HARDY TYPE SPACES ON CERTAIN NONCOMPACT MANIFOLDS ...

arXiv:0812.4209v1 [math.FA] 22 Dec 2008

HARDY TYPE SPACES ON CERTAIN NONCOMPACT MANIFOLDS AND APPLICATIONS GIANCARLO MAUCERI, STEFANO MEDA AND MARIA VALLARINO Abstract. In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below, positive injectivity radius and spectral gap b. We introduce a sequence X 1 (M ), X 2 (M ), . . . of new Hardy spaces on M , the sequence Y 1 (M ), Y 2 (M ), . . . of their dual spaces, and show that these spaces may be used to obtain endpoint estimates for spectral multipliers associated to the Laplace–Beltrami operator L on M . These results complement earlier work of J. Cheeger, M. Gromov and M. Taylor and of the authors, and improve a recent result of A. Carbonaro, Mauceri and Meda. Under the additional condition that the volume of the geodesic √ balls of radius r is controlled by C r α e2 br for some real number α and for all large r, we prove also an endpoint result for first order Riesz transforms ∇L−1/2 . Under stronger geometric assumptions on M we prove an atomic characterisation of the spaces X k (M ): we show that an atom in X k (M ) is an atom in the Hardy space H 1 (M ) introduced by Carbonaro, Mauceri and Meda, satisfying further cancellation conditions.

Contents 1. Introduction 2. New Hardy spaces and interpolation 3. Some lemmata 4. Hardy spaces on Riemannian manifolds 4.1. Background material 4.2. A remark on the wave propagator 4.3. Economical decomposition of atoms 4.4. Operators bounded on H 1 (M ) 4.5. Further properties of Xσk (M ) 4.6. Equivalent norms on X k (M ) 5. Spectral multipliers and Riesz transforms on Riemannian manifolds 6. Atomic decomposition of X k (M ) References

2 6 10 16 17 18 20 22 26 27 29 31 41

Key words and phrases. Spectral multipliers, Laplace–Beltrami operator, imaginary powers, Mihlin type condition, H¨ ormander integral condition, atomic Hardy space, BM O space, noncompact manifolds, isoperimetric property, noncompact symmetric spaces. Work partially supported by the Italian Progetto Cofinanziato “Analisi Armonica”, 2008–2009. 1

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G. MAUCERI, S. MEDA AND M. VALLARINO

1. Introduction Suppose that (X, ρ, µ) is a measured metric space. Assume that T is a bounded operator on L2 (X), that kT is a function on X × X, locally integrable off the diagonal, such that Z kT (x, y) f (y) dµ(y), T f (x) = X

for each continuous function f with compact support and for almost every x not in the support of f . Assume further that kT satisfies the following H¨ ormander’s type integral condition [Ho] Z |kT (x, y) − kT (x, cB )| dµ(x) < ∞; sup sup B∈B y∈B

(2B)c

here B is the collection of all balls in X, cB denotes the centre of B and 2B is the ball with the same centre as B and twice the radius. If the measure µ is doubling, i.e., if there exists a constant C such that (1.1)

µ(2B) ≤ C µ(B)

for all balls B, then T is bounded on Lp (X) for all p in (1, 2] and from the atomic Hardy space H 1 (X) to L1 (X), and it is of weak type 1. In the case where X is the Euclidean space and T is translation invariant the weak type 1 result was proved L. H¨ ormander [Ho] and the H 1 –L1 boundedness by C. Fefferman and E.M. Stein in the seminal paper [FeS]. These results were subsequently generalised to the setting of spaces of homogeneous type by R.R. Coifman and G. Weiss [CW]. It is interesting to speculate whether there are analogues of the aforementioned results on certain (nondoubling) locally doubling spaces. We mainly focus on noncompact Riemannian manifolds with exponential volume growth. As far as weak type 1 estimates in this setting are concerned, we recall the classical results of M. Taylor [T1] and of J. Cheeger, M. Gromov and Taylor [CGT], which have recently been improved by the authors [MMV1] and by Taylor [T4]. These results are based on Taylor’s method of subordinating functions of the Laplace–Beltrami operator to the wave propagator, or on refinements thereof. In this paper we shall concentrate on H 1 –L1 boundedness results and related estimates involving new function spaces of Hardy type. Implicitly or explicitly, the proofs of some of these estimates are inspired by the method of Taylor. An extension of the H 1 –L1 results of Coifman and Weiss to certain (nondoubling) locally doubling measured metric spaces has recently been obtained by A. Carbonaro, Mauceri and Meda [CMM1, Thm 8.2] (see also [CMM2, Thm 6.1] for the case of finite measure). Following up earlier works of E. Russ [Ru], who defined local atomic Hardy spaces on possibly nondoubling Riemannian manifolds, and of A.D. Ionescu [I1], who defined a BM O space on rank one symmetric spaces of the noncompact type and proved some interpolation results, they defined an atomic Hardy space H 1 (X) and a space of functions of bounded mean oscillation BM O(X) on measured metric spaces satisfying three geometric assumptions: µ is locally doubling, i.e., for each positive s there exists a constant C such that (1.1) holds for all balls B of radius at most s, X possesses a certain isoperimetric property (I) and the approximate midpoint property (see Section 2). Both H 1 (X) and BM O(X) are defined much as in the classical case of spaces of homogeneous type, the only

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difference being that atoms in the definition of H 1 (X) are supported in balls with radius at most 1, and that in the definition of BM O(X) averages are taken only on balls of radius at most 1. As a consequence, they proved that if T is bounded on L2 (X) and its kernel kT satisfies the following local H¨ ormander’s type condition Z (1.2) sup sup |kT (x, y) − kT (x, cB )| dµ(x) < ∞, B∈B1 y∈B

(2B)c

where B1 denotes the collection of all balls in X of radius at most 1, then T is bounded on Lp (X) for all p in (1, 2] and from the atomic Hardy space H 1 (X) to L1 (X). A large class of nondoubling Riemannian manifolds to which this theory applies may be obtained as follows. Suppose that M is a complete connected noncompact Riemannian manifold with Ricci curvature bounded from below and positive injectivity radius. Denote by −L the Laplace–Beltrami operator on M : L is a symmetric operator on Cc∞ (M ) (the space of compactly supported smooth complex-valued functions on M ). Its closure is a self adjoint operator on L2 (M ) which, with a slight abuse of notation, we still denote by L. We assume throughout that the bottom b of the spectrum of L is strictly positive. Important examples of manifolds with these properties are nonamenable connected unimodular Lie groups equipped with a left invariant Riemannian distance, and symmetric spaces of the noncompact type with the Killing metric. It is known [CMM1, Section 8] that for manifolds with Ricci curvature bounded from below the assumption b > 0 is equivalent to the isoperimetric property (I), and that this property implies that M has exponential volume growth, ergo µ is nondoubling and the classical theory of Coifman and Weiss does not apply. An important class of operators whose kernels satisfy (1.2), hence to which [CMM1, Thm 8.2] applies, is the following. Denote by {Pλ } the spectral resolution of the identity for which Z ∞ Lf = λ dPλ f b

for every √f in the domain of L. For notational convenience, denote by D the  operator L − b. For each W in R+ , denote by SW the strip z ∈ C : Im z ∈ (−W, W ) , and by H ∞ (SW ) the space of all bounded holomorphic even functions on SW . Recall that the lower bound for the Ricci curvature implies an upper bound of the volume growth of M . Indeed, there are positive constants α, β and C such that
(1.3) µ B(p, r) ≤ C rα e2β r ∀r ∈ [1, ∞) ∀p ∈ M,
where µ B(p, r) denotes the Riemannian volume of the geodesic ball with centre p and radius r. M. Taylor [T1, Thm 1.6], following up earlier work of J. Cheeger, M. Gromov and Taylor [CGT], proved that if M is a manifold with C ∞ bounded geometry, β is as in (1.3), m is in H ∞ (Sβ ) and satisfies estimates of the form
−j (1.4) |Dj m(ζ)| ≤ C 1 + |ζ| ∀ζ ∈ Sβ ∀j ∈ {0, 1, . . . , N }, where N is a sufficiently large integer depending on the dimension n of M , then the operator m(D) is bounded on Lp (M ) for p in (1, ∞), and of weak type 1. Carbonaro, Mauceri and Meda [CMM1, Thm 10.2] complemented Taylor’s result by proving that if m satisfies (1.4), then the operator m(D) maps H 1 (M ) to L1 (M )

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G. MAUCERI, S. MEDA AND M. VALLARINO

and L∞ (M ) to BM O(M ). See also [MMV1], where we prove weak type 1 estimates for m(D) under weaker geometric assumptions. One of the motivations of this work, though not the most important, is to extend [CMM1, Thm 10.2] by assuming only a lower bound of the Ricci curvature of M . We prove the desired extension in Section 4, as a consequence of a H 1 (M ) boundedness result for operators of the form m(D), where m satisfies (1.4). Now we discuss the main motivation of this paper. It is worth observing that there are many interesting operators of the form m(D), where m does not satisfy (1.4), but rather the following Mihlin type estimate (1.5)

|Dj m(ζ)| ≤ C max |ζ − iβ|−j , |ζ + iβ|−j




∀ζ ∈ Sβ .

To the best of our knowledge weak type 1 estimates for operators satisfying (1.5) in the case where b = β 2 (recall that b > 0 implies b ≤ β 2 by a result of R. Brooks [Br, Thm 1]) are known only when L is the Laplace–Beltrami operator associated to the Killing metric on Riemannian symmetric spaces of noncompact type [I2, MV]. It is worth observing that the proof of this fact is quite intricate and hardly extendable to the general setting described above. It is perhaps surprising that in general operators satisfying (1.5) do not extend to bounded operators from H 1 (M ) to L1 (M ). Consider, in particular, the purely imaginary powers of L in the case where b = β 2 and L is the Laplace–Beltrami operator associated to the Killing metric on Riemannian symmetric spaces of the noncompact type. Then for each u in R \ {0} the operator Liu is unbounded from H 1 (M ) to L1 (M ) (hence its Schwartz kernel kLiu does not satisfy (1.2)). The proof of this fact hinges on quite delicate estimates of the inverse spherical Fourier transform of the associated multiplier, and will appear in [MMV2]. A class of interesting operators for which the weak type 1 estimate fails, and other endpoint estimates for p = 1 are unknown, are the Riesz transforms of order ≥ 3 and the Riesz potentials L−α when α is a complex number with Re α > 1 on noncompact symmetric spaces [A1, AJ, MV]. The purpose of this paper is to introduce a sequence X 1 (M ), X 2 (M ), . . . of new spaces of Hardy type on M , and the sequence Y 1 (M ), Y 2 (M ), . . . of their dual spaces, and show that these spaces may be used to obtain endpoint estimates for operators of the form m(D), where m satisfies (1.5) or even more general conditions (see Definition 5.1). We emphasise the fact that X k (M ) and Y k (M ) will depend on the operator L. The strategy of constructing ad hoc Hardy spaces to obtain (endpoint) estimates for a given operator has already been used in various settings. To mention a few, see [NTV, To, Ve] for Hardy spaces associated to integral operators satisfying Calder´ on–Zygmund estimates on (possibly nondoubling) spaces with polynomial growth, [MM1] for an Hardy space associated to the Ornstein–Uhlenbeck operator, and [DY1, DY2, HM, HLMMY, AMR] for Hardy spaces associated to a given elliptic operator on Rn and to operators satisfying Gaffney–Davies estimates on spaces of homogeneous type in the sense of Coifman and Weiss. The main idea of this paper is that the space H 1 (M ) may be used as a stepping stone to construct smaller spaces X k (M ), each of which is an isometric copy of H 1 (M ). For each σ in R+ denote by Uσ the operator L (σI +L)−1 . It is straightforward to check that Uσ is a bounded injective operator on L1 (M ) + L2 (M ). Denote

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by Xσk (M ) the range of the restriction of Uσk to H 1 (M ), endowed with the norm kf kXσk (M) = kUσ−k f kH 1 (M) .

