L−boundedness of oscillating spectral multipliers on Riemannian manifolds

We prove endpoint estimates for operators given by oscillating spectral multipliers on Riemannian manifolds with C∞-bounded geometry and nonnegative Ricci curvature.


Introduction and statement of the results
Let M be an n−dimensional, complete, noncompact Riemannian manifold with nonnegative Ricci curvature and let us assume that it has C ∞ -bounded geometry, that is, the injectivity radius is positive and every covariant derivative of the curvature tensor is bounded (cf.[25]).Let d(., .)denote the Riemannian distance on M , dx its volume element.Let us denote by B(x, r) the ball of radius r > 0 centered at x ∈ M and by |B(x, r)| its volume.By the Bishop comparison theorem (cf.[5]), the assumption that M has nonnegative Ricci curvature implies that and hence This is the so called 'doubling volume property' and makes M a 'space of homogeneous type' in the sense of Coifman and Weiss [8].Thus we can define the atomic Hardy space H 1 (M ) and BM O (M ), the space of functions of bounded mean oscillation, in the standard way (cf.[8]).Further, by Theorem B of [8], BM O (M ) is the dual of H 1 (M ).
Let L be the Laplace-Beltrami operator.It admits a selfadjoint extension on L 2 (M ), also denoted by L and hence the spectral resolution Given a bounded measurable function m(λ), we can define, by the spectral theorem, the operator This operator is bounded on L 2 (M ).The function m(λ) is called multiplier.
In this article we shall prove some endpoint results concerning the L p boundedness of the family of operators We have the following: Theorem 1.1: Let m α,β be as above and let α ∈ (0, 1).The following hold: (i).If β = αn 2 , then m α,β (L) is bounded from H 1 (M ) to L 1 (M ), on L p (M ), 1 < p < ∞ and from L ∞ (M ) to BM O (M ).
For the proof of the H 1 − L 1 boundedness of m α, αn 2 (L), we follow the strategy that Alexopoulos sketches at the end of the paper [1].The idea, which is due to M. Taylor, is to express m α,β (L) in terms of the wave operator cos t √ L and then use the Hadamard parametrix method to get very precise estimates of its kernel near the diagonal.Away from the diagonal, we use the finite propagation speed property of cos t √ L and the fast decay of the multiplier at infinity to obtain that m α,β (L) is bounded on L p , p ≥ 1.
To prove that the operator m α,β (L) is bounded on with the imaginary powers of the Laplacian, which are bounded on H 1 , (cf.[19]), and then use the H 1 − L 1 boundedness of m α, αn 2 (L) and complex interpolation.
We shall apply Theorem 1.1 in order to obtain similar results for the Riesz means associated with the Schrödinger type group e isL α/2 i.e. for the family of operators We have the following Theorem 1.2:For any α ∈ (0, 1), the following hold: In the context of R n , the operators I k,α (L) are studied for example in [27] and [22].According to [27], the results above, for k ≤ n/2, are optimal.The operators I k,α (L) have also been studied in more abstract contexts, see for example [1,2,17,18,4,6].

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It is worth mentioning that our approach is valid only for α ∈ (0, 1).This is due to the fact that the estimates of the multiplier m α,β (λ) are available only for α ∈ (0, 1), (cf.[31] and Section 5).
The paper is organized as follows.In Section 2 we recall some known facts about the Hardy space H 1 and BM O (Subsection 2.1), the wave operator and the construction of its parametrix (Subsection 2.2).In Section 3 the estimates of the Fourier transform of the derivatives of the multiplier m α,β (λ) are given.In Section 4 we give the estimates of the kernel of the operator m α,β (L) near the diagonal and in Section 5 we establish its L p -boundedness when β > n/2.In Section 6 we prove the H 1 − L 1 boundedness of the operator m α, αn 2 (L) and in Section 7 we finish the proofs of Theorems 1.1 and 1.2.
Throughout this article the different constants will always be denoted by the same letter c.When their dependence or independence is significant, it will be clearly stated.

The Hardy space H 1 and BM O
Let us recall that a complex-valued function a on M is an atom if it is supported in a ball B (y 0 , r) and satisfies a ∞ ≤ |B(y 0 , r)| −1 and M a (x) dx = 0.
A function f on M belongs to the Hardy space H 1 (M ) if there exist (λ m ) m∈N ∈ 1 and a sequence of atoms (a m ) m∈N such that where the series converges in L 1 (M ).The norm f H 1 is the infimum of m∈N |λ m | for all such decompositions of f .A function f belongs to BM O(M ), if there exists a constant c > 0 such that for all balls B(x, r), The smallest of all such constants c is the BM O norm of f .Finally we note that the dual of H 1 (M ) is BM O(M ), (cf.[8], Theorem B, p. 593).

