Improved critical eigenfunction estimates on manifolds of nonpositive curvature
aa r X i v : . [ m a t h . C A ] D ec IMPROVED CRITICAL EIGENFUNCTION ESTIMATES ONMANIFOLDS OF NONPOSITIVE CURVATURE
CHRISTOPHER D. SOGGE
Abstract.
We prove new improved endpoint, L p c , p c = n +1) n − , estimates (the“kink point”) for eigenfunctions on manifolds of nonpositive curvature. We do this byusing energy and dispersive estimates for the wave equation as well as new improved L p , 2 < p < p c , bounds of Blair and the author [4], [6] and the classical improvedsup-norm estimates of B´erard [3]. Our proof uses Bourgain’s [7] proof of weak-typeestimates for the Stein-Tomas Fourier restriction theorem [42]–[43] as a template tobe able to obtain improved weak-type L p c estimates under this geometric assumption.We can then use these estimates and the (local) improved Lorentz space estimates ofBak and Seeger [2] (valid for all manifolds) to obtain our improved estimates for thecritical space under the assumption of nonpositive sectional curvatures. Introduction.
Let (
M, g ) be a compact n -dimensional Riemannian manifold and let ∆ g be the as-sociated Laplace-Beltrami operator. We shall consider L -normalized eigenfunctions offrequency λ , i.e., − ∆ g e λ = λ e λ , Z M | e λ | dV g = 1 , with dV g denoting the volume element.The author showed in [28] that one has the following bounds for a given 2 < p ≤ ∞ and λ ≥ k e λ k L p ( M ) ≤ Cλ µ ( p ) , µ ( p ) = max (cid:0) n − ( − p ) , n ( − p ) − (cid:1) . These estimates are saturated on the round sphere by zonal functions, Z λ , for p ≥ n +1) n − = p c and for 2 < p ≤ p c by the highest weight spherical harmonics Q λ = λ n − ( x + ix ) k , if λ = λ k = p ( k + n − k . See [27]. The zonal functions have themaximal concentration at points allowed by the sharp Weyl formula, while the highestweight spherical harmonics have the maximal concentration near periodic geodesics thatis allowed by (1.1).Over the years there has been considerable work devoted to determining when (1.1)can be improved. Although not explicitly stated, this started in the work of B´erard [3],which implies that for manifolds of nonpositive curvature the estimate for p = ∞ canbe improved by a (log λ ) − factor (see [31, Proposition 3.6.2]). By interpolation withthe special case of p = p c in (1.1), one obtains improvement for all exponents p c
Zelditch [35] showed that for generic manifolds one can obtain o ( λ µ ( p ) ) bounds for k e λ k L p if p c < p ≤ ∞ . These results were improved in [34] and in [39] and [40]. In the latter twoarticles, a necessary and sufficient condition in the real analytic setting was obtained forsuch bounds for exponents larger than the critical one, p c .The estimate for the complementary range of 2 < p < p c has also garnered much at-tention of late. In works of Bourgain [8] and the author [30] for n = 2, it was shown thatimprovements of (1.1) for this range is equivalent to improvements of the geodesic re-striction estimates of Burq, G´erard and Tzvetkov [9], as well as natural Kakeya-Nikodymbounds introduced in [30] measuring L -concentration of eigenfunctions on λ − tubesabout unit-length geodesics. This is all very natural in view of the properties of thehighest weight spherical harmonics (see [30] and [32] for further discussion). Using thisequivalence and improved geodesic restriction estimates, the author and Zelditch showedin [38] that k e λ k L p = o ( λ µ ( p ) ) for 2 < p < p c if n = 2 under the assumption of nonpositivecurvature, and similar improved bounds in higher dimensions and the equivalence of thisproblem and improved Kakeya-Nikodym estimates were obtained by Blair and the authorin [5]. Very recently, in [4] and [6], we were able to obtain logarithmic improvements forthis range of exponents in all dimensions under the assumption of nonpositive curva-ture using microlocal analysis and the classical Toponogov triangle comparison theoremin Riemannian geometry. In addition to relationships with geodesic concentration andquantum ergodicity, improvements of (1.1) for 2 < p ≤ p c are of interest because of theirconnection with nodal problems for eigenfunctions (see, e.g., [5], [4], [11], [15], [17], [36]and [37]).Despite the success in obtaining improvements of (1.1) for the ranges 2 < p < p c and p c < p ≤ ∞ , improvements for the critical space where p = p c = n +1) n − have proven tobe elusive. The special case of (1.1) for this exponent reads as follows:(1.1 ′ ) k e λ k L n +1) n − ( M ) ≤ Cλ n − n +1) , and by interpolating with the trivial L estimate and the sup-norm estimate k e λ k L ∞ = O ( λ n − ), which is implicit in Avakumovi´c [1] and Levitan [23], one obtains all of theother bounds in (1.1).Improving (1.1 ′ ) has been challenging in part because it detects both point concentra-tion and concentration along periodic geodesics (as we mentioned for the sphere). Thetechniques developed for improving (1.1) for p > p c focused on the former and the morerecent ones for 2 < p < p c focused on the latter. To date the only improvements of (1.1 ′ )are recent ones of Hezari and Rivi`ere [15] who used small-scale variants of the classicalquantum ergodic results of Colin de Verdi`ere [12], Snirelman [26] and Zelditch [44] (seealso [45]) to show that for manifolds of strictly negative sectional curvature there is adensity one sequence of eigenfunctions for which (1.1 ′ ) can be logarithmically improved.The L -improvements for small balls that were used had been obtained independently byHan [13] earlier, and, in a companion article [33] to [15], the author showed that, underthe weaker assumption of ergodic geodesic flow, one can improve (1.1 ′ ) for a density onesequence of eigenfunctions.Our main result here is that, under the assumption of nonpositive curvature, one canobtain improved L p c estimates for all eigenfunctions: MPROVED CRITICAL EIGENFUNCTION ESTIMATES 3
Theorem 1.1.
