Kakeya-Nikodym averages and L p -norms of eigenfunctions
KKAKEYA-NIKODYM AVERAGES AND L p -NORMS OF EIGENFUNCTIONS CHRISTOPHER D. SOGGE
Abstract.
We provide a necessary and sufficient condition that L p -norms, 2 < p <
6, of eigenfunctions of the square root of minus the Laplacian on two-dimensionalcompact boundaryless Riemannian manifolds M are small compared to a naturalpower of the eigenvalue λ . The condition that ensures this is that their L -normsover O ( λ − / ) neighborhoods of arbitrary unit geodesics are small when λ is large(which is not the case for the highest weight spherical harmonics on S for instance).The proof exploits Gauss’ lemma and the fact that the bilinear oscillatory integrals inH¨ormander’s proof of the Carleson-Sj¨olin theorem become better and better behavedaway from the diagonal. Our results are related to a recent work of Bourgain whoshowed that L -averages over geodesics of eigenfunctions are small compared to anatural power of the eigenvalue λ provided that the L ( M ) norms are similarly small.Our results imply that QUE cannot hold on a compact boundaryless Riemannianmanifold ( M, g ) of dimension two if L p -norms are saturated for a given 2 < p <
6. Wealso show that eigenfunctions cannot have a maximal rate of L -mass concentratingalong unit portions of geodesics that are not smoothly closed. Introduction.
The main purpose of this paper is to slightly sharpen a recent result of Bourgain [5]concerning two-dimensional compact boundaryless Riemannian manifolds. By doing sowe shall be able to provide a natural necessary and sufficient condition concerning thegrowth rate of L p -norms of eigenfunctions for 2 < p < L -concentration aboutgeodesics.There are different ways of measuring the concentration of eigenfunctions. One is bymeans of the size of their L p -norms for various values of p >
2. If M is a compact bound-aryless manifold with Riemannian metric g = g jk ( x ) and if ∆ g is the associated Laplace-Beltrami operator, then the eigenfunctions solve the equation − ∆ g e λ j ( x ) = λ j e λ j ( x ) fora sequence of eigenvalues 0 = λ ≤ λ ≤ λ . . . . Thus, we are normalizing things so that λ j are the eigenvalues of the first-order operator (cid:112) − ∆ g . We shall also usually assumethat the e λ j have L -norm one, in which case { e λ j } provides an orthonormal basis of L ( M, dx ) where dx is the volume element coming from the metric. Earlier, in the two-dimensional case, we showed in [26] that if M is fixed then there is a uniform constant C so that for 2 ≤ p ≤ ∞ and j = 1 , , , . . . (1.1) (cid:107) e λ j (cid:107) L p ( M ) ≤ Cλ δ ( p ) j (cid:107) e λ j (cid:107) L ( M ) , Mathematics Subject Classification.
Primary: 35P99, 35L20; Secondary: 42C99.The author was supported in part by NSF Grant DMS-0555162. a r X i v : . [ m a t h . A P ] J un CHRISTOPHER D. SOGGE with δ ( p ) =
12 ( 12 − p ) , ≤ p ≤ , − p , ≤ p ≤ ∞ . These estimates are sharp for the round sphere S , and in this case they detect twotypes of concentration of eigenfunctions that occur there. Recall that on S with thecanonical metric the distinct eigenvalues are √ k + k , k = 0 , , , . . . , which repeat withmultiplicity d k = 2 k + 1. If H k , the space of spherical harmonics of degree k , is thespace of all eigenfunctions with eigenvalue √ k + k , and if H k ( x, y ) is the kernel ofthe projection operator onto H k , then the k -th zonal function at x ∈ S is Z k ( y ) =( H k ( x , x )) − / H k ( x , y ). Its L -norm is one but its mass is highly concentrated at ± x where it takes on the value (cid:112) d k / π . Explicit calculations show that (cid:107) Z k (cid:107) L p ( S ) ≈ k δ ( p ) for p ≥ M = S with the roundmetric (1.1) cannot be improved for this range of exponents. Another extreme type ofconcentration is provided by the highest weight spherical harmonics which have massconcentrated on the equators of S , which are its geodesics. The ones concentrated onthe equator γ = { ( x , x , x + x = 1 } are the functions Q k , which are the restrictionsof the R harmonic polynomials k / ( x + ix ) k to S = { x ; | x | = 1 } . One can check thatthe Q k have L -norms comparable to one and L p -norms comparable to k ( − p ) when2 ≤ p ≤ Q k have Gaussian type concentration aboutthe equator γ . Specifically, if T k − / ( γ ) denotes all points on S of distance smaller than k − / from γ then one can check that(1.2) lim inf k →∞ (cid:90) T k − / ( γ ) | Q k ( x ) | dx > . For future reference, obviously the Q k also have the related property that(1.3) (cid:90) γ | Q k | ds ≈ k / , if ds is the measure on γ induced by the volume element.Thus, the sequence of highest weight spherical harmonics shows that the norms in(1.1) (for 2 < p < L -normalized eigen-functions one has(1.4) lim sup j →∞ λ − δ ( p ) j (cid:107) e λ j (cid:107) L p ( M ) = 0for every 2 < p ≤ ∞ . This was verified for exponents p > x through which a positive measure ofgeodesics starting at x loop back through x then (cid:107) e λ (cid:107) ∞ = o ( λ / ). By interpolating withthe estimate (1.1) for p = 6, this yields (1.4) for all p >
6. Corresponding results werealso obtained in [30] for higher dimensions. Recently, these results were strengthenedby Toth, Zelditch and the author [29] to allow similar results for quasimodes under the
AKEYA-NIKODYM AVERAGES AND L p -NORMS OF EIGENFUNCTIONS 3 weaker condition that at every point x the set of recurrent directions for the first returnmap for geodesic flow has measure zero in the cosphere bundle S ∗ x M over x .Other than the partial results in Bourgain [5], there do not seem to be any resultsaddressing when (1.4) holds for a given 2 < p < L -normalized eigenfunctions have uniformly bounded L -norms). Further-more, there do not seem to be results addressing the interesting endpoint case of p = 6,where one expects both types of concentration mentioned before to be relevant.Recently authors have studied the L norms of eigenfunctions over unit-length geodesics.Burq, G´erard and Tzvetkov [6] showed that if Π is the collection of all unit lengthgeodesics then(1.5) sup γ ∈ Π (cid:90) γ | e λ j | ds (cid:46) λ / j (cid:107) e λ j (cid:107) L ( M ) , j = 1 , , , . . . , which is sharp in view of (1.3). Related results for hyperbolic surfaces were obtainedearlier by Reznikov [20], who opened up the present line of investigation. The proof of(1.5) boils down to bounds for certain Fourier integral operators with folding singularities(cf. Greenleaf and Seeger [12], Tataru [32]). In §