Clearly Uσk is an isometric isomorphism between H 1 (M ) and Xσk (M ), whence Uσ−k is an isometric isomorphism between Xσk (M ) and H 1 (M ). As a consequence, its ad
∗ joint map Uσ−k is an isometric isomorphism between the dual of H 1 (M ), i.e., the space BM O(M ) of functions of bounded mean oscillation introduced in [CMM1], and the dual of Xσk (M ), which we denote by Yσk (M ). The definitions of the spaces Xσk (M ) and Yσk (M ) are fairly abstract, but have the advantage that useful interpolation results follow comparatively easily from them. In Section 2 we
prove that if θ is in (0, 1), then the complex interpolation space Xσk (M ), L2 (M ) [θ] is Lp (M )
where 1/p = 1 − θ/2, and the complex interpolation space L2 (M ), Yσk (M ) [θ] is Lq (M ) where 1/q = (1 − θ)/2. It may be worth observing that the theory developed in Section 2 holds, in fact, in a quite general setting, and M need not be a manifold. More concrete descriptions of Xσk (M ) and Yσk (M ) in the case where M is a manifold will be given in Section 6. Further properties of Xσk (M ) are established in Section 4, where we assume that M is a Riemannian manifold with the properties described at the beginning of the Introduction. We prove that Xσk (M ) is independent of σ, as long as σ belongs to (β 2 − b, ∞). Then we denote Xβk2 (M ) simply by X k (M ). A sharper result in the case where M is a Lie group or a symmetric space will be proved in [MMV2]. In Theorem 4.9 we prove an H 1 (M ) boundedness result for operators of the form m(D), where m satisfies (1.4). This result is sharp, in the sense that the width of the strip of holomorphy of m is best possible, as the case of noncompact symmetric spaces shows. A key step in proving this result is an economical decomposition of H 1 -atoms proved in Lemma 4.6. As a consequence of Theorem 4.9 we shall prove that H 1 (M ) ⊃ X 1 (M ) ⊃ X 2 (M ) ⊃ · · · (see Section 4), with proper inclusions. Applications of the theory developed in Sections 2–4 to spectral multipliers are contained in Section 5. One of the main result of this paper, Theorem 5.3 below, states that if J is a sufficiently large integer, and m satisfies (1.5), or even the more general condition (5.1) with k > τ + J, then m(D) is bounded from H 1 (M ) to L1 (M ) and from L∞ (M ) to BM O(M ) in the case where b < β 2 and from X k (M ) to H 1 (M ) and from BM O(M ) to Y k (M ) in the case where b = β 2 . This provides, in the case where b = β 2 , endpoint estimates for operators of the form Liu , but also for “more singular operators”, such as Liu−τ (I + L)τ , whose kernels have a comparatively slow decay at infinity. We give applications also to first order Riesz transforms. It follows from work of T. Coulhon and X.T. Duong [CD] that, in our setting, the first order Riesz transform ∇L−1/2 is bounded on Lp (M ) for all p in (1, 2] and that the translated Riesz transform ∇(I + L)−1/2 is of weak type 1. Russ complemented this result by showing that ∇(I + L)−1/2 map H 1 (M ) into L1 (M ). Observe that if we consider the part off the diagonal of the kernel of ∇(I + L)−1/2 , then the corresponding integral operator is bounded on L1 (M ). This is no longer true for the kernel of the Riesz transform, which decays much slower at infinity. Despite this, we prove that if b = β 2 , then ∇L−1/2 is bounded from X k (M ) to L1 (M ) for large k. Applications of these spaces to higher order Riesz transforms associated to the Laplace–Beltrami operator on noncompact symmetric spaces and to multipliers for the spherical Fourier transform will be considered in a forthcoming paper [MMV2].

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The last part of this paper is devoted to the proof of an atomic characterisation of X k (M ). Since X k (M ) is continuously included in H 1 (M ), each function in X k (M ) admits an atomic decomposition in terms of H 1 -atoms. It is natural to speculate whether functions in X k (M ) may be characterised as those functions in H 1 (M ) that admit a decomposition in terms of “special” atoms in H 1 (M ). We say that A is a X k -atom if A is a H 1 -atom supported in a ball B of radius at most 1 and is orthogonal in L2 (B) to the space QkB of all functions V in L2 (M ) such that Lk V is constant on a neighbourhood of B. In Section 6 we prove that if the first 2k − 2 covariant derivatives of the Ricci tensor of M are uniformly bounded, then a function F is in X k (M ) if and only if it admits a decomposition of the form P k j λj Aj , where {λj } is a summable sequence and each Aj is a X -atom. As in the Euclidean case, an atomic characterisation of X k (M ) naturally leads to a concrete description of the action on atoms of continuous linear functionals in Y k (M ). This is done in the last part of Section 6. When the writing up of this paper was about to be completed, Michael Taylor kindly sent us the paper [T3] in which he introduces the spaces h1 (M ) and bmo(M ) on Riemannian manifolds with C ∞ bounded geometry. These are generalisations to Riemannian manifolds of the local Hardy and BM O spaces introduced by D. Goldberg [G] on Euclidean spaces. Note that the assumption b > 0 is not required in [T3]. We remark that when both h1 (M ) and H 1 (M ) are defined, then h1 (M ) contains 1 H (M ) and BM O(M ) contains bmo(M ) properly (the distance function from a reference point o is in BM O(M ), but not in bmo(M )). Despite this, various functions of L have similar boundedness properties on h1 (M ) and H 1 (M ). To illustrate this, suppose that u is real and that c > 2β. Then on the one hand the operators (L + c)iu are bounded both on h1 (M ) [T3, Proposition 6.1] and on H 1 (M ) (Theorem 4.9 below). Note however, that Theorem 4.9 is sharper, because it yields boundedness of (L + c)iu on H 1 (M ) also for c > β 2 − b. On the other hand, we have already observed that when u 6= 0 and L is the Laplace–Beltrami operator associated to the Killing metric on Riemannian symmetric spaces of the noncompact type, then Liu is unbounded from H 1 (M ) to L1 (M ), and, a fortiori, from h1 (M ) to L1 (M ), but it is bounded, at least when b = β 2 , from X k (M ) to H 1 (M ) if k is sufficiently large (see Theorem 5.3). Presumably the same construction that starting from H 1 (M ) leads to X k (M ) may be applied to h1 (M ) to generate a family of spaces Xk (M ) with properties similar to the spaces X k (M ) on manifolds with C ∞ bounded geometry. We do not prove this in this paper. We will use the “variable constant convention”, and denote by C, possibly with sub- or superscripts, a constant that may vary from place to place and may depend on any factor quantified (implicitly or explicitly) before its occurrence, but not on factors quantified afterwards. 2. New Hardy spaces and interpolation In this section we define new Hardy spaces in a fairly general framework, and show that they have natural interpolation properties. Suppose that (M, d, µ) is a measured metric space, and denote by B the family of all balls on M . We assume that µ(M ) > 0 and that every ball has finite measure. For each B in B we denote by cB and rB the centre and the radius of B respectively.

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Furthermore, we denote by c B the ball with centre cB and radius c rB . For each scale parameter s in R+ , we denote by Bs the family of all balls B in B such that rB ≤ s. Basic assumptions 2.1. We assume throughout that M is unbounded and possesses the following properties: (i) local doubling property (LD): for every s in R+ there exists a constant Ds such that

(2.1) µ 2B ≤ Ds µ B ∀B ∈ Bs ;

(ii) isoperimetric property (I): there exist κ0 and C in R+ such that for every bounded open set A   µ x ∈ A : d(x, Ac ) ≤ κ ≥ C κ µ(A) ∀κ ∈ (0, κ0 ];

(iii) approximate midpoint property (AM): there exist R0 in [0, ∞) and γ in (1/2, 1) such that for every pair of points x and y in M with d(x, y) > R0 there exists a point z in M such that d(x, z) < γ d(x, y) and d(y, z) < γ d(x, y); (iv) there is a semigroup of linear operators {Ht } acting on L1 (M ) + L2 (M ) such that (a) the restriction of {Ht } to L1 (M ) is a strongly continuous semigroup of contractions; (b) the restriction of {Ht } to L2 (M ) is strongly continuous, and has spectral gap b > 0, i.e. kHt f k2 ≤ e−bt kf k2

∀f ∈ L2 (M ) ∀t ∈ R+ ;

(c) {Ht } is ultracontractive, i.e. for every t in R+ the operator Ht maps L1 (M ) into L∞ (M ). Remark 2.2. Assumption (ii) forces µ(M ) = ∞. In fact, it forces M to have exponential volume growth (see [CMM1, Proposition 2.5 (i)] for details). Remark 2.3. Assumption (iv) has the following straightforward consequences: (i) {Ht } is a strongly continuous semigroup of contractions on L1 (M )+L2(M ); (ii) since for each p in [1, 2] the space Lp (M ) is continuously embedded in L1 (M ) + L2 (M ), we may consider the restriction Hpt of the operator Ht to Lp (M ). Then {Hpt } is strongly continuous on Lp (M ), and satisfies the estimate (2.2)

kHpt f kp ≤ e−2b (1−1/p) t kf kp

∀f ∈ Lp (M )

∀t ∈ R+ ;

(iii) by (iv) (a) and (iv) (c) above, for each t in R+ the operator Ht maps L1 (M ) into L1 (M )∩L2 (M ). Hence Ht maps L1 (M ) into Lp (M ) for each p in [1, 2]. Denote by −G the infinitesimal generator of {Ht } on L1 (M )+L2 (M ). Since {Ht } is contractive on L1 (M ) + L2 (M ), the spectrum of G is contained in the right half plane. Then, for every σ in R+ we may consider the resolvent operator (σI + G)−1 of {Ht }, that we denote by Rσ . We denote by Rσ,p the restriction of Rσ to Lp (M ), and by −Gp the generator of {Hpt }. Obviously Rσ,p is the resolvent of {Hpt } and
−Gp is the restriction of −G to Dom(Gp ), which coincides with Rσ Lp (M ) .