Assume that ( M, g ) is of nonpositive curvature. Then there is a constant C = C ( M, g ) so that for λ ≫ k e λ k L n +1) n − ( M ) ≤ Cλ n − n +1) (cid:0) log log λ (cid:1) − n +1)2 . Additionally, (1.3) (cid:13)(cid:13) χ [ λ,λ +(log λ ) − ] f (cid:13)(cid:13) L n +1) n − ( M ) ≤ Cλ n − n +1) (cid:0) log log λ (cid:1) − n +1)2 k f k L ( M ) . Here if 0 = λ < λ ≤ λ ≤ · · · are the eigenvalues of p − ∆ g counted with respect tomultiplicity and if { e j } is an associated orthonormal basis of eigenfunctions, if I ⊂ [0 , ∞ ) χ I f = X λ j ∈ I E j f, where E j f ( x ) = (cid:0)Z M f e j dV g (cid:1) × e j ( x ) , denotes the projection onto the j th eigenspace. Thus, (1.3) implies (1.2).By interpolation and an application of a Bernstein inequality, this bound implies thatfor all exponents p ∈ (2 , ∞ ] one can improve (1.1) by a power of (log log λ ) − . Althoughstronger log-improvements are in [3], [4], [6] and [14] for p = p c , (1.2) represents thefirst improvement involving all eigenfunctions for the critical exponent. Also, besides theearlier improved geodesic eigenfunction restriction estimates for n = 2 of Chen and theauthor [10], this result seems to be the first improvement of estimates that are saturatedby both the zonal functions and highest weight spherical harmonics on spheres.The main step in proving these L p c -bounds will be to show that one has the followingrelated weak-type estimates: Proposition 1.2.
Assume, as above, that ( M, g ) is a fixed manifold of nonpositive cur-vature. Then there is a uniform constant C so that for λ ≫ we have (1.3 ′ ) (cid:12)(cid:12)(cid:8) x ∈ M : (cid:12)(cid:12) χ [ λ,λ +(log λ ) − ] f ( x ) (cid:12)(cid:12) > α (cid:9)(cid:12)(cid:12) ≤ Cλ (cid:0) log log λ (cid:1) − n − α − n +1) n − ,α > , if k f k L ( M ) = 1 . Here | Ω | denotes the dV g measure of a subset Ω of M . Note that, by Chebyshev’s inequality (1.3) implies an inequality of the type (1.3 ′ ),but with a less favorable exponent for the log log λ factor. The inequality says that χ [ λ,λ +(log λ ) − ] sends L ( M ) into L p c , ∞ ( M ), i.e., weak- L p c , with norm satisfying(1.3 ′′ ) k χ [ λ,λ +(log λ ) − ] k L ( M ) → L n +1) n − , ∞ ( M ) = O (cid:0) λ n − n +1) / (log log λ ) n +1 ) . After we obtain this weak-type L p c estimate, we shall be able to obtain (1.3) by, ineffect, interpolating it with another improved L p c estimate of Bak and Seeger [2], whichsays that the operators χ [ λ,λ +1] map L ( M ) into the Lorentz space L p c , ( M ) (see § O ( λ n − n +1) ). This “local” estimate holds for all manifolds—nocurvature assumption is needed. CHRISTOPHER D. SOGGE
Before turning to the proofs, let us point out that the weak-type bound (1.3 ′ ) cannothold for S n . There there are two special values of α that cause problems. The zonalfunctions are sensitive to α ≈ λ n − , and |{ x ∈ S n : | Z λ ( x ) | > α }| ≈ λ − n ≈ λα − n +1) n − , if α = cλ n − , with c > Q λ ,are sensitive to α ≈ λ n − in that |{ x ∈ S n : | Q λ ( x ) | > α }| ≈ λ − n − ≈ λα − n +1) n − , if α = cλ n − , and c > ′ ) and Chebyshev’s inequality, wealways have, on any ( M, g ),(1.4) |{ x ∈ M : | e λ ( x ) | > α }| . λα − n +1) n − , and so the zonal functions and the highest weight spherical harmonics saturate this weak-type estimate. We shall give a simple proof of (1.4) in the next section that will serve asa model for the proof of the improved weak-type bounds in Proposition 1.2. It is basedon a modification of Bourgain’s [7] proof of a weak-type version of the critical Fourierrestriction estimate of Stein and Tomas [42]–[43].Let us give an overview of why are able to obtain (1.3 ′ ) and (1.3). As we mentionedbefore, the potentially dangerous values of α for the former are α ≈ λ n − and α ≈ λ n − .The aforementioned sup-norm estimates of B´erard [3] provide log-improvements over(1.4) for α ≥ λ n − / (log λ ) , while the recent log-improved L p estimates, 2 < p < p c ,of Blair and the author [4], [6] yield log-improvements for α near the other dangerousvalue λ n − . Specifically, we are able to obtain improvements when α ≤ λ n − (log λ ) δ n for some δ n >
0. We can cut and paste these improvements into the aforementionedargument of Bourgain [7] to obtain (1.3 ′ ). We then can upgrade the weak-type estimatesthat we obtain (at the expense of less favorable powers of (log log λ ) − ) to a standard L p c estimate using the result of Bak and Seeger [2]. Thus, we combine the earlier “global”results of [3], [4], and [6] with “local” harmonic analysis techniques to obtain our mainestimate (1.3).The paper is organized as follows. In the next section we shall give the variation of theargument from [7] that yields (1.4). In § § §
5, we shall state some natural problems related to ourapproach. Also, in what follows whenever we write A . B , we mean that A is dominatedby an unimportant constant multiplied by B .2. The model local argument.
