectral projector. This approach gives a more explicit explanation for the L^∞ improvements from defect measures. In Section 4 we sketch the proof of Theorem 1.5 in the case that µx₀ ⊥ Hⁿ_x0 using the spectral projector.

The idea behind our proof is to estimate the absolute value ofuatx0in terms of the degree to which energy concentrates along each bicharacteristic passing through Σx₀. Either too much localization or too little localization will yield an improvement over the naive bound. By covering Λ_x0 with appropriate cutoffs to tubes around bicharacteristics we are then able to give o(h¹⁻ⁿ² ) bounds whenever µ_x0 ⊥ Hⁿ_x

0. The proof relies, roughly, on the fact that if a compactly microlocalized function u on R^m has defect measure supported at (y0, η0), then kukL^∞ = o(h^−m/2) rather than the standard estimateO(h^−m/2).

2. Consequences of Theorem 1.5

We first formulate a local result matching those in [19, 20] more closely.

Corollary 2.1. — Let x0 ∈ M and P ∈ Ψ^m(M) satisfying the as- sumption of Theorem 1.5. Then there exists a constantC_n depending only onnwith the following property. Suppose that uis a compactly microlo- calized quasimode forP, and has defect measureµ. Define ρx₀ ⊥ Hⁿ_x

0 and f ∈L¹(Λx0;H_xⁿ₀)by

µx₀ =:ρx₀+fdHⁿ_x0.

Then for allε >0, there exists a neighborhoodN(ε)ofx0 andh0(ε)such that for0< h < h₀(ε),

kuk_L^∞_(N(ε)) 6h⁻ⁿ⁻¹² Cn

Z

Σx0

pf

s|ν(Hp)|

|∂_ξp|_g d volΣ_x0+ε

! .

Proof that Theorem 1.5 implies Corollary 2.1. — Let Aex₀:=Cn

Z

Σx0

pf s

ν(Hp)

|∂_ξp|_gd volΣ_x0

and suppose that there existsε >0 such that for allr >0,

(2.1) lim sup

h→0

h¹⁻ⁿ² kuhk_L∞(B(x₀,r))>Aex₀+ε.

Fixr₀>0. Then by (2.1) there existsx∈B(x₀, r₀),h₀>0 so that

|u_h0(x)|h

n−1 2

0 >Ae_x0+ε 2. Assume that there exist{hj}^N_j=0 and{xj}^N_j=0so that

hj6 h_j−1

2 , xj∈B(x0, r02^−j), h

n−1 2

j |u(xj)|>Aex₀+ε 2. By (2.1), there existshk ↓0 andxk ∈B(x0, r02^−N−1) such that

h

1−n 2

k |uhk(x_k)|>Ae_x0+ε 2.

Therefore, we can choose k0 large enough so that hk₀ 6 ^h2^N and let (h_N+1, x_N+1) = (h_k0, x_k0), Hence, by induction, there exists h_j ↓0,x_j → x0 such that

h

n−1 2

j |uhj(x_j)|>Ae_x0+ε 2,

contradicting Theorem 1.5.

Proof that Theorem 1.5 implies Theorem 1.4. — Compactness of M together with Corollary 2.1 withf ≡0 implies the contrapositive of Theo- rem 1.4, in particular, ifµx₀⊥ Hⁿ_x0 for allx0, thenkukL^∞ =o(h¹⁻ⁿ² ).

2.1. Proof of Corollaries 1.9 and 1.10 from Theorem 1.5

Lemma 2.2. — Fixx0 ∈M and suppose that uis compactly microlo- calized withP u=o_L2(h). Defineρ_x0 ⊥ Hⁿ_x

0 andf ∈L¹(Λ_x0;Hⁿ_x

0)by µx₀ =ρx₀+fdHⁿ_x0.

Thenf|Σ_x0 ∈L¹(volΣ_x0)andf|Σ_x0(1−1R_x0) = 0almost everywhere.

Proof. — Forξ0∈Σx₀ and ε >0 letB(ξ0, ε)⊂Σx₀ be the open ball of radiusεand

V := [

−2δ<t<2δ

Gt(B(ξ0, ε)).

Observe that by Theorem 1.5 the triple (Λx0, fdHⁿ_x0, G_t) forms a measure preserving dynamical system. The Poincaré recurrence theorem [6, Propo- sitions 4.2.1, 4.2.2] implies that for fdHⁿ_x

0 a.e. (x, ξ) ∈ V there exists t^±_n → ±∞ so thatG_t^±

n(x, ξ) ∈V. By the definition of V, there exists s^±_n with|s^±_n−t^±_n|<2δsuch thatG_s^±

n(x, ξ)∈B(ξ0, ε). In particular, forfdH_xⁿ₀ a.e. (x, ξ)∈V,

(2.2) \

T >0

[

t>T

Gt(x, ξ)∩B(ξ0, ε)6=∅, \

T >0

[

t>T

G−t(x, ξ)∩B(ξ0, ε)6=∅.