The wave operator
Let G t (x, y) be the kernel of the wave operator cos t √ L. Note that G t (x, y) is also the solution of the wave equation In this article we shall exploit the fact that G t (x, y) propagates with finite propagation speed (cf.[7,29]): Next we shall recall some facts about the Hadamard parametrix construction for the kernel G t (x, y), (cf.[3,4,15]).
Let δ ∈ (0, r 0 ), to be fixed later, and let us consider, for every ball B(x, δ), x ∈ M , the exponential normal coordinates centered at x. Let g ij (x, y), y ∈ B(x, δ), be the metric tensor expressed in these coordinates and let us denote by (g ij (x, y)) its inverse matrix.We have the following Taylor expansion of g ij : where the k A ij... are universal polynomials in the components of the curvature tensor and its first k − 2 covariant derivatives at the point x, (cf.[24], p. 85).By the term "universal" we mean that the coefficients of the polynomials k A ij... depend only on the dimension of the manifold.It follows from (2.3) and the assumption of C ∞ -bounded geometry that for any multi-index α there exists a positive constant c α such that In what follows, we shall fix a δ ∈ (0, min(1, r 0 )) such that (2.5) is satisfied.
From (2.4) and (2.5) we also have that there is c α > 0 such that Then, the Laplace-Beltrami operator L can be written as follows: Note that by (2.4), (2.5) and (2.6), the Laplacian can also be written as with the coefficients satisfying for all x ∈ M , y ∈ B(x, δ) and any multi-index β.
Let us consider the following smooth functions: and where y s , s ∈ [0, 1], is the geodesic from x to y and L 2 denotes the Laplacian acting on the second variable.Note that In what follows, we always assume that |t| ≤ δ and y ∈ B(x, δ), x ∈ M .Let us consider the kernels where C 0 is a normalizing constant.They satisfy (cf.[3]) (2.9) Now, let us observe that by (2.4), (2.5) and (2.7) there exists a c > 0 such that (2.10) These also imply that for any k ∈ N there is c > 0 such that If k ≥ n+1 2 , then (2.11) and the fact that (2.12) From (2.8) and (2.12) we get that E N (t, x, y) converges uniformly as N → ∞ and (2.9), (2.11) and (2.1) that the limit is G t (x, y).Thus we have the expansion the convergence being uniform for |t| ≤ δ and y ∈ B(y, δ).

Estimates of the multiplier and of its derivatives
In this section we shall give some estimates for the derivatives of the Fourier transform of the multiplier m α,β .

M. Marias
Let us consider the function Let r 0 be the injectivity radius of M and us fix δ ∈ (0, r 0 ).Let χ δ (t) be a smooth and nonnegative function such that χ δ (t) = 1 for |t| ≤ δ/2 and 0 for |t| ≥ δ.Set In this article we shall need the following: ) Before proceed to the proof of Lemma 3.1, let us recall the following estimates from Wainger [31], Theorem 9.For any α ∈ (0, 1) and > 0, consider the function We have that where J m (z) is the Bessel function.Making use of this formula, Wainger proved that the limit exists and it is continuous for where a 0 = 0, ξ α is real and ξ α = 0; C is a continuous function.
Furthermore  Since we have The integral in brackets above is the same as the integral f ,α,b (t) in formula (3.4), with k = 3 and b = β − 2l.This gives, as → 0, the Fourier transform of the multiplier f α,b (λ) in R 3 .Therefore, the estimates ∂ 2l+1 fα,β (t) follow again from (3.5) and (3.6).

The estimates of the kernel near the diagonal
Let us express the operator m α,β (L) in terms of the wave operator cos t ) and since f α,β is an even function, by the Fourier inversion formula we have that Let m α,β (x, y) be the kernel of m α,β (L).Then by the finite propagation speed property (2.2) This kernel is singular near the diagonal and integrable at infinity.We want to split m α,β (x, y) into these two parts and treat them separately.This can be done by considering the operators where f 0 α,β and f ∞ α,β are defined in (3.1).We have Let m 0 α,β (x, y) and m ∞ α,β (x, y) denote the kernels of m 0 α,β (L) and m ∞ α,β (L), respectively.Then and In the present section we deal with the kernel m 0 α,β (x, y).This kernel contains the singular part of the kernel m α,β (x, y) and from (4.1) it follows that We shall obtain very good L ∞ estimates for m 0 α,β (x, y) by using the Hadamard parametrix construction for G t (x, y).These estimates allow us to prove in Section 6 that m α,β (L) is bounded from We have the following: Lemma 4.1: Let α ∈ (0, 1).Then for all ε ≥ 0, there exists a constant c > 0 such that for all x, y ∈ M m 0 α, αn ) 2 + ε and k = −1, 0, 1,..., we set Lemma 4.1 is a consequence of the expansion (2.13) of G t (x, y) and of the following: 2 , then there is a c > 0 such that , ∀x, y ∈ M. Proof of (4.5) for n = 2p + 1.This is the simpler case.If we put t = ud, then we have (cf. [13], p. 56), we have .
Proof of (4.5), for n = 2p.In this case we have The calculations now are more complicated because k − p − 1 2 is no more an integer.If we put t = du and v = u + 1, then Since f 0 α,β is an even function We shall only treat the term I 0 which is the most singular near v = 0.The integrals I k , k > 0, can be treated similarly.We have By replacing the term (v + 2) −p− 1 2 by its Taylor's expansion at v = 0, we can see that the most singular part of I 0 is the integral Let us observe that fα,β (d(v + 1)) is the Fourier transform of the function Also, the Fourier transform of the distribution v , (cf.[13], p. 172).So, (4.10)We shall only treat J 0,1 .The term J 0,2 can be treated similarly.We have Now L 1 is the Fourier transform of the even function . So, by (3.5), with k = 1, we get that (4.12) By the formula sin x = πx 2 J 1 2 (x), we have The integral in the brackets above is the same as the integral f ,α,b in (3.4) with k = 3 and b = αn 2 + ε − p + 3 2 .Therefore, by (3.5), with k = 3, we get that, for It follows from (4.11), (4.12) and (4.13) that Putting all together, from (4.9) to (4.14), we which proves (4.5), for n = 2p.Proof of (4.6).If k > n+1 2 , then by (3.2) and (3.3) we get Proof of (4.7).We shall only treat the case n = 2p + 1.The case n = 2p can be treated similarly.As in the proof of (4.5), we have to estimate the integral Proof of Lemma 4.1: (i).It is a consequence of (2.11) and Lemma 4.2.