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For every σ in R+ denote by Uσ the operator GRσ . Observe that 1

Uσ = I − σ Rσ ,

so that Uσ is bounded on L (M ) + L2 (M ), and its restriction Uσ,p to Lp (M ) is bounded on Lp (M ) for every p ∈ [1, 2]. Moreover Uσ and Ht commute for every t in R+ . Proposition 2.4. For each positive integer k the following hold: k (i) if p is in (1, 2], then the operator Uσ,p is an isomorphism of Lp (M ); k 1 (ii) the operator Uσ is injective on L (M ) + L2 (M ). Proof. First we prove (i). Clearly, it suffices to show that Uσ,p is an isomorphism of Lp (M ). By (2.2) the bottom of spectrum of Gp is positive. Thus Gp−1 and σ Gp−1 + I −1 are bounded. Since Uσ,p = Gp−1 (σI + Gp ) and Gp−1 (σI + Gp ) = σ Gp−1 + I, (i) is proved. Next we prove (ii). It suffices to prove the result in the case where k = 1, since the general case follows by induction. Suppose that f
is a function in L1 (M ) +
t 2 L (M ) such that Uσ f = 0. Then Uσ H f = Ht Uσ f = 0 for all t in R+ . By the ultracontractivity of Ht , and the fact that the restriction of Ht to L2 (M
) is bounded
on L2 (M ), the function Ht f is in L2 (M ) for all t in R+ . Thus Uσ Ht f = Uσ,2 Ht f = 0. Hence Ht f = 0, because Uσ,2 is an isomorphism. Since {Ht } is strongly continuous on L1 (M ) + L2 (M ) by Remark 2.3 (i), Ht f tends to f in L1 (M ) + L2 (M ) as t tends to 0, and (ii) follows.  We recall the definitions of the atomic Hardy space H 1 (M ) and its dual space BM O(M ) given in [CMM1]. We observe that property (iv) above is not needed for these definitions and is not used in the proof of Theorem 2.10. Definition 2.5. An H 1 -atom a is a function in L1 (M ) supported in a ball B with the following properties: R (i) B a dµ = 0; (ii) kak2 ≤ µ(B)−1/2 .

Definition 2.6. Suppose that s is in R+ . The Hardy space Hs1 (M ) is the space of all functions g in L1 (M ) that admit a decomposition of the form ∞ X (2.3) g= λk ak , k=1

P∞ where ak is a H 1 -atom supportedP in a ball B of Bs , and k=1 |λk | < ∞. The norm ∞ kgkHs1 (M) of g is the infimum of k=1 |λk | over all decompositions (2.3) of g.
The vector space Hs1 (M ) is independent of s in R0 /(1 − γ), ∞ , where R0 and γ are as in Basic assumptions 2.1 (iii) (see [CMM1, Proposition 5.1]). Further
more, given s1 and s2 in R0 /(1 − γ), ∞ , the norms k·kHs1 (M) and k·kHs1 (M) are 2 1 equivalent. Notation 2.7. We shall denote the space Hs1 (M ) simply by H
1 (M ), and we endow H 1 (M ) with the norm Hs10 (M ), where s0 = max R0 /(1 − γ), 1 . We note explicitly that if R0 = 0, then s0 = 1.

The Banach dual of H 1 (M ) is isomorphic [CMM1, Thm 5.1] to the space BM O(M ), which we now define.

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Definition 2.8. The space BM O(M ) is the space of all locally integrable functions f such that N (f ) < ∞, where Z 1 |f − fB | dµ, N (f ) = sup B∈Bs0 µ(B) B and fB denotes the average of f over B. We endow BM O(M ) with the “norm” kf kBMO(M) = N (f ). Remark 2.9. It is straightforward to check that f is in BM O(M ) if and only if its sharp maximal function f ♯ , defined by Z 1 f ♯ (x) = sup |f − fB | dµ ∀x ∈ M, B∈Bs0 (x) µ(B) B is in L∞ (M ). Here Bs0 (x) denotes the family of all balls in Bs0 that contain the point x. Recall the following interpolation result [CMM1]. Given a compatible couple of Banach spaces X0 and X1 we denote by (X0 , X1 )[θ] its complex interpolation space, also denoted by Xθ . Theorem 2.10. Suppose that (M, d, µ) is a measured metric space that possesses properties (I) and (LD). Suppose that θ is in (0, 1). The following hold:
(i) if 1/p = 1 − θ/2, then H 1 (M ), L2 (M ) [θ] = Lp (M );
(ii) if 1/q = (1 − θ)/2, then L2 (M ), BM O(M ) [θ] = Lq (M ).

In the last part of this section we define the new spaces Xσk (M ) of Hardy type and their dual spaces Yσk (M ), and prove an interpolation result, which is relevant for later developments. Definition 2.11. For each positive integer k and for each σ in R+ we denote by Xσk (M ) the Banach space of all L1 (M ) functions f such that Uσ−k f is in H 1 (M ), endowed with the norm kf kXσk (M) = kUσ−k f kH 1 (M) . Note that Uσ−k is, by definition, an isometric isomorphism between Xσk (M ) and H 1 (M ). In Subsection 4.5, we shall show that Xσk (M ) may be characterised as the image of H 1 (M ) under a wide class of maps V k . Remark 2.12. Note that the space Xσk (M ) is continuously included in L1 (M ). Indeed, suppose that f is in Xσk (M ). Then





kf k1 = Uσk Uσ−k f 1 ≤ Uσk 1 Uσ−k f 1 ≤ Uσk 1 Uσ−k f H 1 (M) , = U k f k σ

1

Xσ (M)

as required. Note that the last inequality is a consequence of the fact that H 1 (M ) is continuously included in L1 (M ). Definition 2.13. For each positive integer k, and for each σ in R+ we denote by Yσk (M ) the Banach dual of Xσk (M ).

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Remark 2.14. Since U
σ−k is an isometric isomorphism between Xσk (M ) and H 1 (M ), ∗ its adjoint map Uσ−k is an isometric isomorphism between the dual of H 1 (M ), i.e., BM O(M ), and Yσk (M ). Hence

−k
∗

Uσ f Y k (M) = kf kBMO(M) . σ

Proposition 2.15. Suppose that (X 0 , X 1 ) and (Y 0 , Y 1 ) are interpolation pairs of Banach spaces. Suppose further that T is a bounded linear map from X 0 + X 1 to Y 0 + Y 1 , such that the restrictions T : X 0 → Y 0 and T : X 1 → Y 1 are isomorphisms. Then for every θ in (0, 1) the restriction T : Xθ → Yθ is an isomorphism. Proof. For every θ in [0, 1] denote by Tθ the restriction of T to Xθ . Define S : Y0 + Y1 → X0 + X1 by setting S(y0 + y1 ) = T0−1 y0 + T1−1 y1 .

It is straightforward to check that the operator S is well defined, bounded and linear. Moreover ST is the identity on X0 + X1 and T S is the identity on Y0 + Y1 . Thus S = T −1 . Hence Sθ = Tθ−1 . Finally, Sθ : Yθ → Xθ is bounded by interpolation. This concludes the proof of the proposition.  Proposition 2.16. Suppose that σ is in R+ , k is a positive integer, and θ is in (0, 1). The following hold:
(i) if 1/p = 1 − θ/2, then Xσk (M ), L2 (M ) [θ] = Lp (M ) with equivalent norms;
(ii) if 1/q = (1 − θ)/2, then L2 (M ), Yσk (M ) [θ] = Lq (M ) with equivalent norms. Proof. To prove (i), we first observe that Uσk is an isomorphism of H 1 (M ) + L2 (M ) onto Xσk (M ) + L2 (M ). Then we may apply Proposition 2.15 with Uσk in place of T , X 0 = H 1 (M ), Y 0 = Xσk (M ), X 1 = L2 (M ) = Y 1 . By Theorem 2.10
H 1 (M ), L2 (M ) [θ] = Lp (M ).

By Proposition 2.15, the restriction of Uσk to Lp (M ) is an isomorphism between
k p k 2 , L (M ) and Xσ (M ), L (M ) [θ] . But the restriction of Uσk to Lp (M ) is just Uσ,p
p k 2 which is an isomorphism of L (M ) by Proposition 2.4. Hence Xσ (M ), L (M ) [θ] and Lp (M ) are isomorphic Banach spaces, as required. Now (ii) follows from (i) by the duality theorem.  3. Some lemmata

This section contains a few technical lemmata concerning one-dimensional Fourier analysis. Some of the material in this section is taken from [MMV1, Subsection 2.3], and, for the reader’s convenience, is included here, without proofs. We refer the reader to [MMV1, Subsection 2.3] for a discussion of the motivations behind this rather technical development. For every f in L1 (R) define its Fourier transform fb by Z ∞ f (s) e−ist ds ∀t ∈ R. fb(t) = −∞

HARDY SPACES ON NONCOMPACT MANIFOLDS

11

Suppose that f is a function on R, and that λ is in R+ . We denote by f λ and fλ the λ-dilates of f , defined by (3.1)

f λ (x) = f (λx)

fλ (x) = λ−1 f (x/λ)

and

∀x ∈ R.

For each ν ≥ −1/2, denote by Jν : R \ {0} → C the modified Bessel function of order ν, defined by Jν (t) Jν (t) = ν , t where Jν denotes the standard Bessel function of the first kind and order ν (see, for instance, [L, formula (5.10.2), p. 114] for the definition). Recall that r r 2 2 sin t cos t and that J1/2 (t) = . J−1/2 (t) = π π t For each positive integer ℓ, we denote by Oℓ the differential operator tℓ Dℓ on the real line. Lemma 3.1. For every positive integer k there exists a polynomial Pk+1 of degree k + 1 without constant term, such that Z ∞ Z ∞ (3.2) f (t) cos(vt) dt = Pk+1 (O)f (t) Jk+1/2 (tv) dt, −∞

−∞

ℓ

1

for all functions f such that O f ∈ L (R) ∩ C0 (R) for all ℓ in {0, 1, . . . , k + 1}. Proof. The proof uses the definition and some properties of the generalised Riesz means Rd,z , introduced in [CM, Section 1]. We refer the reader to [MMV1, Section 2] for all the prerequisites needed here. In particular, recall that R3+2k,0 = R3+2k,−k R3,k by [MMV1, Lemma 2.3 (i)]. Now, by integrating by parts and using [MMV1, Lemma 2.3 (i) and (ii)], Z ∞ Z ∞ sin(vt) f (t) cos(vt) dt = − Of (t) dt vt −∞ −∞ r Z ∞
π v (t) dt Of (t) R3+2k,0 J1/2 =− 2 −∞ r Z ∞

π v R∗ (t) dt Of (t) R3,k J1/2 =− 2 −∞ 3+2k,−k

for all v in R. Furthermore, the definitions of R3,k and of J1/2 and an integration by parts show that r Z
2 1 1 2 2 k−1 R3,k J1/2 (u) = sin(su) ds s (1 − s ) π Γ(k) u 0 r Z 1 1 2 (1 − s2 )k cos(su) ds = π Γ(k + 1) 0 = 2k Jk+1/2 (u).


∗ By [MMV1, Lemma 2.4 (i)] there exist constants cℓ such that R3+2k,−k Of = Pk ℓ+1 f , so that ℓ=0 cℓ O Z ∞ Z ∞ k X
′ f (t) cos(vt) dt = cℓ Oℓ Of (t) Jk+1/2 (tv) dt, −∞

ℓ=0

−∞

12

G. MAUCERI, S. MEDA AND M. VALLARINO

and the required formula, with Pk+1 (s) =

Pk

′ ℓ+1 , ℓ=0 cℓ s

follows.



Remark 3.2. We shall denote by Pk+1 (O)∗ the formal adjoint of the operator Pk+1 (O), i.e. the operator defined by Z ∞ Z ∞ f (t) Pk+1 (O)∗ g(t) dt = Pk+1 (O)f (t) g(t) dt ∀f, g ∈ Cc∞ (R). −∞

−∞

Note that Pk+1 (O)∗ is still a polynomial of degree k + 1 in O and that Pk+1 (O)∗ Jk+1/2 (vt) = cos(vt), by (3.2).