Let

µ_Σx

0 :=f|_Σx

0|ν(Hp)||_Σx

0d vol_Σx

0.

We next show that (2.2) holds for µΣx0 a.e. point in B(ξ0, ε). To do so, suppose the opposite. Then there existsA⊂B(ξ0, ε) withµΣ_x0(A)>0 so that for each (x, ξ)∈A, there existsT >0 with

(2.3)







 [

t>T

Gt(x, ξ)



 [



 [

t>T

G−t(x, ξ)









\B(ξ0, ε) =∅.

Let

Aδ :=

δ

[

t=−δ

Gt(A).

ThenAδ ⊂V and for all (x, ξ)∈Aδ, there existsT >0 so that (2.3) holds.

Moreover, invariance offdHⁿ_x0 under G_ttogether with Lemma 3.4 implies that

(fdH_xⁿ₀)(Aδ) = 2δµΣ_x0(A)>0 which contradicts (2.2). Thus (2.2) holds forµ_Σx

0 a.e. point inB(ξ₀, ε).

Let {B(ξi, εi)} be a countable basis for the topology on Σx₀. Then for eachi, there is a subset of full measure,Be_i⊂B(ξ_i, ε_i) so that for every point ofBe_i(2.2) holds withξ₀=ξ_i,ε=ε_i. Noting thatX_i=Be_i∪(Σ_x0\B(ξ_i, ε_i)) has full measure, we conclude thatΣe_x0=∩_iX_i ⊂ R_x0has full measure and thus,µΣ_x0(Rx₀) =µΣ_x0(Σx₀), finishing the proof of the lemma.

Proof of Corollary 1.9. — LetusolveP u=o_L2(h). Then we can extract a subsequence with a defect measureµ. By Lemma 2.2,µx₀ =ρx₀+fdH_xⁿ₀ withρx₀ ⊥ Hⁿ_x0 and suppf|Σ_x0 ⊂ Rx₀. Now, if volΣ_x0(Rx₀) = 0,

Z

Σx0

pfd vol_Σx

0 = 0.

Plugging this into Theorem 1.5 proves the corollary.

Proof of Corollary 1.10. — Let u solve P u = o_L2(h). Then we can extract a subsequence with a defect measureµ. By Lemma 2.2 and The- orem 1.5, µ_x0 = ρ_x0 +fdHⁿ_x

0 where ρ_x0 ⊥ Hⁿ_x

0, suppf|_Σx

0 ⊂ R_x0, and fdH_xⁿ₀ isGtinvariant.

LetTx₀ be as in (1.7). FixT <∞and suppose A⊂ΩT :={η∈Σx₀ |Tx₀(η)6T}.

Write (0, T] =FN(ε)

i=1 (T_i−ε, T_i+ε] and ΩT =

N(ε)

G

i=1

Ω^ε_i, Ω^ε_i :=T_x⁻¹₀ ((Ti−ε, Ti+ε]).

Then, by Lemma 3.4 (using that in the case of−h²∆g−1,|ν(Hp)| ≡2) for any 0< δ small enough

Z

21Afd vol_Σx

0= 1 2δ

Z 1Sδ

−δGt(A)fdHⁿ_x

0= 1 2δ

X

i

Z 1Sδ

−δGt(A∩Ω^ε_i)fdHⁿ_x

0. Next, using invariance offdHⁿ_x

0 under G_t, we have 1

2δ X

i

Z 1Sδ

−δG_t(A∩Ω^ε_i)fdHⁿ_x0=X

i

1 2δ

Z

1S_Ti+δ

Ti−δG_t(A∩Ω^ε_i)fdHⁿ_x0 Then, by the definition of Ω^ε_i, forq∈Ω^ε_i, |Tx0(q)−T_i|< εand

X

i

1STi+δ

Ti−δG_t(A∩Ω^ε_i)−→

ε→01Sδ

−δG_t(η_x(A)) fdH_xⁿ₀ a.e.

In particular, by the dominated convergence theorem

ε→0lim X

i

1 2δ

Z

1STi+δ

Ti−δG_t(A∩Ω^ε_i)fdHⁿ_x0= 1 2δ

Z 1Sδ

−δG_t(η_x(A))fdHⁿ_x0 So, sendingδ→0 gives

2 Z

1Afd volΣx0 = 2 Z

1η_x(A)fd volΣx0

for allA⊂Ω_T measurable. TakingT → ∞then proves this for allA⊂ Lx0

measurable. In particular, changing variables, using that suppf ⊂ Rx₀ ⊂ L_x0, and writingJ_x0(ξ) as in (1.10)

f(ξ)d volΣ_x0(ξ) =f(ηx₀(ξ))·Jx₀(ξ)d volΣ_x0(ξ) which implies Ux₀

√f = √

f where Ux₀ is defined in (1.9). Observe that sincex₀ is dissipative and√

f ∈L²(R_x0,d vol_Σx

0), (1.11) implies√ f = 0.

