Carleson Measures and Logvinenko-Sereda sets on compact manifolds

Given a compact Riemannian manifold $M$ of dimension $m\geq 2$, we study the space of functions of $L^2(M)$ generated by eigenfunctions of eigenvalues less than $L\geq 1$ associated to the Laplace-Beltrami operator on $M$. On these spaces we give a characterization of the Carleson measures and the Logvinenko-Sereda sets.


Introduction and statement of the results
Let (M, g) be a smooth, connected, compact Riemannian manifold without boundary of dimension m ≥ 2. Let dV be the volume element of M associated to the metric g ij . Let ∆ M be the Laplacian on M associated to the metric g ij . It is given in local coordinates by where |g| = det(g ij ) and (g ij ) ij is the inverse matrix of (g ij ) ij . Since M is compact, g ij and all its derivatives are bounded and we assume that the metric g is non-singular at each point of M.
Since M is compact, the spectrum of the Laplacian is discrete and there is a sequence of eigenvalues 0 ≤ λ 1 ≤ λ 2 ≤ · · · → ∞ and an orthonormal basis φ i of smooth real eigenfunctions of the Laplacian i.e. ∆ M φ i = −λ i φ i . So L 2 (M) decomposes into an orthogonal direct sum of eigenfunctions of the Laplacian.
We consider the following spaces of L 2 (M). where L ≥ 1 and k L = dim E L . E L is the space of L 2 (M) generated by eigenfunctions of eigenvalues λ ≤ L. Thus in E L we consider functions in L 2 (M) with a restriction on the support of its Fourier transform. It is, in a sense, a Paley-Wiener type space on M with bandwidth L.
The motivation of this paper is to show that the spaces E L behave like the space defined in S d (d > 1) of spherical harmonics of degree less than √ L. In fact, the space E L is a generalization of the spherical harmonics and the role of them are played by the eigenfunctions. The cases M = S 1 and M = S d (d > 1) have been studied in [14] and [11], respectively.
This similarity between eigenfunctions of the Laplacian and polynomials is not new, for instance, Donnelly and Fefferman in [1,Theorem 1], showed that on a compact manifold, an eigenfunction of eigenvalue λ behaves essentially like a polynomial of degree √ λ. In this direction they proved the following local L ∞ Bernstein inequality.
Theorem (Donnelly-Fefferman). Let M be as above. If u is an eigenfunction of the Laplacian ∆ M u = −λu, then there exists r 0 = r 0 (M) such that for all r < r 0 we have |u|.
The proof of the above estimate is rather delicate. Donnelly and Fefferman conjectured that it is possible to replace λ (m+2)/4 by √ λ in the inequality. If the conjecture holds, we have in particular, a global Bernstein type inequality: In fact, this weaker estimate holds and a proof will be given later. This fact suggests that the right metric to study the space E L should be rescaled by a factor 1/ √ L because in balls of radius 1/ √ λ, a bounded eigenfunction of eigenvalue λ oscillates very little. In the present work we will study for which measures µ = {µ L } L one has with constants independent of f and L.
We will also study the inequalitŷ that defines the Carleson measures and we will present a geometric characterization of them. Inequality (2) will be studied only for the special case dµ L = χ A L dV , where A = {A L } L is a sequence of sets in the manifold. In such case, when (2) holds , we say that A is a sequence of Logvinenko-Sereda sets. Our two main results are the following: In what follows, when we write that A B, A B or A ≈ B we mean that there are constants depending only on the manifold such that A ≤ CB, A ≥ CB or C 1 B ≤ A ≤ C 2 B, respectively. Also, the value of the constants appearing during a proof may change but they will be still denoted with the same letter. We will denote by B(ξ, r) a geodesic ball in M of center ξ and radius r and B(z, r) will denote an Euclidean ball in R m of center z and radius r.
The structure of the paper is the following: in the first section, we will explain the asymptotics of the reproducing kernel of the space E L . In the second section, we shall discuss one of the tools used: the harmonic extension of functions in the space E L . Following this, we will study the Carleson measures associated to M and we will prove Theorem 1. In the last section, Theorem 2 will be proved.
Acknowledgment. We thank professor M. Sanchón for valuable comments on the subject and professor S. Zelditch for providing us appropriate references.
This function is the reproducing kernel of the space E L , i.e.
Note that dim(E L ) = k L = # {λ i ≤ L} . The function K L is also called the spectral function associated to the Laplacian. Hörmander in [5], proved the following estimates.
). In fact, in [5], there are estimates for the spectral function associated to any elliptic operator of order n ≥ 1 with constants depending only on the manifold.
So, for L big enough we have k L ≈ L m/2 and K L (z, ·) 2 2 = K L (z, z) ≈ L m/2 ≈ k L with constants independent of L and z.