In this section we shall present an argument that yields the weak-type estimate (1.4)and serves as a model for the argument that we shall use to prove Theorem 1.1.Let us fix a real-valued function ρ ∈ S ( R ) satisfying(2.1) ρ (0) = 1 , | ρ ( τ ) | ≤ , and supp ˆ ρ ⊂ ( − / , / . MPROVED CRITICAL EIGENFUNCTION ESTIMATES 5
If we set P = p − ∆ g , consider the operators(2.2) ρ ( λ − P ) f ( x ) = ∞ X j =0 ρ ( λ − λ j ) E j f ( x ) , where, as before, 0 = λ < λ ≤ λ ≤ · · · are the eigenvalues counted with respect tomultiplicity and E j denotes projection onto the j th eigenspaceThe “local” analog of Proposition 1.2 then is the following result whose proof we shallmodify in the next section to obtain the “global” weak-type estimates (1.3 ′ ). Proposition 2.1.
For λ ≥ there is a constant C , depending only on ( M, g ) , so that (2.3) (cid:12)(cid:12) { x ∈ M : | ρ ( λ − P ) f ( x ) | > α } (cid:12)(cid:12) ≤ Cλα − n +1) n − k f k n +1) n − L ( M ) , α > . Consequently, (1.4) is valid, and, moreover, if χ λ denotes the unit-band spectral projectionoperators χ λ f = X λ j ∈ [ λ,λ +1] E j f, we have (2.3 ′ ) (cid:12)(cid:12) { x ∈ M : | χ λ f ( x ) | > α } (cid:12)(cid:12) ≤ Cλα − n +1) n − k f k n +1) n − L ( M ) , α > . Since ρ (0) = 1 we have that | ρ ( τ ) | ≥ / | τ | ≤ δ for some δ >
0. Thus, if oneapplies (2.3) with f replaced by P λ j ∈ [ λ,λ + δ ] E j f , one deduces that (cid:12)(cid:12)(cid:8) | X λ j ∈ [ λ,λ + δ ] E j f ( x ) | > α (cid:9)(cid:12)(cid:12) ≤ Cλα − n +1) n − k f k n +1) n − L ( M ) , α > , which implies (2.3 ′ ). So to prove Proposition 2.1, we just need to prove (2.3).To prove (2.3), we require the following lemma which will be useful in the sequel. Weshall assume, as we may, here and in what follows that the injectivity radius of M , Inj M ,satisfies Inj M ≥ . Also, B ( x, r ), r < Inj M , denotes the geodesic ball of radius r about a point x ∈ M withrespect to the Riemannian distance function d g ( · , · ). The result we need then is thefollowing. Lemma 2.2.
Let a ∈ C ∞ (( − , . Then there is a constant C , depending only on ( M, g ) and the size of finitely many derivatives of a , so that for λ − ≤ r ≤ Inj M we have (2.4) (cid:13)(cid:13)(cid:13)Z a ( t ) e itλ (cid:0) e − itP f (cid:1) dt (cid:13)(cid:13)(cid:13) L ( B ( x,r )) ≤ Cr k f k L ( M ) , and, also, if (cid:0) e − itP (cid:1) ( x, y ) denotes the kernel of the half-wave operators e − itP , we have (2.5) (cid:12)(cid:12)(cid:0) ˆ a ( P − λ ) (cid:1) ( x, y ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)Z a ( t ) e itλ (cid:0) e − itP (cid:1) ( x, y ) dt (cid:12)(cid:12)(cid:12) ≤ Cλ n − (cid:0) d g ( x, y ) + λ − (cid:1) − n − . CHRISTOPHER D. SOGGE
We shall omit the proof of (2.5) since it is well known and follows easily from usingstationary phase and parametrices for the half-wave equation. One can easily obtain (2.5)by adapting the proof of Lemma 5.1.3 in [29].Even though (2.4) is in a recent article of the author [33], for the sake of completeness,we shall present a different simple proof here, which only uses energy estimates andquantitative propagation of singularities estimates for the half-wave operators.We start by introducing a Littlewood-Paley bump function β ∈ C ∞ ( R ) satisfying(2.6) β ( τ ) = 1 , τ ∈ [1 / , , and supp β ⊂ (1 / , . Then standard arguments using the aforementioned parametrix show that for any N , wehave that (cid:13)(cid:13)(cid:13)Z a ( t ) e itλ (cid:0) I − β ( P/λ ) (cid:1) ◦ e − itP dt (cid:13)(cid:13)(cid:13) L ( M ) → L ( M ) = O ( λ − N ) , where for each N ∈ N the constants depend only on finitely many derivatives of a . Thus,to prove (2.4), it suffices to prove the variant where e − itP is replaced by β ( P/λ ) ◦ e − itP .By a routine T T ∗ argument, this in turn is equivalent to showing that(2.5 ′ ) (cid:13)(cid:13)(cid:13)Z b ( t ) e itλ (cid:0) β ( P/λ ) ◦ e − itP (cid:1) h dt (cid:13)(cid:13)(cid:13) L ( B ( x,r )) ≤ Cr k h k L ( B ( x,r )) , if supp h ⊂ B ( x, r ) and λ − ≤ r ≤ Inj M, with b = a ( · ) ∗ a ( − · ) . By Minkowski’s inequality, the left side of (2.5 ′ ) is dominated by Z | t |≤ r | b ( t ) | (cid:13)(cid:13)(cid:0) β ( P/λ ) ◦ e − itP (cid:1) h (cid:13)(cid:13) L ( B ( x,r )) dt + Z | t |≥ r | b ( t ) | (cid:13)(cid:13)(cid:0) β ( P/λ ) ◦ e − itP (cid:1) h (cid:13)(cid:13) L ( B ( x,r )) dt = I + II.