3, we shall use ideas from [12], [32], and[10], [16], [29], [30] to show that if γ ∈ Π andlim sup j →∞ λ − / j (cid:90) γ | e λ j | ds > , then the geodesic extension of γ must be a smoothly closed geodesic. Presumably it alsohas to be stable, but we cannot prove this. Further recent work on L -concentrationalong curves can be found in Toth [33].In a recent paper [5], Bourgain proved an estimate that partially links the norms in(1.1) and (1.5), namely that for all p ≥ γ ∈ Π (cid:90) γ | e λ j | ds (cid:46) λ /pj (cid:107) e λ j (cid:107) L p ( M ) . Of course for p = 2, this is just (1.5); however, an interesting feature of (1.6) is thatthe estimate for a given 2 < p ≤ e λ jk is asequence of eigenfunctions with (relatively) small L p ( M ) norms for a given 2 < p ≤
6, itfollows that its L -norms over unit geodesics must also be (relatively) small. Bourgain[5] also came close to establishing the equivalence of these two things by showing thatgiven ε > C ε so that for j = 1 , , . . . (1.7) (cid:107) e λ j (cid:107) L ( M ) ≤ C ε (cid:16) λ / εj (cid:107) e λ j (cid:107) L ( M ) (cid:17) / (cid:104) λ − / j sup γ ∈ Π (cid:90) γ | e λ j | ds (cid:105) / . Since δ (4) = 1 / ε = 0 one would obtainthe linkage of the size of the norms in (1.5) for large energy with the size of the L ( M )norms. Our main estimate in Theorem 1.1 is that a variant of (1.7) holds, which is strongenough to complete the linkage.Bourgain’s approach in proving (1.7) was to employ ideas going back to C´ordoba [9]and Fefferman [11] that were used to give a proof of the Carleson-Sj¨olin theorem [7].The key object that arose in C´ordoba’s work [9] was what he called the Kakeya maximal CHRISTOPHER D. SOGGE function in R , namely,(1.8) M f ( x ) = sup x ∈T λ − / |T λ − / | − (cid:90) T λ − / | f ( y ) | dy, f ∈ L ( R ) , with the supremum taken over all λ − / -neighborhoods T λ − / of unit line segmentscontaining x , and |T λ − / | ≈ λ − / denoting its area. The above maximal operator isnow more commonly called the Nikodym maximal operator as this is the terminology inBourgain’s important papers [2]–[4] which established highly nontrivial progress towardsestablishing the higher dimensional version of the Carleson-Sj¨olin theorem for Euclideanspaces R n , n ≥ γ ∈ Π is a unit geodesic, one could consider the λ − / -tubeabout it given by T λ − / ( γ ) = { y ∈ M ; inf x ∈ γ d g ( x, y ) < λ − / } , with d g ( x, y ) being the geodesic distance between x and y . Then if Vol g ( T λ − / ( γ ))denotes the measure of this tube, the analog of (1.8) would be M f ( x ) = sup x ∈ γ ∈ Π g ( T λ − / ( γ )) (cid:90) T λ − / | f ( y ) | dy. These operators have been studied before because of their applications in harmonic anal-ysis on manifolds. See e.g. [18], [28]. As was shown in [17], following the earlier paper[4], they are much better behaved in 2-dimensions compared to higher dimensions.As (1.7) suggests, it is not the size of the L -norm of M f for f ∈ L ( M ) that isrelevant for estimating L ( M )-norms of eigenfunctions but rather the sup-norm of thisquantity with f = | e λ j | , which up to the normalizing factor in front of the integral isthe quanitity sup γ ∈ Π (cid:90) T λ − / ( γ ) | e λ j ( x ) | dx. If the e λ j are L -normalized this is trivially bounded by one. In rough terms our resultssay that beating this trivial bound is equivalent to beating the bounds in (1.1) for a given2 < p < Theorem 1.1.
Fix a two-dimensional compact boundaryless Riemannian manifold ( M, g ) .Then given ε > there is a constant C ε so that for eigenfunctions e λ of (cid:112) − ∆ g witheigenvalues λ ≥ we have (1.9) (cid:107) e λ (cid:107) L ( M ) ≤ ελ / (cid:107) e λ (cid:107) L ( M ) + C ε λ / (cid:107) e λ (cid:107) L ( M ) sup γ ∈ Π (cid:90) T λ − / γ ) | e λ ( x ) | dx + C (cid:107) e λ (cid:107) L ( M ) , with C being a fixed constant which is independent of λ and ε . We shall prove this not by adapting C´ordoba’s [9] proof of the Carleson-Sj¨olin theorembut rather that of H¨ormander [15]. He obtained sharp oscillatory integral bounds in R that provided sharp B¨ochner-Riesz estimates for L ( R ) (i.e. the Carleson-Sj¨olin AKEYA-NIKODYM AVERAGES AND L p -NORMS OF EIGENFUNCTIONS 5 theorem), which turns out to be the endpoint case for this problem in 2-dimensions.H¨ormander’s approach was to turn this L -problem into an L -problem by squaring theoscillatory integrals and then estimating their L -norms. As his proof shows, the resultingbilinear operators that arise are better and better behaved away from the diagonal, andthis fact is what allows us to take the constant in front of the first term in the right sideof (1.9) to be arbitrarily small (at the expense of the 2nd term).Stein [31] provided a generalization of H¨ormander’s oscillatory integral theorem tohigher dimensions in a way that proved to be sharp because of a later constructionof Bourgain [4]. Bourgain’s example and related ones in [17] suggest that extendingthe results of this paper to higher dimensions (where the range of exponents would be2 < p < n +1) / ( n − ≤ C N λ − N (cid:107) e λ (cid:107) for any N , but this is not important for our applications. Also, weremark that the proof of the Theorem will show that the constant C ε in (1.9) can betaken to be O ( ε − ) as ε → L -norms of eigenfunctions is equivalent to size of L -mass near geodesics. Corollary 1.2.