Theorem 1.5 then completes the proof.

2.2. Spectral cluster estimates for −∆g

Let (M, g) be a smooth, compact, boundaryless Riemannian manifold of dimensionn,p=|ξ|²_g−1,Gt= exp(tHp) and

Ax:= Cn

2

vol_Σx

0(Rx0) inf_ξ∈Rx

0T_x0(ξ) ^1/2

whereT_x0 is as in (1.7) andC_n is the constant in Theorem 1.5. We consider an orthonormal basis{uλ_j}^∞_j=1of eigenfunctions of−∆g(i.e. solving (1.1)) and let

Π_[λ,λ+δ] := 1_[λ,λ+δ](p

−∆g).

Corollary 2.3. — For allε >0,x0∈M, there existsδ=δ(x0, ε)>0, a neighborhoodN(x0, ε)ofx0, andλ0=λ0(x0, ε)>0 so that forλ > λ0, (2.4) kΠ[λ,λ+δ]k²_L2(M)→L^∞(N(x0,ε))

= sup

y∈N(x0,ε)

X

λ_j∈[λ,λ+δ]

|u_λj(y)|²6(A²_x

0+ε)λⁿ⁻¹. Note that sinceGt|S^∗M parametrizes the speed 2 geodesic flow and there- fore

ξ∈Rinfx0

Tx₀(ξ)> 1

2L(x0, M)>inj(M), L(x0, M) := inf

t >0

there exists a geodesic of lengtht starting and ending atx0

,

and inj(M) denotes the injectivity radius ofM. Therefore, we could replace A_x0 in (2.4) by either of

A⁰_x0 =Cn

vol_Σx

0(R_x0) 2·L(x₀, M)

^1/2

, A⁰⁰_x0 =Cn

vol_Σx

0(R_x0) 4·inj(M)

^1/2 .

to obtain a weaker, but more easily understood statement. Corollary 2.3 is closely related to the work of Donnelly [7] and gives explicit dependence of the constant in the Hörmander bound in terms of geometric quantities.

Proof. — We start from the fact that forU ⊂M (2.5) kΠ_[λ,λ+δ]k²_L2(M)→L^∞(U)= sup

x∈U

X

λj∈[λ,λ+δ]

|uλ_j(x)|². Forw∈L²(M),

(2.6) k(−∆g−λ²)Π[λ,λ+δ]wkL² 62λδkΠ[λ,λ+δ]wkL².

Suppose that for some ε > 0 no δ, N(x₀), and λ₀ exist so that (2.4) holds. Then for allδ >0,r >0,

lim sup

λ→∞

λ¹⁻ⁿ² kΠ_[λ,λ+δ]k_L2(M)→L^∞(B(x0,r)) > Ax₀+ε.

Therefore, for all 0< m∈Z, there existsλ_k,m↑ ∞so that

(2.7) λ

1−n 2

k,mkΠ_{[λk,m,λk,m+m}−1]k_L2(M)→L^∞(B(x₀,r))> A_x0+ε.

Moreover, we may assume that for m₁< m₂, λ_k,m2 > λ_k,m1. Indeed, as- sume we have chosen such λk,m for m < M. Then there exists λk,M >

max(λ_k,M−1, λ_k−1,M) so that (2.7) holds withm=M. By convention, we letλ−1,m= 0. Now, for m16m2,

kΠ_[λ,λ+m⁻¹

2 ]kL²(M)→L^∞(B(x₀,r))6kΠ_[λ,λ+m⁻¹

1 ]kL²(M)→L^∞(B(x₀,r)), lettingλ_l=λ_l,l,λ_l→ ∞and

λ

1−n 2

l kΠ_[λl,λl+l−1]k_L2(M)→L^∞(B(x₀,r)) > Ax₀+ε.

By (2.6) for w∈L²(M)

k(−λ⁻²_l ∆g−1)Π_[λl,λl+l−1]wk_L2→L² =o(λ⁻¹_l )kΠ_[λl,λl+l−1]wk_L2→L². Fixw_l∈L²(M) withkw_lk_L2 = 1, so that

λ

1−n 2

l kvlkL^∞(B(x₀,r))> A_x0+ε, v_l:= Π[λ_l,λ_l+l⁻¹]w_l.

Then extracting a further subsequence if necessary, we may assume thatvl

has defect measureµwithµx₀ =ρx₀+fdHⁿ_x0 and hence that Corollary 2.1 applies tov_l. Furthermore, sincekvlkL² 6kwlkL² = 1,

(2.8)

Z

Λ_x0

fdH_xⁿ₀ 61.