Denote by ω an even function in Cc∞ (R) which is supported in [−3/4, 3/4], is equal to 1 in [−1/4, 1/4], and satisfies X ω(t − j) = 1 ∀t ∈ R. j∈Z

1/4

Denote by φ the function ω − ω, where ω 1/4 denotes the 1/4-dilate of ω. Then φ is smooth, even and vanishes in the complement of the set {t ∈ R : 1/4 ≤ |t| ≤ 4}. For a fixed R in (0, 1] and for each positive integer i, denote by Ei the set {t ∈ R : P∞ i i 4i−1 R ≤ |t| ≤ 4i+1 R}. Clearly φ1/(4 R) is supported in Ei , and i=1 φ1/(4 R) = 1 in R \ (−R, R). Denote by d the integer [log4 (3/R)]] + 1. To avoid cumbersome notation, we write ρi instead of 1/(4i R). Then (3.3)

ω ρ0 +

d X

φρi = 1

on [−3, 3].

i=1

Definition 3.3. We say that a function g : R → C satisfies a Mihlin condition [Ho] of order J at infinity on the real line if there exists a constant C such that (3.4)

|Dℓ g(t)| ≤ C (1 + |t|)−ℓ

∀t ∈ R

∀ℓ ∈ {0, . . . , J}.

We denote by kgkMih(J) the infimum of all constants C for which (3.4) holds. Lemma 3.4. Suppose that k is a nonnegative integer, and that K is an even b Mih(k+2) is finite. The following hold: tempered distribution on R such that kKk

(i) for each ℓ in {0, . . . , k} the function t Oℓ K is in L∞ (R), and there exists a constant C such that b Mih(k+2) kt Oℓ Kk∞ ≤ C kKk ∀ℓ ∈ {0, . . . , k}; Pd b (ii) if k ≥ 1 and the support of K is contained in [−1, 1], then K(λ) = i=0 Si (λ), where the functions Si : R → C are defined by (3.5) Z ∞ k X b cj,k K(t) Oj ω(ρ0 t) Ok−j Jk+1/2 (λt) dt + S0 (λ) = (b ωρ0 ∗ K)(λ) j=1

−∞

for suitable constants cj,k , and, for i in {1, . . . , d}, Z ∞ 1 (3.6) Si (λ) = φρi (t) Pk+1 (O)K(t) Jk+1/2 (λt) dt. 2π −∞

(iii) if the support of K is contained in [−1, 1], then there exists a constant C such that b Mih(2) . kS0 k∞ ≤ C kKk

HARDY SPACES ON NONCOMPACT MANIFOLDS

13

b satisfies a Mihlin Proof. First we prove (i) in the case where k = 0. Since K 2 b 1 condition of order 2 at infinity, D K is in L (R) (see (3.4)), and we may define F : R → C by Z ∞ b F (t) = D2 K(ζ) eiζt dζ. −∞

By elementary Fourier analysis tK(t) = −t−1 F (t). Observe that F (0) = 0, because Z A b F (0) = lim D2 K(ζ) dζ A→∞

−A

b = 2 lim DK(A) A→∞

= 0,

b vanishes at infinity, because where we have used the fact that K is even and DK b Mih(2) is finite. Furthermore kKk F (t) = F (t) − F (0) Z ∞ b (eiζt − 1) dζ. = D2 K(ζ) −∞

Suppose that t is positive. Then we write the last integral as the sum of the integrals over the sets {ζ ∈ R : |ζ| ≤ 1/t} and {ζ ∈ R : |ζ| > 1/t}, and estimate them separately. To treat the first we integrate by parts, and obtain Z b D2 K(ζ) (eiζt − 1) dζ |ζ|≤1/t Z i −i b b b DK(ζ) eiζt dζ. = DK(1/t) (e − 1) − DK(−1/t) (e − 1) − it |ζ|≤1/t

b is odd, its integral over [−1/t, 1/t] vanishes, so that the last integral may Since DK be rewritten as Z b DK(ζ) (eiζt − 1) dζ. Hence Z

|ζ|≤1/t

|ζ|≤1/t

b b Mih(2) D2 K(ζ) (eiζt − 1) dζ ≤ C kKk

|t| + 1 + |t| + C t2

b Mih(2) |t| ≤ C kKk

|ζ|≤1/t

∀t ∈ R+ .

To estimate the second, write Z Z 2 b iζt b D K(ζ) (e − 1) dζ ≤ C k Kk Mih(2) |ζ|>1/t

Finally, since K is even,

Z

|ζ|>1/t

b Mih(2) |t| ≤ C kKk

|F (t)| |t| t∈R b ≤ C kKkMih(2) ,

ktKk∞ ≤ sup

b |ζ DK(ζ)| dζ

1 dζ 1 + ζ2

∀t ∈ R+ .

14

G. MAUCERI, S. MEDA AND M. VALLARINO

as required to conclude the proof of (i) in the case where k = 0. Next we assume that k ≥ 1. By the case k = 0 applied to Oℓ K, we see that [ ℓ Kk kt Oℓ Kk∞ ≤ C kO Mih(2) .

Pℓ j b [ ℓK = Since O j=0 αj,ℓ O K for suitable constants αj,ℓ , [ ℓ Kk kO Mih(2) ≤ C

ℓ X j=0

b Mih(2) kOj Kk

b Mih(2+ℓ) , ≤ C kKk

b Mih(k+2) , as required to conclude the proof of which is clearly dominated by C kKk (i). b Now we prove (ii). Suppose that ε is in (0, 1). Clearly K(λ) is the limit of b (b ω ε K)(λ) as ε tends to 0. By Fourier inversion formula and Lemma 3.1 Z ∞ 1 ε b (b ω K)(λ) = ωε ∗ K(t) cos(λt) dt 2π −∞ Z ∞ 1 Pk+1 (O)(ωε ∗ K)(t) Jk+1/2 (λt) dt ∀λ ∈ R. = 2π −∞ Pd We write the right-hand side as i=0 Si (λ; ε), where Z ∞ 1 (3.7) S0 (λ; ε) = ∀λ ∈ R, ω ρ0 (t) Pk+1 (O)(ωε ∗ K)(t) Jk+1/2 (λt) dt 2π −∞ and, for each i in {1, . . . , d}, Z ∞ 1 Si (λ; ε) = φρi (t) Pk+1 (O)(ωε ∗ K)(t) Jk+1/2 (λt) dt 2π −∞ Observe that 1 S0 (λ; ε) = 2π ∗

Note that Pk+1 (O) (ω

ρ0

Z

∞

−∞

∀λ ∈ R.

λ )(t) dt. (ωε ∗ K)(t) Pk+1 (O)∗ (ω ρ0 Jk+1/2

λ Jk+1/2 )

may be written as

λ )+ ω ρ0 Pk+1 (O)∗ (Jk+1/2

k X j=1

c′j,k (Oj ω)ρ0 (Ok−j Jk+1/2 )λ ,

c′j,k ,

λ for suitable constants and that Pk+1 (O)∗ (Jk+1/2 )(t) = cos(tλ), by Remark 3.2. Hence   b (λ)+ ω ε K) S0 (λ; ε) = ω bρ0 ∗ (b Z ∞ k X cj,k (ωε ∗ K)(t) (Oj ω)ρ0 (t) (Ok−j Jk+1/2 )λ (t) dt. + j=1

−∞

Note that for each positive integer j the function Oj ω vanishes in [−1/4, 1/4], and that the restriction of K to [−1/4, 1/4]c is a bounded function by (i) (with k = 0). Then it is straightforward to check that S0 (λ; ε) tends to S0 (λ) for all λ in R.

HARDY SPACES ON NONCOMPACT MANIFOLDS

15

To prove that Si (λ; ε) tends to Si (λ) for all λ in R and all i in {1, . . . , d}, observe that E D λ , Pk+1 (O)(ωε ∗ K) 2π Si (λ; ε) = φρi Jk+1/2 E D λ ), ωε ∗ K , = Pk+1 (O)∗ (φρi Jk+1/2

where h·, ·i denotes the duality between test functions and distributions on R. Now we let ε tend to 0 and obtain E D λ ), K 2π Si (λ; ε) → Pk+1 (O)∗ (φρi Jk+1/2 E D λ , Pk+1 (O)K . = φρi Jk+1/2 By (i) the distribution Pk+1 (O)K is a bounded function on the support of φρi , so that the right hand side is exactly 2π Si (λ), thereby concluding the proof of (ii). Finally, to prove (iii), observe that |S0 (λ)| ≤ |(b ωρ0

b ∗ K)(λ)| +C

k Z X

b Mih(2) ≤ C kKk

−∞

j=1

b ∞ + C ktKk∞ ≤ C kKk

∞

k Z X j=1

|K(t)| |(Oj ω)ρ0 (t)| dt ∞

−∞

|t|−1 |(Oj ω)ρ0 (t)| dt

∀λ ∈ R,

as required. We have used (i) (with k = 0) in the second inequality above.



Remark 3.5. The differential operator t−1 Pk+1 (O) will occur frequently in the sequel (Pk+1 is defined in Lemma 3.1). For notational convenience we shall write σk+1 (t, D) instead of t−1 Pk+1 (O). Note that there exist constants cℓ such that σk+1 (t, D) =

k X

cℓ tℓ Dℓ+1 .

ℓ=0

Suppose that c is in R+ , and denote by r the function defined by r(λ) =

1 c2 + λ2

∀λ ∈ C \ {±ic}.

Lemma 3.6. Suppose that N is a positive integer, and let S0 be as in Lemma 3.4 with K = ω rb, and k = N − 1. The following hold:

(i) the norm kσN (·, D)b r k∞ is finite; (ii) if N ≥ 3, then there exists a constant C, independent of R in (0, 1], such that sup (λ2 + 1) |S0 (λ)| ≤ C kσN (·, D)b r k∞ . λ≥0

Proof. Recall that σN (·, D) is a finite linear combination of operators of the form tℓ Dℓ+1 , where ℓ is in {0, . . . , N − 1} (see Remark 3.5). Thus, to prove (i) it suffices to show that sup |tℓ Dℓ+1 rb(t)| < ∞ t∈R

∀ℓ ∈ {0, . . . , N − 1}.

16

G. MAUCERI, S. MEDA AND M. VALLARINO

This is a standard estimate in Fourier analysis. Recall that rb(t) = (1/c) e−c|t| . It is straightforward to check that Db r = −c rb sgn, and that for each k ≥ 1 D2k rb = c2k rb − 2

Hence

c2(k−1−j) D2j δ0

j=0

D2k+1 rb = − c2k+1 rb · sgn − 2

t2k−1 D2k rb(t) = c2k t2k−1 rb(t)

so that

k−1 X

k−1 X

c2(k−1−j) D2j+1 δ0 .

j=0

and t2k D2k+1 rb(t) = −c2k+1 t2k sgn(t) · rb(t),

|tℓ Dℓ+1 rb(t)| = cℓ |t|ℓ e−c|t|

∀t ∈ R,

and the required estimate follows. To prove (ii), observe preliminary that in the case where K = ω rb the function S0 has an expression simpler than (3.5), namely Z ∞ 1 S0 (λ) = ∀λ ∈ R. ω ρ0 (t) PN (O)(ω rb)(t) JN −1/2 (λ t) dt 2π −∞

Indeed, it is straighforward to check that the right hand side of this formula is the limit as ε tends to 0 of S0 (λ; ε), where S0 (λ, ε) is as in (3.7), with ω rb instead of K. Then, on the one hand, Z ∞ |S0 (λ)| ≤ kωk∞ kσN (·, D)(ω rb)k∞ |t| |JN −1/2 (tλ)| dt −∞

≤ kωk∞ kσN (·, D)(ω rb)k∞ λ−2

∀λ ∈ [0, ∞).