Let e λ jk be a sequence of eigenfunctions with eigenvalues λ j ≤ λ j ≤ . . . and unit L ( M ) -norms. Then (1.10) lim sup k →∞ sup γ ∈ Π (cid:90) T λ − / jk ( γ ) | e λ jk ( x ) | dx = 0 if and only if (1.11) lim sup k →∞ λ − / j k (cid:107) e λ jk (cid:107) L ( M ) = 0 . To prove this, we first notice that if we assume (1.10), then (1.11) must hold becauseof (1.9). Also, by H¨older’s inequality (cid:16) (cid:90) T λ − / ( γ ) | e λ ( x ) | dx (cid:17) / ≤ (cid:0) Vol g ( T λ − / ( γ )) (cid:1) / (cid:107) e λ (cid:107) L ( M ) (cid:46) λ − / (cid:107) e λ (cid:107) L ( M ) , and so (1.11) trivially implies (1.10).If we use Bourgain’s estimate (1.6) and (1.1) we can say a bit more. CHRISTOPHER D. SOGGE
Corollary 1.3.
Let { e λ jk } ∞ k =1 be as above and suppose that < p < . Then the followingare equivalent lim sup k →∞ λ − / j k sup γ ∈ Π (cid:90) γ | e λ jk ( s ) | ds = 0(1.12) lim sup k →∞ sup γ ∈ Π (cid:90) T λ − / jk ( γ ) | e λ jk ( x ) | dx = 0(1.13) lim sup k →∞ λ − δ ( p ) j k (cid:107) e λ jk (cid:107) L p ( M ) = 0 . (1.14)To prove this result, we first note that, by the M. Riesz interpolation theorem and (1.1)for p = 2 and p = 6, (1.14) holds for a given 2 < p < p = 4,which we just showed is equivalent to (1.13). Clearly (1.12) implies (1.13). Finally, sinceBourgain’s estimate (1.6) shows that (1.14) implies (1.12), the proof of Corollary 1.3 iscomplete.Let us conclude this section by describing one more application. Recall that a sequenceof L -normalized eigenfunctions { e λ jk } ∞ k =1 satisfies the quantum unique ergodicity prop-erty (QUE) if the associated Wigner measures | e λ jk | dx tend to the Liouville measure on S ∗ M . If this is the case, then one certainly cannot havelim sup k →∞ sup γ ∈ Π (cid:90) T λ − / jk ( γ ) | e λ jk ( x ) | dx > , since the tubes are shrinking.In the case where M has negative sectional curvature Schnirelman’s [22] theorem,proved by Zelditch [35], says there is a density one subsequence { e λ jk } ∞ k =1 of all the { e λ j } satisfying QUE. Rudnick and Sarnak [21] conjectured that in the negatively curved casethere should be no exceptional subsequences violating QUE, i.e., in this case QUE shouldhold for the full sequence { e λ j } of L -normalized eigenfunctions. On the other hand, byCorollary 1.3, we have the following. Corollary 1.4.
Let M be a two-dimensional compact boundaryless Riemannian manifold.Then QUE cannot hold for M if for a given < p < there is saturation of L p norms,i.e. lim sup j →∞ λ − δ ( p ) j (cid:107) e λ j (cid:107) L p ( M ) > , with e λ j being the L -normalized eigenfunctions. See e.g. [36] for connections between QUE and the Lindel¨of hypothesis, and see [8] forrecent developments regarding the QUE conjecture.2.
Proof of Theorem 1: Gauss’ lemma and the Carleson-Sj¨olin condition.
As in [5] and [6] we shall prove our estimate by using certain convenient operators thatreproduce eigenfunctions. Specifically, we shall use a slight variant of a result from [27],Chapter 5 that was presented in [6].
AKEYA-NIKODYM AVERAGES AND L p -NORMS OF EIGENFUNCTIONS 7 Lemma 2.1.
Let δ > be smaller than half of the injectivity radius of ( M, g ) . Thenthere is a function χ ∈ S ( R ) with χ (0) = 1 so that if d g ( x, y ) is the geodesic distancebetween x, y ∈ M (2.1) χ λ f ( x ) = χ (cid:0)(cid:112) − ∆ g − λ (cid:1) f ( x ) = λ / (cid:90) M e iλd g ( x,y ) α ( x, y, λ ) f ( y ) dy + R λ f ( x ) , where (cid:107) R λ f (cid:107) L ∞ ( M ) ≤ C N λ − N (cid:107) f (cid:107) L ( M ) , for all N = 1 , , . . . , and α ∈ C ∞ has the property that | ∂ αx,y α ( x, y, λ ) | ≤ C α , for all α, and, moreover, (2.2) α ( x, y, λ ) = 0 if d g ( x, y ) / ∈ ( δ/ , δ ) . Since χ λ e λ = e λ and since the 4th power of the L -norm of R λ e λ is dominated by thelast term in (1.9), we conclude that in order to prove Theorem 1.1 it is enough to showthat, given ε > C ε so that when λ ≥ (cid:90) M (cid:12)(cid:12)(cid:12)(cid:12) λ / (cid:90) M e iλd g ( x,y ) α ( x, y, λ ) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) | f ( x ) | dx ≤ ελ / (cid:107) f (cid:107) L ( M ) (cid:107) f (cid:107) L ( M ) + C ε λ / (cid:107) f (cid:107) L ( M ) sup γ ∈ Π (cid:90) T λ − / ( γ ) | f ( x ) | dx, for, if f = e λ , the first term in the right is bounded by a fixed constant times ελ / (cid:107) e λ (cid:107) L ( M ) ,because of (1.1).After applying a partition of unity (and abusing notation a bit), we may assume that inaddition to (2.2), α ( x, y, λ ) vanishes unless x is in a small neighborhood of some x ∈ M and y is in a small neighborhood of some y ∈ M with δ/ < d g ( x , y ) < δ . We mayassume both of these neighborhoods are contained in the geodesic ball B ( x , δ ) = { y ∈ M ; d g ( x , y ) < δ } . As mentioned before, we are also at liberty to take δ > γ which passes through x and is perpendicular to the geodesic connecting x and y . Thesecoordinates will be well defined on B ( x , δ ) if δ is small. Furthermore, we may assumethat the image of γ ∩ B ( x , δ ) in the resulting coordinates is a line segment which isparallel to the 2nd coordinate axis and that all horizontal line segments s → { ( s, t ) } aregeodesic with the property that d g (( s , t ) , ( s , t )) = | s − s | . See Figure 1 below.If we use these coordinates and apply Schwarz’s inequality, we conclude that, in orderto prove (2.3), it suffices to show that given ε > C ε < ∞ so that when λ ≥ (cid:90) (cid:18) (cid:90) (cid:12)(cid:12)(cid:12) λ / (cid:90) e iλd g ( x, ( s,t )) α ( x, ( s, t ) , λ ) f ( s, t ) dt (cid:12)(cid:12)(cid:12) | f ( x ) | dx (cid:19) ds ≤ ελ / (cid:107) f (cid:107) L ( M ) (cid:107) f (cid:107) L ( M ) + C ε λ / (cid:107) f (cid:107) L ( M ) sup γ ∈ Π (cid:90) T λ − / ( γ ) | f ( x ) | dx. CHRISTOPHER D. SOGGE γ Figure 1.