By computing in normal geodesic coordinates at x₀, observe that for p=|ξ|²_g−1,|ν(Hp)|=|∂ξp|g= 2. Thus, Corollary 2.1, implies the existence ofr >0 small enough so that

A_x0+ε6lim sup

l→∞

λ

1−n 2

l kvlkL^∞(B(x₀,r))6C_n Z

Σ_x0

pfd vol_Σx (2.9) 0

Finally, by Lemma 2.2 and (2.8), suppf ⊂ Rx0 and kfkL¹(Λ_x0,Hⁿ_x0) 61.

Therefore, Cn

Z

Σ_x0

pfd volΣ_x0

6Cn

1 2

Z

Σ_x0

f|ν(Hp)|d volΣ_x0

!^1/2

volΣ_x0(Rx₀

1/2

=C_n





1

4·infξ∈R_x0(Tx₀(ξ)) Z

Λ_x0,infR

x0Tx0(ξ)

fdHⁿ_x0





1/2

vol_Σx

0(Rx0

1/2

6Cn

2

vol_Σx

0(R_x0) inf_ξ∈Rx

0(T_x0(ξ)) ^1/2

=Ax₀,

contradicting (2.9).

Compactness of M, the fact that volΣ_x0(Rx₀)6vol(Sⁿ⁻¹), and Corol- lary 2.3 imply Corollary 1.7.

3. Dynamical and measure theoretic preliminaries 3.1. Dynamical preliminaries

The following lemma gives an estimate on how much spreading the geo- desic flow has near a point.

Lemma 3.1. — Fixx0 ∈M. Then there exists δM,p >0 small enough andC₁>0so that uniformly fort∈[−δM,p, δM,p],

(3.1) 1

2d (x₀, ξ₁),(x₀, ξ₂)

−C₁d (x₀, ξ₁),(x₀, ξ₂)² 6d G_t(x₀, ξ₂), G_t(x₀, ξ₁) 62d (x₀, ξ₁),(x₀, ξ₂)

+C₁d (x₀, ξ₁),(x₀, ξ₂)² where d is the distance induced by the Sasaki metric. Furthermore if Gt(x0, ξi) = (xi(t), ξi(t)),

(3.2) d_M(x₁(t), x₂(t))6C₁d (x₀, ξ₁),(x₀, ξ₂) δ_M,p

wheredM is the distance induce by the metricM.

Proof. — By Taylor’s theorem Gt(x0, ξ1)−Gt(x0, ξ2)

=d_ξG_t(x₀, ξ₂)(ξ₁−ξ₂) +O_C^∞

sup

q∈Σ

|d²_ξG_t(q)|(ξ1−ξ₂)²

Now,

G_t(x₀, ξ) = (x₀, ξ) + (∂_ξp(x₀, ξ)t,−∂_xp(x₀, ξ)t) +O(t²) so

dξGt(x₀, ξ) = (0, I) +t(∂_ξ²p,−∂_ξx² p) +O(t²) In particular,

Gt(x0, ξ1)−Gt(x0, ξ2) = ((0, I) +O(t))(ξ1−ξ2) +O((ξ1−ξ2)²) and choosingδ_M,p>0 small enough gives the result.

3.2. Measure theoretic preliminaries

We will need a few measure theoretic lemmas to prove our main theorem.

Lemma 3.2. — Suppose that µx₀ =ρx₀+fdHⁿ_x0 is a finite Borel mea- sure invariant underGtandρx₀⊥ Hⁿ_x0. Thenρx₀ andfdHⁿ_x0 are invariant underG_t.

Proof. — Sinceρx₀ ⊥ Hⁿ_x0, there exist disjointN, P such thatρx₀(P) = Hⁿ_x

0(N) = 0 and Λ_x0=N∪P. SupposeAis Borel. Then the invariance of µx₀ implies

(3.3)

Z

(1_A◦G_−t−1_A)dρ_x0 = Z

(1_A−1_A◦G_−t)fdH_xⁿ₀.

Now, if A ⊂ N then the fact that Gt is a diffeomorphism implies that it maps 0 Hausdorff measure sets to 0 Hausdorff measure sets and hence Hⁿ_x0(A) =Hⁿ_x0(Gt(A)) = 0. Therefore,

(3.4) ρx0(A) =ρx0(Gt(A)), A⊂N In particular,

ρx₀(N) =ρx₀(Gt(N)) =ρx₀(Λx₀).

Using again that fort∈R, Gt: Σ→Σ is a diffeomorphism, we have ρx0(Gt(P)) =ρx0(Λx0\Gt(N)) =ρx0(Λx0)−ρx0(Gt(N)) = 0.

So, in particular,

(3.5) ρx₀(Gt(A)) = 0, A⊂P.