On the other hand, the function JN −1/2 is bounded, so that Z ∞ |S0 (λ)| ≤ C kσN (·, D)(ω rb)k∞ ω ρ0 (t) |t| dt ≤ C kσN (·, D)(ω rb)k∞

−∞

∀λ ∈ [0, ∞).

We have used the fact that R ≤ 1 in the last inequality. To conclude the proof of (ii) it suffices to observe that there exists a constant C such that kσN (·, D)(ω rb)k∞ ≤ C kσN (·, D)b r k∞ .

The proof of the lemma is complete.



4. Hardy spaces on Riemannian manifolds Suppose that M is a connected n-dimensional Riemannian manifold of infinite volume with Riemannian measure µ. Denote by −L the Laplace–Beltrami 2 operator on M , by b the and set β =
 bottom of the L (M ) spectrum of L, lim supr→∞ log µ B(o, r) /(2r). By a result of Brooks [Br] b ≤ β 2 . Basic assumptions 4.1. We make the following assumptions on M :

(i) b > 0; (ii) Ric ≥ −κ2 for some positive κ and the injectivity radius is positive.

HARDY SPACES ON NONCOMPACT MANIFOLDS

17

Remark 4.2. It is well known that manifolds with properties (i)-(ii) above satisfy the uniform ball size condition, i.e., 

inf µ B(p, r) : p ∈ M > 0 and sup µ B(p, r) : p ∈ M < ∞. See, for instance, [CMP], where complete references are given.

Note that manifolds satisfying the assumptions above also satisfy the Basic assumptions 2.1. Indeed, every length metric space satisfies the approximate midpoint property (AM), and, by standard comparison theorems [Ch, Thm 3.10], the measure µ is locally doubling. Furthermore, it is known [CMM1, Section 8] that for manifolds with Ricci curvature bounded from below the assumption b > 0 is equivalent to the isoperimetric property (I). Finally, the heat semigroup {Ht } possesses the properties (iv) (a)–(c) of the Basic Assumptions 2.1, as mentioned at the end of Subsection 4.1 below. The aim of this section is to complement the theory developed in Section 2 by proving further properties of the spaces Xσk (M ) and Yσk (M ). One of the main results of this section, Theorem 4.11, states that, for fixed k the spaces {Xσk (M ) : σ ∈ (β 2 −b, ∞)} are isomorphic Banach spaces and they agree as vector spaces. The main step in establishing this theorem is a H 1 (M ) boundedness result for certain spectral multipliers of L (Theorem 4.9 below). The proof of this multiplier result is rather technical and requires some background material and a few preliminary results, some of which are of independent interest. This long section is subdivided into the following subsections: Subsection 4.1, which contains √ estimates for the kernels of certain functions of the operator D (recall that D = L − b) that will be directly used in the proof of Theorem 4.9; Subsection 4.2, where we explain the rˆole of the wave propagator in the decomposition into atoms of the image T a of an H 1 -atom a by an operator T ; Subsection 4.3, where we prove an economical decomposition of H 1 -atoms with “big” support into H 1 -atoms with support in balls in B1 ; Subsection 4.4, where we prove Theorem 4.9; Subsection 4.5, in which we prove Theorem 4.11 and Subsection 4.6 where we characterise X k (M ) as the image of H 1 (M ) under a certain class of operators V k different from U k. 4.1. Background material. The material in Subsection 4.1 is contained in [MMV1, Section 2] and is included here, without proofs, for the reader’s convenience, √ Notation. For notational convenience, we denote by D1 the operator L − b + κ2 (κ is defined in the Basic assumptions 4.1). Suppose that T is an operator bounded on L2 (M ). We denote by kT its Schwartz kernel (with respect to the Riemannian density µ). Now we recall some estimates for kF (tD) , kF (tD1 ) and kdF (tD1 )∗ , when the function F decays sufficiently fast at infinity. Denote by {H2t } the semigroup generated by L. It is well known that for every p in [1, ∞), the operator H2t extends from a map from L2 (M ) ∩ Lp (M ) into L2 (M ) to a contraction on Lp (M ), denoted by Hpt . Furthermore, {H1t } is ultracontractive, i.e., it maps L1 (M ) into L∞ (M ) for every t in R+ . We denote simply by {Ht } the semigroup on L1 (M ) + L∞ (M ) whose restriction to Lp (M ) is {Hpt }. We abuse the notation, and write |||Ht |||p;q < ∞ for the operator norm of the restriction of {Ht } to Lp (M ) as an operator from Lp (M ) to Lq (M ).

18

G. MAUCERI, S. MEDA AND M. VALLARINO

Recall that the heat semigroup Ht satisfies the following ultracontractivity estimate [Gr, Section 7.5] t H ≤ C e−bt t−n/4 (1 + t)n/4−δ/2 (4.1) ∀t ∈ R+ 1;2

for some δ in [0, ∞). Note that if M is a symmetric space of the noncompact type and Ht denotes the semigroup associated to the Killing metric, then δ is equal to the sum of the cardinality of the positive indivisible restricted roots and half the rank of M [CGM, Thm 3.2 (iii)]. Proposition 4.3. Assume that γ is in (n/2 + 1, ∞), and that F is a bounded function on [0, ∞) such that
γ sup | 1 + λ F (λ)| < ∞. λ∈R+

+

Then for every t in R the operators F (tD), F (tD1 ) and dF (tD1 )∗ are bounded from L1 (M ) to L2 (M ). Furthermore, there exists a constant C such that for all t in R+

(i) supy∈M kF (tD) (·, y) 2 ≤ C t−n/2 (1 + t)n/2−δ ;

(ii) supy∈M kF (tD1 ) (·, y) 2 ≤ C t−n/2 (1 + t)n/2−δ ;

(iii) supy∈M d2 kF (tD1 ) (·, y) 2 ≤ C t−n/2−1 (1 + t)n/2+1−δ , where d2 denotes exterior differentiation with respect to the second variable. Proof. See [MMV1, Proposition 2.2].



4.2. A remark on the wave propagator. We shall need to prove that certain operators map H 1 -atoms into H 1 (M ). In particular, we need to show that the image of an atom a has integral 0. Proposition 4.4. Suppose that ν is in [−1/2, ∞), that w is in L1 (R), and that a is a H 1 -atom. Define the operator Wν on L2 (M ) spectrally by Z ∞ w(t) Jν (tD)f dt ∀f ∈ L2 (M ). Wν f = −∞

The following hold: R (i) RM Wν a dµ = 0; (ii) M S0 (D)a dµ = 0 (S0 is defined in (3.5)). The same conclusions hold if we replace the operator D by the operator D1 . Proof. We observe preliminarly that if a is a H 1 -atom, then Z (4.2) cos(tD)a dµ = 0 ∀t ∈ R+ . M

Indeed, cos(tD)a is in L2 (M ), because cos(tD) is bounded on L2 (M ), and is supported in a ball of radius t + rB , where B is any ball that contains the support of a. Therefore, cos(tD)a is in L1 (M ), and Z Z cos(tD)a dµ = lim 1B(cB ,N ) cos(tD)a dµ. N →∞ M M
Now, the last integral is the inner product cos(tD)a, 1B(cB ,N ) in L2 (M ), and
is equal to a, cos(tD)1B(cB ,N ) , because cos(tD) is self adjoint. Observe that √ cos(tD)1B(cB ,N ) is equal to cosh( bt) on B(cB , N − t), because both functions are solutions of the wave equation ∂t2 u + Lu = bu in B(cB , N ) × (0, ∞) and satisfy

HARDY SPACES ON NONCOMPACT MANIFOLDS

19

the same initial conditions u(x, 0) = 1, ∂t u(x, 0) = 0 in B(cB , N ). Hence, they coincide in {(x, t) : d(x, cB ) < N − t}, by standard energy estimates [T2]. If N is so big that B(cB , N − t) contains the support of a, then Z √
a dµ = 0, a, cos(tD)1B(cB ,N ) = cosh( bt) M

and (4.2) follows. A straightforward consequence of (4.2) is that for any ν in (−1/2, ∞) and for every H 1 -atom a Z (4.3) Jν (tD)a dµ = 0 ∀t ∈ R+ . M

Indeed, Jν (tD)a = √

ν+2 π Γ(ν + 1/2)

Z

1

0

(1 − s2 )ν−1/2 cos(stD)a ds,

and the required conclusion follows from Fubini’s Theorem. It is straightforward to check that similar considerations apply to the operator D1 , so that for each ν in [−1/2, ∞) Z Jν (tD1 )a dµ = 0 ∀t ∈ R+ . M

To prove (i) we just observe that Z Z Z ∞ Wν (D)a dµ = dµ w(t) cos(tD)a dt M M −∞ Z ∞ Z = dt w(t) cos(tD)a dµ = 0, −∞

M

where the change of the order of integration is justified by Fubini’s theorem. Next we prove (ii). By (3.5), the function S0 (D)a may be written as the sum of b (b ωρ0 ∗ K)(D)a

and

k X j=1

cj,k

Z

∞

−∞

K(t) Oj ω(ρ0 t) Ok−j Jk+1/2 (tD)a dt,

b is bounded and where K is a compactly supported distribution on R such that K ∞ tK is in L (R). It is a straightforward consequence of (i) that the integral of each summand of the sum above is equal to 0. Thus, to prove that the integral b makes sense of S0 (D)a is 0, it suffices to show that the integral of (b ωρ0 ∗ K)(D)a ε b b and is equal to 0. Since K is bounded, ω K tends pointwise and boundedly to b b tends pointwise and boundedly to ω b as ε tends to 0. Then ω bρ0 ∗ K K bρ0 ∗ (ω ε K) as ε tends to 0 by the Lebesgue dominated convergence theorem. Therefore the b b in the strong operator tends to the operator ω bρ0 ∗ K(D) operator ω bρ0 ∗ (ω ε K)(D) ε b 2 b in L2 (M ) bρ0 ∗ K(D)a topology of L (M ). Consequently ω bρ0 ∗ (ω K)(D)a tends to ω as ε tends to 0. Suppose that the support of a is contained in the ball B. Since the function ω ρ0 (b ωε ∗ K) is in L1 (R), Z ∞   b (D)a = 1 ωε ∗ K)(t) cos(tD)a dt. ω ρ0 (t) (b ω bρ0 ∗ (ω ε K) 2π −∞

20

G. MAUCERI, S. MEDA AND M. VALLARINO

 ωε ∗ K) is contained in [−1, 1], all the functions ω bρ0 ∗ Since the support of ω ρ0 (b  ε b (ω K) (D)a are supported in the ball B(cB , rB + 1) by finite propagation speed, and Z   b (D)a dµ = 0 ω bρ0 ∗ (ω ε K) M

b is also supported in B(cB , rB + 1). Hence by (i). Thus, the function ω bρ0 ∗ K(D)a ε b b bρ0 ∗ K(D)a in L1 (M ) as ε tends to 0, so that ω bρ0 ∗ (ω K)(D)a tends to ω Z Z b b dµ = lim (b ωρ0 ∗ K)(D)a dµ = 0, ω bρ0 ∗ (ω ε K)(D)a ε→0

M

M

as required to conclude the proof of (ii).