By energy estimates, we trivially have I . r k h k L , as desired, and we do not need to use our support assumptions in (2.5 ′ ) here.To handle II , though, we do need to make use of them. We also need the routinedyadic estimates(2.7) (cid:12)(cid:12)(cid:12)(cid:0) β ( P/λ ) ◦ e − itP (cid:1) ( w, z ) (cid:12)(cid:12)(cid:12) = O (cid:0) λ n (1 + λ | t | (cid:1) − N (cid:1) ∀ N, if d g ( w, z ) ≤ | t | / , which also follows easily from an integration by parts argument using the parametrix for e − itP . From (2.7) we immediately get (cid:12)(cid:12)(cid:12)(cid:0) β ( P/λ ) ◦ e − itP (cid:1) ( w, z ) (cid:12)(cid:12)(cid:12) = O (cid:0) λ n (1 + λ | t | ) − N (cid:1) ∀ N, if w, z ∈ B ( x, r ) and | t | ≥ r. As a result, by Schwarz’s inequality, we have that if, as in (2.5 ′ ), supp h ⊂ B ( x, r ), II . ( rλ ) n (cid:16)Z | t |≥ r (cid:0) λ | t | (cid:1) − n dt (cid:17) × k h k L ≈ r k h k L , as desired, completing the proof of (2.5 ′ ). MPROVED CRITICAL EIGENFUNCTION ESTIMATES 7
Proof of Proposition 2.1.
To prove (2.3) it suffices to show that if Ω is a relatively com-pact subset of a coordinate patch Ω for M then we have(2.8) (cid:12)(cid:12)(cid:8) x ∈ Ω : | ρ ( λ − P ) f ( x ) | > α (cid:9)(cid:12)(cid:12) ≤ Cλα − n +1) n − , α > , assuming that(2.9) k f k L ( M ) = 1 . We shall work in these local coordinates to make the decomposition we require.Let A = { x ∈ Ω : | ρ ( λ − P ) f ( x ) | > α } denote the set in (2.8). Our decomposition will be based on the scale(2.10) r = λα − n − , which is motivated by an argument in Bourgain [7]. Note that, since the sup-normestimates of Avakumovi´c [1] and Levitan [23] give k ρ ( λ − P ) f k L ∞ = O ( λ n − ) , the estimate (2.8) is trivial when r is smaller than a multiple of λ − , which allows us touse (2.4).Write A = [ A j , where A j = A ∩ Q j and Q j denote a nonoverlapping lattice of cubes of sidelength r inour coordinates. At the expense of replacing A by a set of proportional measure, we mayassume that(2.11) dist ( A j , A k ) > C r, j = k, for a constant C to be specified later. Also, let(2.12) ψ λ ( x ) = ( ρ ( λ − P ) f ( x ) / | ρ ( λ − P ) f ( x ) | , if ρ ( λ − P ) f ( x ) = 01 , otherwise , so that ψ λ , of modulus one, is the signum function of ρ ( λ − P ) f .We then have, by Chebyshev’s inequality, (2.9) and the Cauchy-Schwarz inequality, α | A | ≤ (cid:12)(cid:12)(cid:12)Z ρ ( λ − P ) f ψ λ A dV g (cid:12)(cid:12)(cid:12) ≤ (cid:16)Z (cid:12)(cid:12)X j ρ ( λ − P ) a j (cid:12)(cid:12) dV g (cid:17) , where 1 A denotes the indicator function of A and a j denotes ψ λ times the indicatorfunction of A j . As a result, if S λ = (cid:0) ρ ( λ − P ) ∗ ◦ ρ ( λ − P ) (cid:1) = ρ ( λ − P ), α | A | ≤ X j Z | ρ ( λ − P ) a j | dV g + X j = k Z ρ ( λ − P ) a j ρ ( λ − P ) a k dV g = X j Z | ρ ( λ − P ) a j | dV g + X j = k Z S λ a j a k dV g = I + II.
CHRISTOPHER D. SOGGE
Since a j is supported in a ball of radius ≈ r , by (2.1) and the dual version of (2.4)with a = ˆ ρ , we have Z | ρ ( λ − P ) a j | dV g ≤ Cr Z | a j | dV g = Cr | A j | . Whence, by (2.10) I . r | A | = λα − n − | A | . To estimate II , we note that by (2.5) with a = ˆ ρ ( · ) ∗ ˆ ρ ( − · ), we have that the kernel K λ ( x, y ) of S λ satisfies(2.13) | K λ ( x, y ) | ≤ Cλ n − (cid:0) d g ( x, y ) + λ − (cid:1) − n − . Therefore, by (2.12), II . X j = k Z Z | K λ ( x, y ) | | a j ( x ) | | a k ( y ) | dV g ( x ) dV g ( y ) . λ n − (cid:0) C r (cid:1) − n − X j = k k a j k L k a k k L ≤ C − n − α | A | . Thus, α | A | . λα − n − | A | + C − n − α | A | , and so, if C in (2.11) is large enough, the last term can be absorbed in the left side. Weconclude that | A | . λα − − n − = λα − n +1) n − , which is (2.8). (cid:3) Proof of improved weak-type estimates.
We shall now prove Proposition 1.2. Repeating the arguments from the previoussection shows that if ρ ∈ S ( R ) is as in (2.1) then it suffices to show that we have thefollowing Proposition 3.1.