Harmonic extension
In what follows, given f ∈ E L , we will note by h the harmonic extension of f in N = M × R. The metric that we consider in N is the product metric, i.e. if we denote it byg ij for i = 1, . . . , m + 1 then Using this matrix, we can calculate the gradient and the Laplacian for N. If h(z, t) is a function defined on N then Observe that h(z, 0) = f (z). Moreover The function h is harmonic in N because For the harmonic extension, we don't have the mean-value property, because it is not true for all manifolds (only for the harmonic manifolds, see [17] for a complete characterization of them). What is always true is a "submean value property" (with a uniform constant) for positive subharmonic functions, see for example [15, Chapter II, Section 6]).
Observe that since h is harmonic on N, |h| 2 is a positive subharmonic function on N. In fact, |h| p is subharmonic for all p ≥ 1 (for a proof see [3,Proposition 1]). Therefore, we know that for all r < R 0 (M) where I r = I r (0). The following result relates the L 2 −norm of f and h.
with constants depending only on the manifold M.
Proof. Using the orthogonality of {φ i } i we have Similarly, one can prove the left hand side inequality of (4).
We recall now a result proved by Schoen and Yau that estimates the gradient of harmonic functions.
Theorem (Schoen-Yau). Let N be a complete Riemannian manifold with Ricci curvature bounded below by -(n-1)K (n is the dimension of N and K a positive constant). Suppose B a is a geodesic ball in N with radius a and h is an harmonic function on B a . Then where C n is a constant depending only on the dimension of N.
Remark 1. We will use Schoen and Yau's estimate in the following context. Take N = M × R. Observe that Ricc(N) = Ricc(M) which is bounded below because M is compact. Note that N is complete because it is a product of two complete manifolds. We will take a = r/ √ L (r < R 0 (M)) and B a = B(z, r/ √ L) × I r (note that this is not the ball of center (z, 0) ∈ N and radius r/ √ L, but it contains and it is contained in such ball of comparable radius).
Using Schoen and Yau's theorem, we deduce the global Bernstein inequality for a single eigenfunction.

Corollary 1 (Bernstein inequality).
If u is an eigenfunction of eigenvalue λ, then We conjecture that in inequality (6), one can replace u by the func- ∇f ∞ √ L f ∞ . For instance, as a direct consequence of Green's formula, we have the L 2 −Bernstein inequality for the space E L : For our purpose, it is sufficient a weaker Bernstein type inequality that compares the L ∞ norm of the gradient with the L 2 norm of the function.
Proposition 2. Let f ∈ E L . Then there exists a universal constant C such that For the proof, we need the following lemma.
Proof. Using inequality (5) and the submean-value inequality for |h| 2 , we have Proposition 2. Using Lemma 1, given 0 < r < R 0 (M)/2 there exists a constant C r such that

Characterization of Carleson measures
Remark 2. Condition (7) can be viewed as First, we prove the following result.
Proof. Obviously, the first condition implies the second one since Let's see the converse. The manifold M is covered by the union of balls of center ξ ∈ M and radius c/ √ L. Taking into account the 5-r covering lemma (see [13, Chapter 2, page 23] for more details), we get a family of disjoint balls, denoted by , such that M is covered by the union of 5B i . This union may be finite or countable. Let ξ ∈ M and consider B := B(ξ, 1/ √ L). Let n be the number of ballsB i such thatB ∩5 B i = ∅. SinceB is compact, we have a finite number of these balls, so that We claim that n is independent of L. Hence we will get our statement because We only need to check that n is independent of L. Using the triangle inequality, for all i = 1, . . . , n Therefore, where the union is a disjoint union of balls. Now, Hence, n (10/c) m and we can choose it independently of L.