Remark 4.5. Note that for every ν in [−1/2, ∞) the function λ 7→ Jν (tλ) is even and of entire of exponential type t, so that kernel kJν (tD) of the operator Jν (tD) is supported in the set {(x, y) ∈ M × M : d(x, y) ≤ t} by the finite propagation speed. A similar remark applies to the kernel of the operator Jν (tD1 ). 4.3. Economical decomposition of atoms. The following lemma produces an economical decomposition of atoms supported in “big” balls as finite linear combination of atoms supported in balls of radius at most 1, and is key to prove Theorem 4.9 below. The idea is “to transport charges along geodesics”. Lemma 4.6. There exists a constant C such that for every H 1 -atom a supported in a ball B of radius rB > 1 kakH 1 (M) ≤ C rB ,

where kakH 1 (M) is the atomic norm in H 1 (M ) associated to the scale 1. Proof. Denote by S a 1/3-discretisation of M , i.e. a set of points in M that is maximal with respect to the property min{d(z, w) : z, w ∈ S, z 6= w} > 1/3,

and d(S, x) ≤ 1/3

∀x ∈ M.

The family {B(z, 1) : z ∈ S} is a covering of M which is uniformly locally finite, by the uniform ball size and the locally doubling properties. By the same token, the set B ∩ S is finite and has at most N points z1 , . . . , zN , with N ≤ C µ(B), where C is a constant which does not depend on B. Denote by Bj the ball with centre zj and radius 1, and by {ψj : j = 1, . . . , N } a partition of unity on B subordinated to the covering {Bj : j = 1, . . . , N }. N Fix j in {1, . . . , N } and denote by zj0 , . . . , zj j points on a minimizing geodesic N

joining zj and cB , with the property that zj0 = zj , zj j = cB , and d(zjh , zjh+1 ) is approximately equal to 1/3. Note that Nj ≤ 4rB . Denote by Bjh the ball B(zjh , 1/12), for j = 1, . . . , N and h = 0, . . . , Nj . Then the balls Bjh are disjoint, N

Bjh ⊂ B(zjh , 1) ∩ B(zjh+1 , 1) and Bj j = B(cB , 1/12). Denote by φhj a nonnegative function in Cc∞ (Bjh ) that has integral 1. By the uniform ball size property we may choose the functions φhj so that there exists a constant A such that kφhj k2 ≤ A for all h and j. Now, denote by a0j the function a ψj . Clearly a=

N X j=1

ψj a =

N X j=1

a0j .

HARDY SPACES ON NONCOMPACT MANIFOLDS

21

Next, define a1j

=

a0j

−

φ0j

Z

M

a0j

ahj

dµ and

=

(φh−2 j

−

φh−1 ) j

Z

a0j dµ,

M

2 ≤ h ≤ Nj + 1.

Then, for every h in {1, . . . , Nj }, the support of ahj is contained in B(zjh−1 , 1), the integral of ahj vanishes and Z kahj k2 ≤ 2A |a0j | dµ M

≤ C ka0j k2 µ(Bj )1/2

≤ C ka0j k2 µ(Bjh )−1/2 .

In the last two inequalities we have used the fact that for each r in R+ the supremum of µ(B) over all balls B of radius r is finite by the uniform ball size property. Hence there exists a constant C, independent of j and h, such that kahj kH 1 (M) ≤ C ka0j k2 .

(4.4) Moreover

Nj +1

a0j =

X

Nj

ahj + φj

h=1

Z

M

a0j dµ.

Thus a=

j +1 N NX X

ahj ,

j=1 h=1

because

P R j

R

a0 dµ = M a dµ M j N Bj j = B(cB , 1/12).

coincide, for and conclude that

N

= 0 and all the functions φj j , j = 1, . . . , Nj Now we use (4.4) and the fact that Nj ≤ C rB ,

kakH 1 (M) ≤ C

j +1 N NX X

j=1 h=1 N X

≤ C rB

j=1

ka0j k2

ka0j k2 .

Then we use Schwarz’s inequality and the fact that N ≤ C µ(B), and obtain that N X

kakH 1 (M) ≤ C rB N 1/2

j=1

1/2

≤ C rB µ(B)

≤ C rB .

ka0j k2

 2 1/2

kak2

The last inequality follows because a is a H 1 -atom supported in the ball B. This completes the proof of the lemma.



22

G. MAUCERI, S. MEDA AND M. VALLARINO

4.4. Operators bounded on H 1 (M ). First we define an appropriate function space of holomorphic functions which will be needed in the statement of Theorem 4.9. Then, for the reader’s convenience, we recall one of its properties, which will be key in the proof of Theorem 4.9. Definition 4.7. Suppose that J is a positive integer and that W is in R+ . Denote by SW the strip {ζ ∈ C : Im(ζ) ∈ (−W, W )} and by H ∞ (SW ; J) the vector space of all bounded even holomorphic functions f in SW for which there exists a positive constant C such that (4.5)

|Dj f (ζ)| ≤ C (1 + |ζ|)−j

∀ζ ∈ SW

∀j ∈ {0, 1, . . . , J}.

We denote by kf kSW ;J the infimum of all constants C for which (4.5) holds. Lemma 4.8 ([HMM, Lemma 5.4]). Suppose that J is an integer ≥ 2, and that W is in R+ . Then there exists a positive constant C such that for every function f in
H ∞ SW ; J , and for every positive integer h ≤ J − 2 |Oh fb(t)| ≤ C kf kSW ;J |t|h−J e−W |t|

∀t ∈ R \ {0}.

The following boundedness result of spectral multipliers on H 1 (M ) is one of the main results of this paper. Taylor [T3] proves a similar result, but with the local space h1 (M ) in place of H 1 (M ) and stronger bounded curvature assumptions on M . We emphasise the fact that the strip in [T3, Proposition 6.1] is larger than that in Theorem 4.9, and that the width of the strip in Theorem 4.9 is best possible as the case of symmetric spaces of the noncompact type shows. Theorem 4.9. Assume that α and β are as in (1.3), and δ as in (4.1). Denote by N
the integer [n/2+1]]+1. Suppose that J is an integer > max N +2+α/2−δ, N +1/2 . Then there exists a constant C such that
|||m(D)|||H 1 (M) ≤ C kmkSβ ;J ∀m ∈ H ∞ Sβ ; J .

Proof. For notational convenience, in this proof we shall write J instead of JN −1/2 . Step I: reduction of the problem. We claim that it suffices to prove that for each H 1 -atom a the function m(D) a may be written as the sum of atoms with supports contained in balls of B1 , with ℓ1 norm of the coefficients controlled by C kmkSβ ;J . Indeed, by arguing as in [MSV, Thm 4.1], we may then show that m(D) extends to a bounded operator from H 1 (M ) to L1 (M ), with norm dominated by C kmkSβ ;J . Note that [MSV, Thm 4.1] is stated for spaces of homogeneous type. However, its proof extendsP to the present setting. Now, suppose that f is a functionP in H 1 (M ) and that f = j λj aj is an atomic decomposition of f with kf kH 1 (M) ≥ j |λj |−ε. P Then m(D)f = j λj m(D)aj , where the series is convergent in L1 (M ), because m(D) extendsP to a bounded operator from H 1 (M ) to L1 (M ). But the partial sums of the series j λj m(D)aj is a Cauchy sequence in H 1 (M ), hence the series is convergent in H 1 (M ), and the sum must be the function m(D)f . Then X |λj | km(D)aj kH 1 (M) km(D)f kH 1 (M) ≤ j

≤ C kmkSβ ;J

X j

|λj |

≤ C kmkSβ ;J (kf kH 1 (M) + ε),

HARDY SPACES ON NONCOMPACT MANIFOLDS

23

and the required conclusion follows by taking the infimum of both sides with respect to all admissible decompositions of f . Step II: splitting of the operator. Let ω be the cut-off function defined in Section 3. Clearly ω b ∗ m and m − ω b ∗ m are bounded functions. Define the operators S and T spectrally by S = (b ω ∗ m)(D)

and

T = (m − ω b ∗ m)(D).

Then m(D) = S + T . We analyse the operators S and T in Step III and Step IV respectively. Suppose that a is a H 1 -atom supported in B(p, R) for some p in M and R ≤ 1.

Step III: analysis of S. In the following, we shall need to estimate the L2 (M ) norm of the differential of the kernel of certain operators related to S. To this end, and to be able to apply Proposition 4.3 (iii), we write the √ operator S as a function of the operator D1 , rather than of D. Recall that D1 = D2 + κ2 . Since ω b ∗ m is an even entire function of exponential type 1, the function S, defined by p
S(ζ) = (b ω ∗ m) ζ 2 − κ2 ∀ζ ∈ C, is well defined, and is of exponential type 1. Hence its Fourier transform has support in [−1, 1]. It is straightforward to check that S = S(D1 ),

and that

kSkMih(J) ≤ C kb ω ∗ mkMih(J) , where the constant C does not depend on m. By arguing much as in the proof of [HMM, Proposition 5.3], we may show that kb ω ∗ mkMih(J) ≤ C kmkMih(J) , where C is independent of m. Clearly kmkMih(J) ≤ kmkSβ ;J

Hence there exists a constant C such that (4.6)

kSkMih(J) ≤ C kmkSβ ;J

∀m ∈ H ∞ (Sβ ; J). ∀m ∈ H ∞ (Sβ ; J).

Define the functions Si as in (3.5) and (3.6), but with N − 1 in place of k and P the Fourier transform of S in place of K. We further decompose S as di=0 Si (D1 ), where d is as in (3.3). The function S0 is bounded by Lemma 3.4 (iii), hence S0 (D1 ) is bounded on L2 (M ) by the spectral theorem, and |||S0 (D1 )|||2 ≤ kS0 k∞ ≤ C kSkMih(2) ≤ C kmkSβ ;J .

Observe that the support of the kernel of the operator Si (D1 ) is contained in {(x, y) : d(x, y) ≤ 4i+1 R} by the finite propagation speed. Thus the support of Si (D1 )a is contained in the ball with centre p and radius (4i+1 + 1)R, which henceforth we denote by Bi . In particular S0 (D1 )a is supported in B0 = B(p, 5R), and kS0 (D1 )ak2 ≤ C |||S0 (D1 )|||2 kak2 ≤ C R−n/2 kmkSβ ;J .