Fermi normal coordinates about γ This, in turn would follow if we could show that given ε > (cid:90) (cid:12)(cid:12)(cid:12) λ / (cid:90) e iλd g ( x, ( s,t )) α ( x, ( s, t ) , λ ) h ( t ) dt (cid:12)(cid:12)(cid:12) | f ( x ) | dx ≤ ελ / (cid:107) h (cid:107) L ( dt ) (cid:107) f (cid:107) L ( M ) + C ε λ / (cid:107) h (cid:107) L ( dt ) sup γ ∈ Π (cid:90) T λ − / ( γ ) | f ( x ) | dx, with C ε depending on ε > s or on λ ≥ s ,which, after relabeling, we may assume to be s = 0. Since the proof of (2.4) for this caserelies only on Gauss’ lemma and the related Carleson-Sj¨olin condition, it also yields theuniformity in s , assuming, as we may, that α has small support.To prove this inequality, let us choose a function η ∈ C ∞ ( R ) satisfying η ( t ) = 0, | t | >
1, and (cid:80) ∞ j = −∞ η ( t − j ) ≡
1. Given λ ≥ η j ( t ) = η λ,j ( t ) = η ( λ / t − j ) . Then, given N = 1 , , . . . , we have that(2.5) (cid:12)(cid:12)(cid:12)(cid:12) λ / (cid:90) e iλd g ( x, (0 ,t )) α ( x, (0 , t ) , λ ) h ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ N (cid:88) j (cid:12)(cid:12)(cid:12)(cid:12) λ / (cid:90) e iλd g ( x, (0 ,t )) η j ( t ) α ( x, (0 , t ) , λ ) h ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:90) (cid:90) e iλ ( d g ( x, (0 ,t ))+ d g ( x, (0 ,t (cid:48) )) a N ( x, t, t (cid:48) ) h ( t ) h ( t (cid:48) ) dtdt (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) , AKEYA-NIKODYM AVERAGES AND L p -NORMS OF EIGENFUNCTIONS 9 where a N ( x, t, t (cid:48) ) = (cid:88) | j − k | >N η j ( t ) α ( x, (0 , t ) , λ ) η k ( t (cid:48) ) α ( x, (0 , t (cid:48) ) , λ )vanishes when | t − t (cid:48) | ≤ ( N − λ − / . The first term in the right side of the precedinginequality comes from applying Young’s inequality to handle the double-sum over indiceswith | j − k | ≤ N . Because of (2.5), we conclude that (2.4) would follow if we could showthat there is a constant independent of λ ≥ N = 2 , , . . . so that(2.6) (cid:13)(cid:13)(cid:13)(cid:13) λ (cid:90) (cid:90) e iλ [ d g ( x, (0 ,t )) − d g ( x, (0 ,t (cid:48) )] a N ( x, t, t (cid:48) ) h ( t ) h ( t (cid:48) ) dtdt (cid:48) (cid:13)(cid:13)(cid:13)(cid:13) L ( dx ) ≤ Cλ / N − / (cid:107) h (cid:107) L ( dt ) , and also that there is a constant C independent of j ∈ Z and λ ≥ (cid:90) (cid:12)(cid:12)(cid:12) λ / (cid:90) e iλd g ( x, (0 ,t )) η j ( t ) α ( x, (0 , t ) , λ ) h ( t ) dt (cid:12)(cid:12)(cid:12) | f ( x ) | dx ≤ Cλ / (cid:107) h (cid:107) L ( dt ) sup γ ∈ Π (cid:90) T λ − / ( γ ) | f ( x ) | dx. Indeed, by using the finite overlapping of the supports of the η j , if we set ε = CN − / ,then we see that these two inequalities and (2.5) imply (2.4) with C ε ≈ ε − . Since theproof of (2.7) only uses Gauss’ lemma and the fact that coordinates have been chosen sothat s → ( s, t ) are unit speed geodesics for fixed t , we shall just verify (2.7) for j = 0,as the argument for this case will yield the other cases as well.The next step is to see that these two inequalities are consequences of the followingtwo propositions. Proposition 2.2.
Let a ( x, t, t (cid:48) ) , x ∈ R , t, t (cid:48) ∈ R satisfy | ∂ αx a | ≤ C α for all multi-indices α and a ( x, t, t (cid:48) ) = 0 if | x | > δ or | t − t (cid:48) | > δ where δ > is small. Suppose also that φ ∈ C ∞ ( R × R ) is real and satisfies the Carleson-Sj¨olin condition on the support of a ,i.e., (2.8) det (cid:18) φ (cid:48)(cid:48) x t φ (cid:48)(cid:48) x t φ (cid:48)(cid:48)(cid:48) x tt φ (cid:48)(cid:48)(cid:48) x tt (cid:19) (cid:54) = 0 . Then if the δ > above is sufficiently small, there is a uniform constant C so that when λ, N ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:90) (cid:90) | t − t (cid:48) |≥ Nλ − / e iλ [ φ ( x,t )+ φ ( x,t (cid:48) )] a ( x, t, t (cid:48) ) F ( t, t (cid:48) ) dtdt (cid:48) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( R ) ≤ Cλ − / N − (cid:107) F (cid:107) L ( R ) . To state the next Proposition, we need to introduce one more coordinate system, whichfinally explains where the L norms over small tubular neighborhoods of geodesics comesinto play. Since we are proving (2.7) with j = 0 and since η is supported in the smallinterval [ − λ − / , λ − / ], it is natural to take geodesic normal coordinates about (0 , x → κ ( x ) that preserve this axis (and its orientation). Such a system is unique up to reflectionabout this axis, and we shall just fix one of these two choices. Proposition 2.3.