Let ( M, g ) be an n -dimensional compact Riemannian manifold ofnonpositive curvature. Then for λ ≫ (cid:13)(cid:13) ρ (log λ ( λ − P )) (cid:13)(cid:13) L ( M ) → L n +1) n − , ∞ ( M ) = O (cid:0) λ n − n +1) / (log log λ ) n +1 (cid:1) . The earlier arguments show that (3.1) yields (1.3 ′′ ) and hence Proposition 1.2 assum-ing, as in there and as we shall throughout this section, that the sectional curvatures of( M, g ) are nonpositive.To prove (3.1), as in (2.8), it suffices to show now that if Ω is a relatively compactsubset of a coordinate patch Ω , then(3.2) (cid:12)(cid:12)(cid:8) x ∈ Ω : | ρ (log λ ( λ − P )) f ( x ) | > α (cid:9)(cid:12)(cid:12) ≤ Cα − n +1) n − λ/ (log log λ ) n − , assuming that(3.3) k f k L ( M ) = 1 . MPROVED CRITICAL EIGENFUNCTION ESTIMATES 9
To prove this, in addition to (2.4), we shall require the following two results.
Lemma 3.2.
Let ( M, g ) be as above. Then there is a δ n > so that for λ ≫ and µ ( p ) as in (1.1)(3.4) (cid:13)(cid:13) ρ (log λ ( λ − P )) (cid:13)(cid:13) L ( M ) → L nn − ( M ) = O (cid:0) λ µ (cid:0) nn − (cid:1) / (log λ ) δ n ) . Lemma 3.3. If ( M, g ) is as above then there is a constant C = C ( M, g ) so that for T ≥ and large λ we have the following bounds for the kernel of η ( T ( λ − P )) , η = ρ , (3.5) (cid:12)(cid:12) η (cid:0) T ( λ − P ) (cid:1) ( w, z ) (cid:12)(cid:12) ≤ CT − ( λ/d g ( w, z )) n − + Cλ n − exp( CT ) . The first estimate, (3.4), is a simple consequence of the bounds(3.4 ′ ) (cid:13)(cid:13) χ [ λ,λ +(log λ ) − ] (cid:13)(cid:13) L ( M ) → L p ( M ) ≤ λ µ ( p ) / (log λ ) δ ( p,n ) , < p < n +1) n − , with δ ( p, n ) > p = nn − . Any other exponent between2 and n +1) n − in (3.4 ′ ) would work as well for us. We just chose p = nn − to simplify thecalculations.The other bound, (3.5), is well known and follows from the arguments in B´erard [3].Indeed, it is a simple consequence of inequality (3.6.8) in [31].Let us see how we can use these results to obtain (3.2).We first note that by Lemma 3.2 and the Chebyshev inequality we have that since nn − · µ ( nn − ) = , (cid:12)(cid:12)(cid:8) x ∈ Ω : | ρ (log λ ( λ − P )) f ( x ) | > α (cid:9)(cid:12)(cid:12) ≤ α − nn − Z M | ρ (log λ ( λ − P )) f | nn − dV g (3.6) . α − nn − λ (log λ ) − nn − δ n . To use this, we note that for large λ we have(3.7) α − nn − λ (log λ ) − nn − δ n ≪ α − n +1) n − λ (cid:0) log log λ (cid:1) − n − , if α ≤ λ n − (log λ ) δ n . Thus, by (3.6), we would obtain (3.2) if we could show that for λ ≫ (cid:12)(cid:12)(cid:8) x ∈ Ω : | ρ (log λ ( λ − P )) f ( x ) | > α (cid:9)(cid:12)(cid:12) ≤ Cα − n +1) n − λ (log log λ ) − n − , if α ≥ λ n − (log λ ) δ n . As we mentioned in the introduction, this step is key for us since it has allowed usto use our curvature assumptions and move well past the dangerous heights where α iscomparable to λ n − .At this stage, due to the nature of the pointwise estimates in Lemma 3.3, we needto change the frequency scale we are working with. Instead of effectively working with(log λ ) − windows for frequencies as above, we shall work with wider windows of size T − where T = c log log λ , with c chosen later to deal with the second term in the right sideof (3.5). We claim that we would have (3.8), and therefore be done, if we could show that(3.9) (cid:12)(cid:12)(cid:8) x ∈ Ω : | ρ (cid:0) c log log λ ( λ − P ) (cid:1) h ( x ) | > α (cid:9)(cid:12)(cid:12) . α − n +1) n − λ (log log λ ) − n +1 , if α ≥ λ n − (log λ ) δ n , and k h k L ( M ) ≤ . To verify this claim, we note that since ρ (0) = 1 and ρ ∈ S , for τ ∈ R and for λ ≫ (cid:12)(cid:12)(cid:2) ρ ( c log log λ ( λ − τ )) − (cid:3) ρ (log λ ( λ − τ )) (cid:12)(cid:12) . log log λ log λ (1 + | λ − τ | ) − N , for any N = 1 , , . . . . Thus, by using the fact that by [28] the unit band spectralprojection operators χ λ satisfy k χ λ k L ( M ) → L n +1) n − ( M ) = O ( λ n − n +1) ) , we deduce that (cid:13)(cid:13)(cid:2) ρ ( c log log λ ( λ − P )) − I (cid:3) ◦ ρ (log λ ( λ − P )) f k L n +1) n − ( M ) . log log λ log λ λ n − n +1) , and so, by Chebyshev, for all α > (cid:12)(cid:12)(cid:8) x ∈ M : | [ ρ ( c log log λ ( λ − P )) − I ] ◦ ρ (log λ ( λ − P )) f ( x ) | > α (cid:9)(cid:12)(cid:12) . (cid:0) log log λ log λ (cid:1) n +1) n − λα − n +1) n − , which is much better than the bounds posited in (3.8). If we take h = ρ (log λ ( λ − P )) f in(3.9), we deduce the claim from this since, by (2.1), k ρ (log λ ( λ − P )) k L ( M ) → L ( M ) ≤ A = { x ∈ Ω : | ρ ( c log log λ ( λ − P )) h ( x ) | > α } , and let ψ λ be defined as in (2.12) but with ρ ( λ − P ) replaced by ρ ( c log log( λ − P )).Note that for large λ A = ∅ if λ n − (log log λ ) − . α, since estimates of B´erard [3] (see also [31]) give k ρ ( c log log λ ( λ − P )) k L ( M ) → L ∞ ( M ) . λ n − / (cid:0) log log λ (cid:1) . This will allow us to apply (2.4).Next, as in the proof of Proposition 2.1, we write A = ∪ A j where A j = Q j ∩ A , withthe Q j coming from a lattice of nonoverlapping cubes in our coordinate system, exceptnow, instead of (2.1), we take(3.10) r = λα − n − (log log λ ) − n − . As before, at the expense of replacing A by a set of proportional measure, we may assumethat(3.11) dist ( A j , A k ) > C r, j = k, where C will be specified momentarily. MPROVED CRITICAL EIGENFUNCTION ESTIMATES 11
Let us now collect the two estimates that we need for the proof of (3.9). First, if S λ = η ( c log log λ ( λ − P )), η = ρ , then by (3.5) if c > K λ , satisfies(3.12) | K λ ( w, z ) | ≤ C h (log log λ ) − (cid:16) λd g ( w, z ) (cid:17) n − + λ n − (log λ ) δn i , with C independent of λ ≫ T ≥
1, we have(3.13) (cid:13)(cid:13) ρ ( T ( λ − P )) f k L ( B ( x,r )) ≤ Cr k f k L ( M ) , if λ − ≤ r ≤ Inj M, with C independent of λ ≫
1. Since ρ ( T ( λ − P )) = 12 πT Z ˆ ρ ( t/T ) e itλ e − itP dt, and, by (2.1), ˆ ρ ( t/T ) = 0 if | t | ≥ T , this follows easily from (2.4) and the fact that thehalf-wave operators e − itP are unitary.We now use the proof of Proposition 2.1 to obtain (3.9). We argue as before to seethat if T λ = ρ ( c log log λ ( λ − P )) and a j = ψ λ × A j , then since k h k L ( M ) ≤
1, we have α | A | ≤ X j Z | T λ a j | dV g + X j = k Z S λ a j a k dV g = I + II.
By the dual version of (3.13) and (3.10) I . r X j Z | a j | dV g = r | A | = λ (log log λ ) − n − α − n − | A | . By (3.12) II . h (log log λ ) − λ n − (cid:0) C r (cid:1) − n − + λ n − (log λ ) δn i X j = k k a j k L k a k k L ≤ C − n − α | A | + λ n − (log λ ) δn | A | . Since we are assuming that α ≥ λ n − (log λ ) δ n , the last term is ≪ α | A | if λ is large.This means that we can fix C in (3.11) so that for large λ we have II ≤ α | A | . Hence α | A | ≤ Cλ (log log λ ) − n − α − n − | A | + 12 α | A | , which of course yields the desired estimate | A | . λ (log log λ ) − n − α − − n − = λ (log log λ ) − n − α − n +1) n − , assuming, as we are, that α ≥ λ n − (log λ ) δ n . This concludes the proof of (3.9), Proposition 3.1 and Proposition 1.2. Proof of Theorem 1.1.
Even though (1.3), and hence Theorem 1.1, follows directly from interpolating betweenthe weak-type estimate (1.3 ′ ) and the estimate,(4.1) k χ [ λ,λ +1] k L ( M ) → L pc, ( M ) = O ( λ pc ) , p c = n +1) n − , of Bak and Seeger [2], for the sake of completeness, we shall give the simple argumenthere.Let us start by recalling some basic facts about Lorentz spaces. See § u on M , we let ω ( α ) = (cid:12)(cid:12)(cid:8) x ∈ M : | u ( x ) | > α (cid:9)(cid:12)(cid:12) , α > , denote its distribution function, and u ∗ ( t ) = inf { α : ω ( α ) ≤ t } , t > , the nonincreasing rearrangement of u .Then the Lorentz spaces L p,q ( M ) for 1 ≤ p < ∞ and 1 ≤ q < ∞ are defined as all u so that(4.2) k u k L p,q ( M ) = (cid:18) qp Z ∞ (cid:2) t p u ∗ ( t ) (cid:3) q dtt (cid:19) q < ∞ . By equation (3.9) in Chapter 5 of [41], we then have(4.3) k u k L p,p ( M ) = k u k L p ( M ) , and by Lemma 3.8 there we also havesup t> t p u ∗ ( t ) = sup α> α (cid:2) ω ( α ) (cid:3) p = sup α> α (cid:12)(cid:12)(cid:8) x ∈ M : | u ( x ) | > α (cid:9)(cid:12)(cid:12) p . If we take u = χ [ λ,λ +(log λ ) − ] f and assume from now on that k f k L ( M ) = 1, wetherefore have, by our improved weak-type estimates (1.3 ′ ),(4.4) sup t> t pc u ∗ ( t ) ≤ Cλ pc (cid:0) log log λ (cid:1) − n +1 . Also, for this u we have χ [ λ,λ +1] u = u , and so, by (4.1),(4.5) k u k L pc, ( M ) ≤ Cλ pc k u k L ( M ) ≤ Cλ pc k f k L ( M ) = Cλ pc . MPROVED CRITICAL EIGENFUNCTION ESTIMATES 13
By (4.2)–(4.3) and (4.4)–(4.5), we therefore get k u k L pc ( M ) = (cid:18) Z ∞ (cid:2) t pc u ∗ ( t ) (cid:3) p c dtt (cid:19) pc ≤ ( p c / pc (cid:18) sup t> t pc u ∗ ( t ) (cid:19) pc − pc (cid:18) p c Z ∞ (cid:2) t pc u ∗ ( t ) (cid:3) dtt (cid:19) pc . (cid:2) λ pc (cid:0) log log λ (cid:1) − n +1 (cid:3) pc − pc k u k pc L pc, ( M ) . (cid:2) λ pc (cid:0) log log λ (cid:1) − n +1 (cid:3) pc − pc (cid:16) λ pc (cid:17) pc = λ pc (cid:0) log log λ (cid:1) − n +1)2 , as ( p c − / ( n + 1) p c = 2 / ( n + 1) . Since u = χ [ λ,λ +(log λ ) − ] f and we are assumingthat k f k L ( M ) = 1, we conclude that (1.3) must be valid, which completes the proof ofTheorem 1.1. (cid:3) Concluding remarks.