Furthermore, the integral of S0 (D1 )a vanishes by Proposition 4.4 (ii), so that S0 (D1 ) a is a constant multiple of a H 1 -atom. Denote by kSi (D1 ) the integral kernel of the operator Si (D1 ). Observe that Z   Si (D1 ) a(x) = a(y) kSi (D1 ) (x, y) − kSi (D1 ) (x, p) dµ(y). B(p,R)

24

G. MAUCERI, S. MEDA AND M. VALLARINO

By Minkowski’s integral inequality and the fact that the support of Si (D1 ) a is contained in Bi , we have that kSi (D1 ) ak2 = kSi (D1 ) akL2 (Bi ) Z ≤ |a(y)| Ii (y) dµ(y), B(p,R)

where Ii (y) = kkSi (D1 ) (·, y) − kSi (D1 ) (·, p)kL2 (Bi ) ∀y ∈ B(p, R). To estimate Ii (y), we observe that

Ii (y) ≤ d(y, p) sup d2 kSi (D1 ) (·, z) L2 (B ) . i

z∈M

and, by Lemma 3.4 (ii) (with k = N − 1), Z ∞ 1 b d2 kJ (tD ) (·, z) dt. d2 kSi (D1 ) (·, z) = φρi (t) PN (O)S(t) 1 2π −∞

Recall that φρi is supported in Ei = {t ∈ R : 4i−1 R ≤ |t| ≤ 4i+1 R}, that the support of Sb is contained in [−1, 1] and that d(p, y) < R. Then, by Lemma 4.3 (ii) (with J in place of F ), there exists a constant C, independent of i and R, such that Z ∞

b sup d2 kJ (tD1 ) (·, z) L2 (Bi ) dt Ii (y) ≤ C d(y, p) φρi (t) |PN O)S(t)| z∈M −∞ Z b ∞R |t|−n/2−2 dt ≤ C ktPN (O)Sk i

Thus,

Ei −n/2−1

≤ C kmkSβ ;J R (4 R)

.

kSi (D1 ) ak2 ≤ C kmkSβ ;J 4−i (4i R)−n/2 kak1 ≤ C kmkSβ ;J 4−i µ(Bi )−1/2 .

Furthermore the integral of Si (D1 ) a vanishes by Proposition 4.4 (i), so that 4i Si (D1 ) a is a constant multiple of a H 1 -atom. Thus ∞ X 4−i kS akH 1 (M) ≤ C kmkSβ ;J i=0

≤ C kmkSβ ;J .

Step IV: analysis of T . For each j in {1, 2, 3, . . .}, define ωj by the formula

(4.7)

P∞

ωj (t) = ω(t − j) + ω(t + j)

∀t ∈ R.

Observe that j=1 ωj = 1 − ω and that the support of ωj is contained in the set of all t in R such that j − 3/4
≤ |t| ≤ j + 3/4. Since m is in H ∞ Sβ ; J and J ≥ N + 2, the function m b and its derivatives up to the order N are rapidly decreasing at infinity by Lemma 4.8, so that Oℓ (ωj m) b is in L1 (R) ∩ C0 (R+ ) for all ℓ in {0, . . . , N }, and so does PN (O)(ωj m). b In the rest of this proof, we write Ωj,N instead of PN (O)(ωj m). b Observe that the support of Ωj,N is contained in {t ∈ R : j − 3/4 ≤ |t| ≤ j + 3/4}. Define the function Tj : R → C by Z ∞ Ωj,N (t) J (tλ) dt ∀λ ∈ R. (4.8) Tj (λ) = −∞

HARDY SPACES ON NONCOMPACT MANIFOLDS

25

P∞ We may use the observation that (m − ω b ∗ m)b = j=1 ωj m b and formula (3.2), and write Z ∞
1 1 − ω(t) m(t) b cos(tλ) dt (m − ω b ∗ m)(λ) = 2π −∞ ∞ X Tj (λ). = j=1

Then, by the spectral theorem, Ta=

∞ X

Tj (D)a.

j=1

By the asymptotics of JN −1/2 [L, formula (5.11.6), p. 122] sup |(1 + s)N J (s)| < ∞. s>0

Since N − 1/2 > (n + 1)/2, we may apply Proposition 4.3 (i) and conclude that kJ (tD)ak2 ≤ kak1 J (tD) 1;2

≤ sup kJ (tD) (·, y) 2 y∈M


n/2−δ ≤ C |t|−n/2 1 + |t|

∀t ∈ R \ {0}.

Then J (tD)a is supported in B(p, t + R), and has integral 0 by Proposition 4.4 (i). Observe that Z ∞ kTj (D)ak2 ≤ C |Ωj,N (t)| kJ (tD)ak2 dt −∞ j+3/4

(4.9)

≤C

Z

j−3/4


n/2−δ dt |Ωj,N (t)| |t|−n/2 1 + |t|

≤ C kmkSβ ;J j N −J−δ e−β j

∀j ∈ {1, 2, . . .}.

In the last inequality we have used Lemma 4.8 and Proposition 4.3 (i). Note that j δ+J−N −α/2 Tj (D)a is a constant multiple of a H 1 -atom. Indeed, Tj (D)a is
a function in L2 (M ) with support contained in B p, j + 1 , and has integral 0 by Proposition 4.4 (i). Moreover kj δ+J−N −α/2 Tj (D)ak2 ≤ C kmkSβ ;J j −α/2 e−β j Hence we may write


−1/2 ≤ C kmkSβ ;J µ B(p, j + 1) Ta=

where

a′j

∞ X

λj a′j ,

j=1


is a H -atom supported in B p, j + 1 , and 1

λj = C kmkSβ ;J j N +α/2−J−δ .

∀j ∈ {1, 2, . . .}.

26

G. MAUCERI, S. MEDA AND M. VALLARINO

By Lemma 4.6 we have ka′j kH 1 (M) ≤ C j, so that kT ak

H 1 (M)

≤

∞ X j=1

|λj | ka′j kH 1 (M)

≤ C kmkSβ ;J

∞ X

j 1+N +α/2−J−δ ,

j=1

which is finite (and independent of a) because J > 2 + N + α/2 − δ. Step V: conclusion. By Step III and Step IV there exists a constant C such that for every H 1 -atom a with support contained in a ball of radius at most 1 kSakH 1 (M) + kT akH 1 (M) ≤ C kmkSβ ;J .

Then Step II implies that

km(D)akH 1 (M) ≤ C kmkSβ ;J .

The required conclusion follows from Step I.



A straightforward consequence of Theorem 4.9 is the following. Corollary 4.10. Denote by N the integer [n/2 + 1]] + 1. Suppose that J is an
integer > max N + 2 + α/2 − δ, N + 1/2 . Then there exists a constant C such that
|||m(D)|||H 1 (M);L1 (M) ≤ C kmkSβ ;J ∀m ∈ H ∞ Sβ ; J .

This corollary should be compared with [CMM1, Thm 10.2]. On the one hand the geometric assumptions of Corollary 4.10 are much less stringent than those of [CMM1, Thm 10.2]. On the other hand in [CMM1, Thm 9.2] we require the control of smaller number of derivatives of m. 4.5. Further properties of Xσk (M ). An important consequence of the previous result is that, for fixed k, the spaces Xσk (M ) do not depend on the parameter σ, as σ varies in the range (β 2 − b, ∞).

Theorem 4.11. The following hold: (i) if σ1 and σ2 are in (β 2 − b, ∞), then Xσk1 (M ) and Xσk2 (M ) agree as vector spaces, and their norms are equivalent; (ii) if σ is in (β 2 − b, ∞), then H 1 (M ) ⊃ Xσ1 (M ) ⊃ Xσ2 (M ) ⊃ · · · with continuous inclusions; (iii) the inclusions in (ii) are proper. Proof. First we prove (i). Consider the operator Tσ1 ,σ2 , defined on L2 (M ) by Uσ2 . Tσ1 ,σ2 = Uσ−1 1

. Since both Uσ1 and Uσ2 are isomorphisms on L2 (M ), so are Tσ1 ,σ2 and Tσ−1 1 ,σ2 1 are bounded on H (M ). Indeed, Observe that the operators Tσ1 ,σ2 and Tσ−1 1 ,σ2 Tσ1 ,σ2 = (σ1 I + L) (σ2 I + L)−1 = (σ1 − σ2 ) (σ2 I + L)−1 + I.

Hence the boundedness of Tσ1 ,σ2 on H 1 (M ) is equivalent to that of (σ2 I +L)−1 . To prove that (σ2 I+L)−1 is bounded on H 1 (M ), it suffices to check that the associated spectral multiplier ζ 7→ (σ + b + ζ 2 )−1 satisfies the hypotheses of Theorem 4.9. We is bounded omit the details of this calculation. A similar argument shows that Tσ−1 1 ,σ2 on H 1 (M ).

HARDY SPACES ON NONCOMPACT MANIFOLDS

27

= I, the identity Thus, Tσ1 ,σ2 is an isomorphism of H 1 (M ). Since Uσ1 Tσ1 ,σ2 Uσ−1 2 is an isomorphism between Xσ11 (M ) and Xσ12 (M ), as required to conclude the proof of (i) in the case where k = 1. The proof in the case where k ≥ 2 is similar, and is omitted. Note that (i) is equivalent to the boundedness of Uσ on H 1 (M ). Since Uσ = I − σ (σI + L)−1 , it suffices to prove that the resolvent operator (σI + L)−1 is bounded on H 1 (M ). This has already been done in the proof of (i), and (ii) follows. Finally we prove (i). Choose a function ψ in Cc∞ (M ) with nonvanishing integral. Observe that Lψ is a multiple of a H 1 -atom, hence it is in H 1 (M ). We shall prove that Lk+1 ψ is in Xσk (M ) \ Xσk+1 (M ). Indeed, on the one hand

Uσ−k Lk+1 ψ = (I + β 2 L)k Lψ ,

which again is a multiple of an H 1 -atom, hence is in H 1 (M ). On the other hand
Uσ−(k+1) Lk+1 ψ = (I + β 2 L)k+1 (ψ),

which may be written as a linear combination of ψ and of terms of the form Lj ψ
−(k+1) with j in {1, . . . , k + 1}. Therefore the integral of Uσ Lk+1 ψ does not vanish, hence it is not in H 1 (M ) and Lk+1 ψ is not in Xσk+1 (M ), as required.  Definition 4.12. Suppose that k is a positive integer. The space Xβk2 (M ) will be denoted simply by X k (M ). Remark 4.13. By Theorem 4.11, for any σ in (β 2 − b, ∞) and each positive integer k we have that X k (M ) = Xσk (M ) as vector spaces, and their norms are equivalent. 4.6. Equivalent norms on X k (M ). In this subsection, we prove that X k (M ) may be characterised as the image of H 1 (M ) under a wider class of maps. To prove the main result of this subsection, Proposition 4.15 below, we need the following lemma, which is certainly well known. For the sake of completeness, and for lack of reference, we give a sketch of its proof. √ Lemma 4.14. Suppose that W is in ( b, ∞). Then there exists√ a function η in Cc (R) such that the only zeroes of 1 − ηb in SW are the points ±i b. √ b Proof. Suppose that ψ is a even nonnegative function in Cc∞ (R) such that ψ(i b) = 1. We claim that there exists √ a positive ε such that the only zeroes of 1 − ψb in the strip S√b+ε are the points ±i b. To prove the claim observe that 1 − ψb does not vanish in S√b . Indeed, ψb is even, because ψ is even by assumption, so that for any y in R Z ∞  1 b b b ψ(iy) = ψ(iy) + ψ(−iy) = ψ(t) cosh(y t) dt. 2 −∞ √ √ Thus, for x in R and y in (− b, b) Z ∞ b |ψ(x + iy)| ≤ ψ(t) ey t dt −∞ Z ∞ √ < ψ(t) cosh ( b t) dt, −∞

√ b which is equal to 1, because ψ(i b) = 1 by assumption, so that 1 − ψb does not √ vanish in S b .