Let ψ ( x, t ) = d g (cid:0) x, (0 , t ) (cid:1) , and suppose that ρ ∈ C ∞ ( R × R ) satisfies (2.10) | ∂ mt ρ ( t ; x ) | ≤ C m ( λ / ) m , and , ρ ( t ; x ) = 0 , | t | ≥ λ − / . Suppose also that ρ vanishes when x is outside of a small neighborhood N of a fixed point ( − s , (in the Fermi normal coordinates) with s > . If x → κ ( x ) = ( κ ( x ) , κ ( x )) arethe coordinates described above, assume that points x j ∈ N are chosen so that (2.11) (cid:12)(cid:12)(cid:12) κ ( x j ) | κ ( x j ) | − κ ( x k ) | κ ( x k ) | (cid:12)(cid:12)(cid:12) ≥ cλ − / | j − k | , if | j − k | ≥ , with c > fixed. It then follows that, if N is sufficiently small, then there is a uniformconstant C , which is independent of the { x j } chosen as above, so that (2.12) λ / (cid:90) (cid:12)(cid:12)(cid:12) (cid:88) j e iλψ ( x j ,t ) ρ ( t ; x j ) a j (cid:12)(cid:12)(cid:12) dt ≤ C (cid:88) | a j | . Proposition 2.2 would imply (2.6) if φ ( x, t ) = d g ( x, (0 , t )) satisfies the Carleson-Sj¨olincondition. The fact that this is the case is well known. See e.g., Section 5.1 in [27].It follows from our choice of coordinates and the fact that if x ∈ M is fixed then theset of points {∇ x d g ( x, y ); x = x , d g ( x , y ) ∈ ( δ/ , δ ) } is the cosphere at x , S ∗ x M = { ξ ; (cid:80) g jk ( x ) ξ j ξ k = 1 } , where g jk ( x ) is the cometric (inverse to g jk ( x )). If we choosegeodesic normal coordinates κ ( y ) vanishing at x then the gradient becomes κ ( y ). Thisturns out to be equivalent to the usual formulation of Gauss’ lemma, saying that thisexponential map y → κ ( y ) is a local radial isometry. More specifically, it says thatsmall geodesic spheres centered at x get sent to spheres centered at the origin and smallgeodesic rays through x intersect these geodesic spheres orthogonally and get sent torays through the origin, which is what allows Proposition 2.3 to be true.Let us next see that Proposition 2.3 implies (2.7) for j = 0. If we take ρ ( t ; x ) = η ( t ) α ( x, (0 , t ) , λ ), then ρ satisfies (2.10). Also, if we let S j = { y ; θ ( y ) ∈ ( λ − / j, λ − / ( j + 1)] } , where θ ( y ) ∈ [0 , π ) is defined so that y = | y | (cos θ ( y ) , sin θ ( y )), then, if y = κ ( x ) arethe geodesic normal coordinates about (0 ,
0) in the Proposition 2.3, then the left side of(2.7) is dominated by (cid:88) j (cid:13)(cid:13)(cid:13) λ / (cid:90) e iλψ ( x,t ) ρ ( t ; x ) h ( t ) dt (cid:13)(cid:13)(cid:13) L ∞ ( κ − ( S j )) (cid:107) f (cid:107) L ( κ − ( S j ) ∩ K ) ≤ sup k (cid:107) f (cid:107) L ( κ − ( S k ) ∩ K ) (cid:88) j (cid:13)(cid:13)(cid:13) λ / (cid:90) e iλψ ( x,t ) ρ ( t ; x ) h ( t ) dt (cid:13)(cid:13)(cid:13) L ∞ ( κ − ( S j )) , where K is the x -support of ρ . Since the first factor on the right is dominated by thelast factor in the right hand side of (2.7) (the sup can just be taken over (0 , ∈ γ ∈ Π here), we conclude that we would obtain this inequality if we could show that there is a AKEYA-NIKODYM AVERAGES AND L p -NORMS OF EIGENFUNCTIONS 11 uniform constant so that for all choices of x j ∈ κ − ( S j )(2.13) λ / (cid:88) j (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) e iλψ ( x j ,t ) ρ ( t ; x j ) h ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:107) h (cid:107) L ( dt ) . This inequality is an estimate for an operator from L ( dt ) → (cid:96) . The dual operator isthe one in Proposition 2.3. Therefore since, by duality, (2.13) follows from (2.12) we get(2.7). To verify this assertion, we use the fact that if ρ has small support then the termsin (2.13) with ρ ( t ; x j ) (cid:54) = 0 will fulfill the hypotheses in Proposition 2.3.To finish the proof of Theorem 1.1 we must prove the two propositions. Let us startwith the first one since it is pretty standard. It is based on the well known fact that thebilinear oscillatory integrals arising in H¨ormander’s [15] proof of the Carleson-Sj¨olin [7]theorem become better and better behaved away from the diagonal. Proof of Proposition 2.2:
Let Φ( x ; t, t (cid:48) ) = φ ( x, t ) + φ ( x, t (cid:48) ) be the phase function in(2.9). Then Φ is a symmetric function in the ( t, t (cid:48) ) variables. So if we make the changeof variables u = ( t − t (cid:48) , t + t (cid:48) ) , then since | du/d ( t, t (cid:48) ) | = 2, we see that (2.8) implies that the Hessian determinant of Φsatisfies (cid:12)(cid:12)(cid:12)(cid:12) det (cid:18) ∂ Φ ∂x∂u (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ≥ c | u | , for some c > a , if the latter is small. Since Φ( x ; u ) is an even functionof the diagonal variable u , it must be a C ∞ function of u . So if we make the finalchange of variables v = (cid:16) u , u (cid:17) , then since | dv/du | = | u | , it follows that (cid:12)(cid:12)(cid:12)(cid:12) det (cid:18) ∂ Φ ∂x∂v (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≥ c, for some c >
0. This in turn implies that if v and ˜ v are close then (cid:12)(cid:12) ∇ x (cid:2) Φ( x, v ) − Φ( x, ˜ v ) (cid:3)(cid:12)(cid:12) ≥ c (cid:48) | v − ˜ v | , for some c (cid:48) >
0, and since x, v → Φ is smooth, we also have that (cid:12)(cid:12) ∂ αx (cid:2) Φ( x, v ) − Φ( x, ˜ v ) (cid:3)(cid:12)(cid:12) ≤ C α | v − ˜ v | , for all multi-indices α . Therefore, if we let K λ ( v, ˜ v ) = (cid:90) R a ( x, t, t (cid:48) ) a ( x, ˜ t, ˜ t (cid:48) ) e iλ [Φ( x,v ) − Φ( x, ˜ v )] dx, then by integrating by parts, we find that if the number δ > j = 1 , , , . . . | K λ ( v, ˜ v ) | ≤ C j (1 + λ | v − ˜ v | ) − j (2.14) ≤ C j (1 + λ | ( t + t (cid:48) ) − (˜ t + ˜ t (cid:48) ) | ) − j (1 + λ | ( t − t (cid:48) ) − (˜ t − ˜ t (cid:48) ) | ) − j . Note that the left side of (2.9) equals (cid:90) · · · (cid:90) | t − t (cid:48) | , | ˜ t − ˜ t (cid:48) |≥ Nλ − / K λ ( t, t (cid:48) ; ˜ t, ˜ t (cid:48) ) F ( t, t (cid:48) ) F (˜ t, ˜ t (cid:48) ) dtdt (cid:48) d ˜ td ˜ t (cid:48) . We next claim that there is a uniform constant C so that for λ, N ≥ ˜ t, ˜ t (cid:48) (cid:90) | t − t (cid:48) |≥ Nλ − / | K λ | dtdt (cid:48) , sup t,t (cid:48) (cid:90) | ˜ t − ˜ t (cid:48) |≥ Nλ − / | K λ | d ˜ td ˜ t (cid:48) ≤ Cλ − ( λ / /N ) . This follows from (2.14) and the fact that if τ = s then 2 sds = dτ and so, given τ ∈ R ,we have (cid:90) s ≥ Nλ − / (1 + λ | s − τ | ) − ds = 12 (cid:90) √ τ ≥ Nλ − / (1 + λ | τ − τ | ) − dτ √ τ ≤ ( λ / /N ) (cid:90) + ∞−∞ (1 + λ | τ | ) − dτ ≤ Cλ − ( λ / /N ) . Since (2.15) and Young’s inequality yield (2.9), the proof is complete. (cid:3)
To finish our task we need to prove the other Proposition, which is a straightforwardapplication of Gauss’ lemma.
Proof of Proposition 2.3:
The support assumptions on the amplitude will allow us tolinearize the function t → ψ in the proof, which is a tremendous help. Specifically, ψ ( x, t ) = ψ ( x,
0) + t ( ∂ t ψ ( x, r ( x, t ) , where(2.16) | ∂ mt r ( x, t ) | ≤ C m | t | − m , ≤ m ≤ , and | ∂ mt r | ≤ C m , m ≥ . Our choice of coordinates implies that ∂ t ψ ( x,
0) = (cid:104) ν, κ ( x ) / | κ ( x ) | (cid:105) , where the inner-product is the euclidean one and ν ∈ R is chosen so that (cid:104) ν, ∇(cid:105) is thepushforward of ∂/∂x at (0 ,
0) under the map x → κ ( x )—i.e., tangent vector to the curve t → κ ((0 , t )). Since the pushforward of ∂/∂x is itself under this map, it follows thatthe second coordinate of ν is nonzero. (See Figure 2 below.) Therefore, if N (cid:51) ( s ,
0) issmall enough, then our assumption (2.11) implies that(2.17) | ∂ t ψ ( x j , − ∂ t ψ ( x k , | ≥ c (cid:48) λ − / | j − k | , if | j − k | ≥ , and x j , x k ∈ N , for some constant c (cid:48) > ρ ( x j , x k ; t ) = ρ ( t ; x j ) ρ ( t ; x k ) e iλ ( ψ ( x j , r ( x j ,t )) e − iλ ( ψ ( x k , r ( x k ,t )) , it follows from (2.10) and (2.16) that | ∂ mt ρ ( x j , x k ; t ) | ≤ C m λ m/ , and ρ ( x j , x k ; t ) = 0 , if | t | ≥ λ − / , x j / ∈ N , or x k / ∈ N . AKEYA-NIKODYM AVERAGES AND L p -NORMS OF EIGENFUNCTIONS 13 We can use this since the left side of (2.12) equals λ / (cid:88) j,k a j a k (cid:16) (cid:90) e itλ ( ∂ t ψ ( x j , − ∂ t ψ ( x k , ρ ( x j , x k ; t ) dt (cid:17) , which, after integrating by parts N = 1 , , . . . times, we conclude is dominated by afixed constant C N times (cid:88) j,k | a j a k | (cid:0) | j − k | (cid:1) − N . Since, by Young’s inequality, this is dominated by the right side of (2.12) when N = 2,the proof is complete. (cid:3) ν k j x x Figure 2.
Image of { (0 , t ) } in geodesic normal coordinates about (0 , Local restrictions of eigenfunctions to non-smoothly closed geodesics.
We have shown above that if { e λ jk } ∞ k =1 is a sequence of L -normalized eigenfunctionssatisfying(3.1) lim sup k →∞ sup γ ∈ Π λ − / j k (cid:90) γ | e λ jk | ds = 0 , then λ − δ ( p ) j k (cid:107) e λ jk (cid:107) L p ( M ) = 0, 2 < p <
6. While it seems difficult to determine when thisholds, one can show the following.
Proposition 3.1.