First of all, we were only able to get endpoint results with gains of powers of log log λ instead of powers of log λ due to the estimate (3.5) for the smoothed out spectral pro-jection kernels. Ideally, one would want to be able to use a variant of (3.5) where theexponential factor is not present for the second term in the right. Lower bounds ofJakobson and Polterovich [20]–[21] show that this error term cannot be O ( λ n − ), buttheir bounds do not rule out some improvement over (3.5), which would lead to morefavorable estimates.A better avenue for improvement, though, might be to try to improve the ball-localizedestimates (2.4), where the operators ˆ a ( P − λ ) are replaced by ρ ( T ( λ − P ))) for appropriate T = T ( r ). A seemingly modest improvement where r is replaced by r / (log λ ) ε , forsome ε >
0, if λ − ≤ r ≤ (log λ ) − δ , for some δ > ε = if λ − ≤ r ≪ λ − , but this does not seem veryuseful. On the other hand, assuming that the curvature is strictly negative, Han [13] andHezari and Rivi`ere [15] obtained these types of bounds with ε = n/ δ dependingon the dimension for a density one sequence of eigenfunctions. For toral eigenfunctions,Lester and Rudnick [22] did even better for a density one sequence of eigenfunctions byshowing, for instance, that in when n = 2 one can replace r in (2.4) by r n all the waydown to r being equal to the essentially the wavelength, i.e., λ − o (1) as λ → ∞ . (Seealso [16] for earlier work.)Finally, the arguments we have given could possibly prove new sharp bounds for eigen-functions on manifolds with boundary. Sharp estimates in the two-dimensional case wereobtained by Smith and the author [25], but sharp estimates in higher dimensions are onlyknown for certain exponents. It turns out that the critical exponent for manifolds withboundary should be n +43 n − , which is larger than the one for the boundaryless case, n +1) n − .If one could obtain the analog of (2.5) in this setting with the right hand side replacedby ( λ/ dist ( x, y )) n − + , then one would likely be able to obtain sharp weak-type estimates for p = n +43 n − , whichby interpolation would yield sharp L p estimates for all other p ∈ (2 , ∞ ]. One wouldalso need analogs of (2.4), but these are probably much easier and likely follow fromstretching arguments of Ivri˘ı [19] and Seeley [24]. In the model case involving the Fried-lander model, recently Ivanovici, Lebeau and Planchon [18] obtained dispersive estimatesfor wave equations which have similarities with the types of spectral projection kernelestimates we just described. Acknowledgements
We are grateful for helpful suggestions and comments from our colleagues M. Blair,H. Hezari, A. Seeger and S. Zeldtich.
References [1] V. G. Avakumovi´c, ¨Uber die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten ,Math. Z. (1956), 327–344.[2] J.-G. Bak and A. Seeger, Extensions of the Stein-Tomas theorem , Math. Res. Lett. (2011), no. 4,767–781.[3] P. H. B´erard, On the wave equation on a compact Riemannian manifold without conjugate points ,Math. Z. (1977), no. 3, 249–276.[4] M. D. Blair and C. D. Sogge,
Concerning Toponogov’s theorem and logarithmic improvement ofestimates of eigenfunctions , (2015), arXiv:1510.07726.[5] ,
On Kakeya–Nikodym averages, L p -norms and lower bounds for nodal sets of eigenfunctionsin higher dimensions , J. Eur. Math. Soc. (JEMS) (2015), no. 10, 2513–2543.[6] , Refined and microlocal Kakeya-Nikodym bounds of eigenfunctions in higher dimensions ,(2015), arXiv:1510.07724.[7] J. Bourgain,
Besicovitch type maximal operators and applications to Fourier analysis , Geom. Funct.Anal. (1991), no. 2, 147–187.[8] , Geodesic restrictions and L p -estimates for eigenfunctions of Riemannian surfaces , Linearand complex analysis, Amer. Math. Soc. Transl. Ser. 2, vol. 226, Amer. Math. Soc., Providence, RI,2009, pp. 27–35.[9] N. Burq, P. G´erard, and N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to sub-manifolds , Duke Math. J. (2007), no. 3, 445–486.[10] X. Chen and C. D. Sogge,
A few endpoint geodesic restriction estimates for eigenfunctions , Comm.Math. Phys. (2014), no. 2, 435–459.[11] T. H. Colding and W. P. Minicozzi, II,
Lower bounds for nodal sets of eigenfunctions , Comm. Math.Phys. (2011), no. 3, 777–784.[12] Y. Colin de Verdi`ere,
Ergodicit´e et fonctions propres du laplacien , Comm. Math. Phys. (1985),no. 3, 497–502.[13] X. Han,
Small scale quantum ergodicity in negatively curved manifolds , (2015), arXiv:1410.3911.[14] A. Hassell and M. Tacy,