Now we can prove the characterization of the Carleson measures.
Theorem 3. Assume condition (7). We need to prove the existence of a constant C > 0 (independent of L) such that for each f ∈ E L M |f | 2 dµ L ≤ CˆM |f | 2 dV.
Let f ∈ E L with L and r > 0 (small) fixed.
For the converse, assume that µ is L 2 −Carleson for M. We have to show the existence of a constant C such that for all L ≥ 1 and ξ ∈ M, µ L (B(ξ, c/ √ L)) ≤ C/k L (for some small constant c > 0). We will argue by contradiction, i.e. assume that for all n ∈ N there exists L n and a ball B n of radius c/ √ L n such that µ Ln (B n ) > n/k Ln ≈ n/L m/2 n (c will be chosen later). Let b n be the center of the ball B n . Define F n (w) = K Ln (b n , w). Note that the function L −m/4 n F n ∈ E Ln and F n Now we will study this infimum. Let w ∈ B n = B(b n , c/ √ L n ). Then We pick c small enough so that Finally, we have shown that C n ∀n. This gives us the contradiction.
The following result is a Plancherel-Pólya type theorem but in the context of the Paley-Wiener spaces E L .
Remark 3. The above result is interesting because the inequality (8) means that the sequence of normalized reproducing kernels is a Bessel sequence for E L , i.e.
where K L (·, z Lj ) j are the normalized reproducing kernels. Note that That's the reason why the quantity k L appears in the inequality (8).
Proof. This is a consequence of Theorem 3 applied to the measures

Characterization of Logvinenko-Sereda Sets
Before we state the characterization, we would like to recall some history of these kind of inequalities. The classical Logvinenko-Sereda (L-S) theorem describes some equivalent norms for functions in the Paley-Wiener space P W p Ω . The precise statements is the following: Theorem 5 (L-S). Let Ω be a bounded set and 1 ≤ p < +∞. A set One can find the original proof in [9] and another proof can be found in [4, p. 112-116].
Luecking in [10] studied this notion for the Bergman spaces. Following his ideas, in [12], it has been proved the following result. For the precise definitions and notations see [12]. Using the same ideas, we will prove the above theorem for the case of our arbitrary compact manifold M and the measure given by the volume element.
In what follows, A = {A L } L will be a sequence of sets in M.
Definition 2. We say that A is L-S if there exists a constant C > 0 such that for any L and  Our main statement is the following:

Theorem 7. A is L-S if and only if A is relatively dense.
We shall prove the two implications in the statement separately. First we will show that this condition is necessary. Before proceeding, we construct functions in E L with a desired decay of its L 2 -integral outside a ball. |f L | 2 dV < ǫ ∀L.
(3) For all L ≥ 1 and any subset A ⊂ M, where C 1 is a constant independent of L, ξ and f L .
Remark. In the above Proposition, the R 0 does not depend on the point ξ.
Proof. Given z, ξ ∈ M and L ≥ 1, let S N L (z, ξ) denote the Riesz kernel of index N ∈ N associated to the Laplacian, i.e Note that S 0 L (z, ξ) = K L (z, ξ). The Riesz kernel satisfies the following inequality.
This estimate has its origins in Hörmander's article [6,Theorem 5.3]. Estimate (9) can be found also in [16,Lemma 2.1]. Note that on the diagonal, S N L (z, z) ≈ C N L m/2 . The upper bound is trivial by the definition and the lower bound follows from Similarly we observe that S N L (·, ξ) We will choose the order N later. Each f L belongs to the space E L and has unit L 2 -norm. Let us verify the second property claimed in Proposition 3. Fix a radius R. Using the estimate (9), For any t ≥ 0, consider the following set.
Now we are ready to prove one of the implications in the characterization of the L-S sets.
Let ξ ∈ M be an arbitrary point. Fix ǫ > 0 and consider the R 0 and the functions f L ∈ E L given by Proposition 3. Using the third property of Proposition 3 for the sets A L , we get for all L ≥ 1, where C 1 is a constant independent of L, ξ and f L . Therefore, we have proved that there exist constants c 1 and c 2 such that Hence, A is relatively dense provided ǫ > 0 is small enough.
Before we continue, we will prove a result concerning the uniform limit of harmonic functions with respect to some metric. Lemma 3. Let {H n } n be a family of uniformly bounded real functions defined on the ball B(0, ρ) ⊂ R d for some ρ > 0. Let g be a non-singular C ∞ metric such that g and all its derivatives are uniformly bounded and g ij (0) = δ ij . Define g n (z) = g(z/L n ) (the rescaled metrics) where L n is a sequence of values tending to ∞ as n increases. Assume that the family {H n } n converges uniformly on compact subsets of B(0, ρ) to a limit function H : B(0, ρ) → R and H n is harmonic with respect to the metric g n (i.e. ∆ gn H n = 0). Then, the limit function H is harmonic in the Euclidean sense.
Therefore, the limit function H is harmonic in the weak sense. Applying Weyl's lemma, H is harmonic in the Euclidean sense.
Remark 5. The above argument also holds if we have a sequence of metrics g n converging uniformly to g whose derivatives also converge uniformly to the derivatives of g. In this case, the conclusion would be that the limit is harmonic with respect to the limit metric g.
Now, we shall prove the sufficient condition of the main result.
Proposition 5. If A is relatively dense then it is L-S.
Proof. Fix ǫ > 0 and r > 0. Let D := D ǫ,r,f L be The norm of f L is almost concentrated on D becausê It is enough to prove with constants independent of L and for this, it is sufficient to show that there exists a constant C > 0 such that for all w ∈ D Because then, (11) follows by integrating (12) over D. So we need to prove (12). Assume it is not true. This means that for all n ∈ N there exists L n , functions f n ∈ E Ln and w n ∈ D n = D ǫ,r,fn such that |f n (w n )| 2 > n vol(B(w n , r/ √ L n ))ˆA Ln ∩B(wn,r/ √ Ln) |f n | 2 dV.
By hypothesis, the sequence {A L } L is relatively dense. Taking into account that vol(B(w n , r/ √ L n )) = r m L m/2 n µ n (B(0, 1)), we get that where we have denoted B n ∩ B(0, 1) by B n . Let τ n be such that dτ n = χ Bn dµ n . From a standard argument (τ n are supported in a ball), we know the existence of a weak *-limit of a subsequence of τ n , denoted by τ . This subsequence will be noted as the sequence itself. Using (13), we know that τ is not identically 0. Now we have thatˆB Therefore τ n (B(a, s)) s m for all n. Hence, in the limit τ (B(a, s)) s m . In short, (1) We have sets B n ⊂ B(0, 1) such that ρ ≤ µ n (B n ) ≤ µ n (B(0, 1)) ≈ 1.
(2) We have measures τ n weakly-* converging to τ (not identically 0). and this implies that dim H (suppτ ) ≥ m by Frostman's lemma. So we reach to a contradiction and the proof is complete. We only need to check that H is real analytic. In fact, we will show that H is harmonic. We have the following properties: (1) Observe that the family of measure dµ n converges uniformly to the ordinary Euclidean measure because g ij (exp wn (rz/ √ L n )) → g ij (exp w 0 (0)) = δ ij , where w 0 is the limit point of some subsequence of w n (recall that we are taking normal coordinate charts). (2) If g n (z) := g(rz/ √ L n ) (i.e. g n is the rescaled metric), then ∆ (gn,Id) H n (z, s) = 0 for all (z, s) ∈ B(0, 1)×J 1 , by construction.
(3) The functions H n are uniformly bounded and converge uniformly on compact subsets of B(0, 1) × J 1 . We are in the hypothesis of Lemma 3 that guarantees the harmonicity of H in the Euclidean sense. This concludes the proof of the proposition.