Suppose that γ ∈ Π is not contained in a smoothly closed geodesic.Then if { e λ j } is the full sequence of L -normalized eigenfunctions, we have (3.2) lim sup j →∞ λ − / j (cid:90) γ | e λ j | ds = 0 . In proving this proposition we may assume, after possible multiplying the metric bya constant, that the injectivity radius is more than 10. This will allow us to write downFourier integral operators representing the solution of the wave equation up to times | t | ≤
10. More important, though, is that we shall use an observation of Tataru [32] that the map from Cauchy data to the solution of the wave equation restricted to γ × R is a Fourier integral operator with a one-sided fold. Using this fact and the standardmethod of long-time averages (see e.g. [10], [16], [30], [29]), we shall be able to proveProposition 3.1.To set up our proof, let us choose Fermi normal coordinates about γ so that, inthese coordinates, γ becomes { ( s, ≤ s ≤ } . Note that in these coordinates themetric takes the form g ( x ) dx + dx . As a consequence if p ( x, ξ ) = (cid:112)(cid:80) g jk ( x ) ξ j ξ k is the principal symbol of P = (cid:112) − ∆ g then p (( s, , ξ ) = (cid:112) g (( s, ξ + ξ is an evenfunction of ξ .To proceed, let us fix a real-valued function χ ∈ S ( R ) with χ (0) = 1 and ˆ χ ( t ) = 0, | t | > /
2. Then if e λ is an eigenfunction with eigenvalue λ it follows that χ ( N ( P − λ )) e λ = e λ .Thus, in order to prove (3.2), it would suffice to prove that given λ, N ≥ (cid:13)(cid:13) χ ( N ( P − λ )) f (cid:13)(cid:13) L ( γ ) ≤ CN − / λ / (cid:107) f (cid:107) L ( M ) + C N (cid:107) f (cid:107) L ( M ) . Note that(3.4) χ ( N ( P − λ )) f ( x ) = N − (cid:90) ˆ χ ( t/N ) e − itλ (cid:16) e itP f (cid:17) ( x ) dt, and because of the support properties of the ˆ χ the integrand vanishes when | t | ≥ N/ f → (cid:16) e itP f (cid:17) ( x )is a Fourier operator with canonical relation { ( x, t, ξ, τ ; y, η ); Φ t ( x, ξ ) = ( y, η ) , ± τ = p ( x, ξ ) } , with Φ t : T ∗ M → T ∗ M being geodesic flow on the cotangent bundle and p ( x, ξ ), asabove, being the principal symbol of (cid:112) − ∆ g . Given that we want to restrict the operatorin (3.4) to γ = ( s, ≤ s ≤
1, we really need to also focus on the the Fourier integraloperator f → (cid:16) e itP f (cid:17) ( s, . Given the above, its canonical relation is C = (cid:8) (cid:0) Π γ × R ( x, t, ξ, τ ; y, η (cid:1) ∈ T ∗ ( γ × R ) × T ∗ M ; Φ t ( x, ξ ) = ( y, η ) , ± τ = p ( x , , ξ ) (cid:9) , with Π γ × R being the projection map from T ∗ ( M × R ) to T ∗ ( γ × R ). Note that theprojection from the latter canonical relation to T ∗ ( γ × R ) is the map( s, t, ξ ) → ( s, t, ξ , p (( s, , ξ )) , which has a fold singularity when ξ = 0 but has surjective differential away from thisset (given the aforementioned properties of p ).Because of this, given the explicit formula in Fermi coordinates, if we choose ψ ∈ C ∞ ( M ) equal to one on γ and α ∈ C ∞ ( R ) satisfying α = 1 on [ − / , /
2] but α ( τ ) = 0, | τ | ≥
1, then b ε ( x, ξ ) = ψ ( x ) α ( ξ /ε | ξ | )equals one on a conic neighborhood of the set that projects onto the set where the leftprojection of C has a folding singularity. This means that B ε ( x, ξ ) = ψ ( x ) (cid:0) − α ( ξ /ε | ξ | ) (cid:1) AKEYA-NIKODYM AVERAGES AND L p -NORMS OF EIGENFUNCTIONS 15 has symbol vanishing in a conic neighborhood of this set and consequently the map f → (cid:16) B ε ◦ e itP f (cid:17) (( s, , ≤ s ≤ L boundedness of Fourier integral operators yields (cid:90) N − N (cid:90) (cid:12)(cid:12)(cid:12) (cid:16) B ε ◦ e itP f (cid:17) ( s, (cid:12)(cid:12)(cid:12) dsdt ≤ C N,B ε (cid:107) f (cid:107) L ( M ) . Therefore, an application of Schwarz’s inequality yields (cid:107) χ N,B ε λ f (cid:107) L ( γ ) ≤ C (cid:48) N,B ε (cid:107) f (cid:107) L ( M ) , if χ N,B ε λ f = B ε ◦ χ ( N ( P − λ )) f = N − (cid:90) ˆ χ ( t/N ) e − itλ (cid:16) B ε ◦ e itP (cid:17) f dt. Therefore if we similarly define χ N,b ε λ f = b ε ◦ χ ( N ( P − λ )) f , then χ N,B ε λ f + χ N,b ε λ f = ψχ ( N ( P − λ )) f and since ψ = 1 on γ , the proof of (3.3) would be complete if we couldshow that if ε > N ) then for λ ≥ C independent of ε, N and λ ≥ (cid:107) χ N,b ε λ f (cid:107) L ( γ ) ≤ CN − / λ / (cid:107) f (cid:107) L ( M ) + C N,b ε (cid:107) f (cid:107) L ( M ) . In addition to taking ε > ψ about γ tobe small.It is in proving (3.5) of course where we shall use our assumption that γ is not partof a smoothly closed geodesic. A consequence of this is that, given fixed N , if ε and thesupport of ψ are small enough then(3.6) b ε ( y, η ) = 0 whenever ( y, η ) = Φ t ( x, ξ ) , ( x, ξ ) ∈ supp b ε , ≤ | t | ≤ N. In what follows, we shall assume that ε and ψ have been chosen so that this is the case.The point here is that if γ ( s ), s ∈ R , is the geodesic starting at (0 ,
0) and containing { γ ( s ) = ( s, ≤ s ≤ } , points on the curve γ ( s ), | s | ≤ N + 1 might intersect γ , butthe intersection must be transverse as s → γ ( s ) is not a smoothly closed geodesic. Thenif ε is chosen to be a small multiple of the smallest angle of intersection and if ψ hassmall enough support about γ , then we get (3.6). Using the canonical relation for e itP ,we can deduce from this that(3.7) b ε ◦ e itP ◦ b ∗ ε is a smoothing operator when 2 ≤ | t | ≤ N + 1 , i.e., for such times this operator’s kernel is smooth.Let T be the operator χ N,b ε λ f | γ , i.e., the truncated approximate spectral projectionoperator restricted to γ . Our goal is to show (3.5) which says that (cid:107) T (cid:107) L ( M ) → L ( γ ) ≤ CN − / λ / + C N,b ε . This is equivalent to saying that the dual operator T ∗ : L ( γ ) → L ( M ) with the samenorm, and since (cid:107) T ∗ g (cid:107) L ( M ) = (cid:90) M T ∗ g T ∗ gdx = (cid:90) γ T T ∗ g gds ≤ (cid:107) T T ∗ g (cid:107) L ( γ ) (cid:107) g (cid:107) L ( γ ) , we would be done if we could show that(3.8) (cid:107) T T ∗ g (cid:107) L ( γ ) ≤ (cid:16) CN − λ / + C N,b ε (cid:17) (cid:107) g (cid:107) L ( γ ) . But the kernel of