Improvement of eigenfunction estimates on manifolds of nonpositive cur-vature , Forum Math. (2015), no. 3, 1435–1451.[15] H. Hezari and G. Rivi`ere, L p norms, nodal sets, and quantum ergodicity , Adv. Math. (2016), toappear.[16] , Quantitative equidistribution properties of toral eigenfunctions , J. Spec. Theory (2016), toappear.[17] H. Hezari and C. D. Sogge,
A natural lower bound for the size of nodal sets , Anal. PDE (2012),no. 5, 1133–1137.[18] O. Ivanovici, G. Lebeau, and F. Planchon, Dispersion for the wave equation inside strictly convexdomains I: the Friedlander model case , Ann. of Math. (2) (2014), no. 1, 323–380.[19] V. Ja. Ivri˘ı,
The second term of the spectral asymptotics for a Laplace-Beltrami operator on mani-folds with boundary , Funktsional. Anal. i Prilozhen. (1980), no. 2, 25–34. MPROVED CRITICAL EIGENFUNCTION ESTIMATES 15 [20] D. Jakobson and I. Polterovich,
Lower bounds for the spectral function and for the remainder inlocal Weyl’s law on manifolds , Electron. Res. Announc. Amer. Math. Soc. (2005), 71–77.[21] , Estimates from below for the spectral function and for the remainder in local Weyl’s law ,Geom. Funct. Anal. (2007), no. 3, 806–838.[22] S. Lester and Z. Rudnick, Small scale equidistribution of eigenfunctions on the torus , (2015),arXiv:1508.01074.[23] B. M. Levitan,
On the asymptotic behavior of the spectral function of a self-adjoint differentialequation of the second order , Izvestiya Akad. Nauk SSSR. Ser. Mat. (1952), 325–352.[24] R. Seeley, An estimate near the boundary for the spectral function of the Laplace operator , Amer.J. Math. (1980), no. 5, 869–902.[25] H. F. Smith and C. D. Sogge,
On the L p norm of spectral clusters for compact manifolds withboundary , Acta Math. (2007), no. 1, 107–153.[26] A. I. ˇSnirel ′ man, Ergodic properties of eigenfunctions , Uspehi Mat. Nauk (1974), no. 6(180),181–182.[27] C. D. Sogge, Oscillatory integrals and spherical harmonics , Duke Math. J. (1986), no. 1, 43–65.[28] , Concerning the L p norm of spectral clusters for second-order elliptic operators on compactmanifolds , J. Funct. Anal. (1988), no. 1, 123–138.[29] , Fourier integrals in classical analysis , Cambridge Tracts in Mathematics, vol. 105, Cam-bridge University Press, Cambridge, 1993.[30] ,
Kakeya-Nikodym averages and L p -norms of eigenfunctions , Tohoku Math. J. (2) (2011),no. 4, 519–538.[31] , Hangzhou lectures on eigenfunctions of the Laplacian , Annals of Mathematics Studies, vol.188, Princeton University Press, Princeton, NJ, 2014.[32] ,
Problems related to the concentration of eigenfunctions , (2015), arXiv:1510.07723, JourneesEDP, to appear.[33] ,
Localized L p -estimates of eigenfunctions: A note on an article of Hezari and Rivi`ere , Adv.Math. (2016), 384–396.[34] C. D. Sogge, J. A. Toth, and S. Zelditch, About the blowup of quasimodes on Riemannian manifolds ,J. Geom. Anal. (2011), no. 1, 150–173.[35] C. D. Sogge and S. Zelditch, Riemannian manifolds with maximal eigenfunction growth , Duke Math.J. (2002), no. 3, 387–437.[36] ,
Lower bounds on the Hausdorff measure of nodal sets , Math. Res. Lett. (2011), no. 1,25–37.[37] , Lower bounds on the Hausdorff measure of nodal sets II , Math. Res. Lett. (2012), no. 6,1361–1364.[38] , On eigenfunction restriction estimates and L -bounds for compact surfaces with nonpositivecurvature , Advances in analysis: the legacy of Elias M. Stein, Princeton Math. Ser., vol. 50, PrincetonUniv. Press, Princeton, NJ, 2014, pp. 447–461.[39] , Focal points and sup-norms of eigenfunctions , Rev. Mat. Iberomericana (2015), to appear.[40] ,
Focal points and sup-norms of eigenfunctions II: the two-dimensional case , Rev. Mat.Iberomericana (2015), to appear.[41] E. M. Stein and G. Weiss,
Introduction to Fourier analysis on Euclidean spaces , Princeton UniversityPress, Princeton, N.J., 1971, Princeton Mathematical Series, No. 32.[42] P. A. Tomas,
A restriction theorem for the Fourier transform , Bull. Amer. Math. Soc. (1975),477–478.[43] , Restriction theorems for the Fourier transform , Harmonic analysis in Euclidean spaces(Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, Proc. Sympos.Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–114.[44] S. Zelditch,
Uniform distribution of eigenfunctions on compact hyperbolic surfaces , Duke Math. J. (1987), no. 4, 919–941.[45] , On the rate of quantum ergodicity. I. Upper bounds , Comm. Math. Phys. (1994), no. 1,81–92.
Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218
E-mail address ::