T T ∗ is K ( γ ( s ) , γ ( s (cid:48) )), where K ( x, y ), x, y ∈ M is the kernel of theoperator b ε ◦ ρ ( N ( P − λ )) ◦ b ∗ ε with ρ ( τ ) = ( χ ( τ )) being the square of χ . Its Fouriertransform, ˆ ρ , is the convolution of ˆ χ with itself, and thus ˆ ρ ( t ) = 0, | t | ≥
1. Consequently,we can write(3.9) b ε ◦ ρ ( N ( P − λ )) ◦ b ∗ ε = N − (cid:90) ˆ ρ ( t/N ) e − itλ (cid:16) b ε ◦ e itP ◦ b ∗ ε (cid:17) dt. Thus, if α ∈ C ∞ ( R ) is as above, then by (3.6) and (3.7), the difference of the kernel ofthe operator in (3.9) and the kernel of the operator given by(3.10) N − (cid:90) α ( t/ ρ ( t/N ) e − itλ (cid:16) b ε ◦ e itP ◦ b ∗ ε (cid:17) dt is O ( λ − J ) for any J . Thus, if we restrict the kernel of the difference to γ × γ , it contributesa portion of T T ∗ that maps L ( γ ) → L ( γ ) with norm ≤ C N,b ε .To finish, we need to estimate the remaining piece, which has the kernel of the operatorin (3.10) restricted to γ × γ . Since we are assuming that the injectivity radius of M is10 or more one can use the Hadamard parametrix for the wave equation and standardstationary phase arguments (similar to ones in [27], Chapter 5, or the proof of Lemma 4.1in [6]) to see that the kernel K ( x, y ) of the operator in (3.10) satisfies | K ( x, y ) | ≤ CN − λ / (cid:0) d g ( x, y ) (cid:1) − / + C b ε . The first term comes from the main term in the stationary phase expansion for the kerneland the other one is the resulting remainder term in the one-term expansion. Since thiskernel restricted to γ × γ gives rise to an integral operator satisfying the estimates in(3.8), the proof is complete. (cid:3) Further questions.
While as we explained before the condition that for the L -normalized eigenfunctionslim sup j →∞ sup γ ∈ Π λ − / j (cid:90) γ | e λ j | ds = 0is a natural one to quantify non-concentration, it would be interesting to formulate ageometric condition involving the long-time dynamics of the geodesic flow that wouldimply it and its equivalent version that λ − δ ( p ) j (cid:107) e λ j (cid:107) p →
0, 2 < p <
6. Presumably if γ ∈ Π and(4.1) lim sup j →∞ λ − / j (cid:90) γ | e λ j | ds > , then γ would have to be part of a stable smoothly closed geodesic, and not just a closedgeodesic as we showed above. Toth and Zeldtich made a similar conjecture to this in [34],saying that, in n -dimensions, if γ is a closed stable geodesic then one should be able tofind a sequence of eigenfunctions on which sup-norms are blowing up like λ ( n − / . In[1], [19], it was shown that there is a sequence of quasimodes blowing up at this rate. AKEYA-NIKODYM AVERAGES AND L p -NORMS OF EIGENFUNCTIONS 17 It would also be interesting to formulate a condition that would ensure that (cid:107) e λ (cid:107) L ( M ) = o ( λ δ (6) ) = o ( λ / ), for L -normalized eigenfunctions. Presumably, such a condition wouldhave to involve both ones like those in the present paper and conditions of the type in[29], [30]. Since L is an endpoint for (1.1) one expects that one would need a conditionthat both guarantees that L p bounds for 2 < p < p > L -norms over geodesics might be relevant for theproblem of determining when the L ( M ) norms of eigenfunctions are small. This is in-teresting because the L -norm is the unique L p -norm taken over geodesics that capturesboth the concentration of the highest weight spherical harmonics on geodesics and theconcentration of zonal functions at points. Indeed, the highest weight spherical harmon-ics saturate these norms for 2 ≤ p ≤
4, while the zonal functions saturate them for p ≥ L one. So the results here suggest that size estimates for theKakeya-Nikodym maximal operator associated with broken unit geodesics and appliedto squares of eigenfunctions could be relevant for improving the bounds in [24], whichare known to be sharp in the case of the disk (see [13]). An observation of Grieser[13] involving the Rayleigh whispering gallery modes suggests that in order to obtain avariant of Corollary 1.2 for compact domains one would have to consider L -norms over λ − / j -neighborhoods of broken geodesics. Smith and the author [23] also showed thatfor compact manifolds with geodesically concave boundary one has better estimates thanone does for compact domains in R n . For example, when n = 2 (1.1) holds. Based onthis and the better behavior of the geodesic flow, it seems reasonable that the analog ofCorollary 1.2 might hold (with the same scales) in this setting.Finally, as mentioned before it would be interesting to see to what extent the resultsfor the boundaryless case extend to higher dimensions. The arguments given here and in[5], though, rely very heavily on special features of the two-dimensional case. Acknowlegements:
It is a real pleasure to thank J. Bourgain for sharing an earlyversion of his paper [5] and to also thank W. Minicozzi for helpful conversations and forgoing over a key step in the proof. The author would also like to express his gratitude toJ. Toth and S. Zeldtich for helpful discussions and suggestions.
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