28

G. MAUCERI, S. MEDA AND M. VALLARINO

√ √ Next we prove that the point i b is the only zero of 1− ψb√on the line Im(ζ) = b. b + i b) = 1. Then Indeed, suppose that x is a real number such that ψ(x Z Z ∞ ∞ √ √ √ b + i b) = ψ(t) e bt cos(xt) dt + i ψ(t) e bt sin(xt) dt. 1 = ψ(x

Therefore

R∞

−∞

√

ψ(t) e

√

−∞

bt

−∞

sin(xt) dt = 0, and Z ∞ √   ψ(t) e bt 1 − cos(xt) dt = 0, −∞

b b) = 1. Since ψ is positive in a set of positive measure, this is clearly because ψ(i impossible unless x = 0, as required. Finally, suppose that ε′ > 0, and consider the strip S√b+ε′ . Since ψ is a smooth function with compact support, ψb is a Schwartz function, so that it decays rapidly at infinity, and the decay is uniform in the strip S√b+ε′ . Hence there is a positive constant C such the function 1 − ψb does not vanish in the set {ζ ∈ C : |Re ζ| > √ that ′ C, |Im ζ| ≤ b + ε }. Since 1 − ψb is entire and not identically zero, it has at most √ ′ finitely many zeroes in {ζ ∈ C : |Re ζ| ≤ C, |Im ζ| ≤ b + ε }. Denote by ζ1 , . . . , ζN √ √ the zeroes of 1 − ψb in {ζ ∈ C : |Re ζ| ≤ C, b < |Im ζ| ≤ b + ε′ }, and by ε the minimum of the distances of these points from S√b . Clearly ε is positive. Then the √ only zeroes of 1 − ψb in the strip S√b+ε are the points ±i b, as required to prove the claim. Next observe that the proof of the claim shows that for each ε′ in R+ , the function 1 − ψb has are at most finitely many zeroes repeated √ according their multiplicities, ζ1 , . . .√ , ζN say, in {ζ ∈ C : |Re ζ| ≤ C, |Im ζ| ≤ b + ε′ }. Now, choose ε′ so that W = b + ε′ . Denote by η the tempered distribution on the real line such that   (ζ 2 + 2W 2 )N/2 b ηb(ζ) = 1 − 1 − ψ(ζ) QN j=1 (ζ − ζj )

∀ζ ∈ C \ {ζ1 , . . . , ζN }.

Observe that N is a multiple of 4, by the symmetries of ψ. It is straightforward to check that η possesses the following properties: (i) η is a distribution with compact support, because ηb is an entire function of exponential type, and, in fact, it is √ a continuous function; (ii) the function 1 − ηb vanishes at ±i b, and these are the only points in SW where 1 − ηb vanishes. This concludes the proof of the lemma.  Proposition 4.15. Suppose that k is a positive integer. Assume that η is a function which possesses the properties of the function η in Lemma 4.14 (with W = β + ε for some positive ε). Denote by Vη the operator I − ηb(D). The following hold: (i) the map Vηk is injective on L1 (M ); (ii) Vηk H 1 (M ) = X k (M ) as vector spaces, and the norm on X k (M ), defined by kf kη,k = kVη−k f kH 1 (M)

is equivalent to the norm of X k (M ).

∀f ∈ X k (M ),

Proof. First we prove (i). It suffices to prove the result in the case k = 1, because the case k ≥ 2 follows from this by induction. Suppose that f is in L1 (M ) and that

HARDY SPACES ON NONCOMPACT MANIFOLDS

29


Vη f = 0. Then Ht Vη f = 0 for all t in R+ . Since Ht and Vη commute, we have
that Vη Ht f = 0. Since Ht is ultracontractive, Ht f is in L2 (M ). Observe that I − ηb(D) is an isomorphism of L2 (M ), because 1 − ηb is bounded and bounded
away from 0 on the real line. In particular, I − ηb(D) is injective, so that Vη Ht f = 0 implies Ht f = 0. Since Ht is a strongly continuous semigroup on L1 (M ), i.e., Ht f tends to f as t tends to 0, we may conclude that f = 0, as required. Next we prove (ii). We observe preliminarily that Uβ 2 and Vη are bounded on 1 H (M ) by Theorem 4.9. Recall that both Uβ 2 and Vη are isomorphisms of L2 (M ). 1 2 Hence the map Uβ−1 2 Vη is well defined on H (M ) ∩ L (M ). Observe that Z ∞ 1 − ηb(λ) 2 (β + λ2 + b) dPλ′ , Uβ−1 2 Vη f = 2 −∞ λ + b where {Pλ′ } is the spectral
resolution of D. It is straightforward to check that the function λ 7→ 1 − ηb(λ) (λ2 + b)−1 (β 2 + λ2 + b) satisfies the hypotheses of 1 Theorem 4.9. Hence Uβ−1 Since 2 Vη extends to a bounded operator on H (M ). 1 − ηb does not vanish in SW , also Vη−1 Uβ 2 extends to a bounded operator on H 1 (M ), by a similar argument. The conclusion follows by arguing as in the proof of Theorem 4.11 (ii).  5. Spectral multipliers and Riesz transforms on Riemannian manifolds In this section we apply the results of Section 4 to spectral multipliers and the first order Riesz transform associated to the Laplace–Beltrami operator on complete connected Riemannian manifolds M satisfying the Basic assumptions 4.1. We recall that in Definition 4.7 we introduced the space H ∞ (SW ; J) of functions that are holomorphic and bounded, together with their derivatives up to the order J, in the strip SW , and satisfy a Mihlin-type condition at infinity. Here, to deal with a wider class of operators, we define a larger space of functions that may be singular also at the points ±iW . Definition 5.1. Suppose that J is a positive integer, that τ is in [0, ∞), and that W is in R+ . The space H(SW ; J, τ ) is the vector space of all holomorphic even functions f in the strip SW for which there exists a positive constant C such that
(5.1) |Dj f (ζ)| ≤ C max |ζ 2 + W 2 |−τ −j , |ζ|−j ∀ζ ∈ SW ∀j ∈ {0, 1, . . . , J}.

We denote by kf kSW ;J,τ the infimum of all constants C for which (5.1) holds.

Note that, for each fixed j, the right-hand side of (5.1) is infinite of order −τ − j at ±iW , and vanishes of order j at infinity. Thus, if τ = 0, and f is in H(SW ; J, τ ), then f satisfies Mihlin-type conditions both near the points ±iW and at infinity. In particular, the derivatives of f may be unbounded in any neighbourhood of iW , and of −iW . Finally, if τ is in R+ , and f is in H(SW ; J, τ ), then both f and its derivatives up to the order J may be unbounded in any neighbourhood of iW , and of −iW . Remark 5.2. An interesting example of a function in H(Sβ ; J, τ ) is f (ζ) = (ζ 2 + β 2 )−iu−τ (ζ 2 + β 2 + 1)τ , where τ is in [0, ∞). Note that if b = β 2 , then f (D) = L−iu−τ (L + 1)τ . It is worth observing that there are no endpoint results at p = 1 for this operator in the

30

G. MAUCERI, S. MEDA AND M. VALLARINO

literature when τ > 1. In the case where M is a symmetric space of the noncompact type, it is known [A1, AJ, MV] that f (D) is of weak type 1 if and only if τ ≤ 1, but the proof of this fact uses the spherical Fourier transform and very specific information on the structure of the symmetric space, and it is hardly extendable. Theorem 5.3. Assume that α and β are as in (1.3), and δ as in (4.1). Suppose that τ is in [0, ∞), that J and k are integers, with k > τ + J. The following hold: (i) if b < β 2 and J > max(α + 1, n/2 + 1), then there exists a constant C such that |||m(D)|||H 1 (M);L1 (M) ≤ C kmkSβ;J,τ ∀m ∈ H(Sβ ; J, τ ) and

|||m(D)t |||L∞ (M);BMO(M) ≤ C kmkSβ;J,τ t

∀m ∈ H(Sβ ; J, τ ),

where m(D) denotes the transpose operator of m(D);
(ii) if b = β 2 and J > max N + 2 + α/2 − δ, N + 1/2 , where N denotes the integer [n/2 + 1]] + 1, then there exists a constant C such that |||m(D)|||X k (M);H 1 (M) ≤ C kmkSβ;J,τ

∀m ∈ H(Sβ ; J, τ )

and |||m(D)t |||BMO(M);Y k (M) ≤ C kmkSβ;J,τ t

∀m ∈ H(Sβ ; J, τ ),

where m(D) denotes the transpose operator of m(D). e defined by Proof. First we prove (i). Consider the map U,   Ue = L + (β 2 − b)I (β 2 I + L)−1 .

Observe that Ue = I − b (β 2 I + L)−1 extends to a bounded operator on L1 (M ), because the L1 (M )-spectrum of L is contained in the right half-plane. Similarly, the operator I + b [(β 2 − b)I + L]−1 extends to a bounded operator on L1 (M ); it is straightforward to check that this operator is the inverse of Ue on L1 (M ). Thus, Ue is an isomorphism of L1 (M ), and so is Uek . Consequently, m(D) is bounded from H 1 (M ) to L1 (M ) if and only if Uek m(D) is bounded from H 1 (M ) to L1 (M ). Observe that Uek m(D) = uk (D), where  ζ 2 + β 2 k m(ζ). uk (ζ) = 2 ζ + b + β2 It is straightforward to check that there exists a constant C such that
−j ∀ζ ∈ Sβ ∀j ∈ {0, 1, . . . , J}. |Dj uk (ζ)| ≤ C kmkSβ ;J,τ 1 + |ζ|

Here we use the fact that k > τ + J. Thus, uk (D) is bounded from H 1 (M ) to L1 (M ) by [CMM1, Thm 10.2], as required to prove the first estimate. The second follows from the first by a duality argument. Next we prove (ii). Observe that m(D) = m(D) Uβk2 Uβ−k Since Uβ−k is an 2 . 2 k 1 isometric isomorphism between X (M ) and H (M ), to prove that m(D) is bounded from X k (M ) to H 1 (M ) it suffices to show that the operator m(D) Uβk2 extends to a bounded operator on H 1 (M ). Note that m(D) Uβk2 = vk (D), where  ζ 2 + b k m(ζ). vk (ζ) = 2 ζ + b + β2

HARDY SPACES ON NONCOMPACT MANIFOLDS

31

It is straightforward to check that there exists a constant C such that
−j ∀ζ ∈ Sβ ∀j ∈ {0, 1, . . . , J}. |Dj vk (ζ)| ≤ C kmkSβ ;J,τ 1 + |ζ|

Here we use the fact that k > τ + J. Thus, vk (D) is bounded from H 1 (M ) to H 1 (M ) by Theorem 4.9, as required to prove the first estimate. The second follows from the first by a duality argument. The proof of the theorem is complete.  Corollary 5.4. Suppose that M is a symmetric space of the noncompact type and that −L is the Laplace–Beltrami operator with respect to the Killing metric. If k > n/2 + 3, then Liu is bounded from X k (M ) to H 1 (M ).

Proof. Indeed, it is well known that α = (r − 1)/2, where r is the rank of the symmetric space, and δ = υ + r/2, where υ denotes the cardinality of the indivisible positive restricted roots. Notice that 3/2 + α/2 − δ ≤ 0, so that the hypotheses of Theorem 5.3 are satisfied whenever J > n/2 + 2 and k > J, and the required conclusion follows.  We conclude this section with the following endpoint result for the first order Riesz transform. Our method hinges on the fact that if b = β 2 and k is large enough, then the operator Lk (β 2 I + L)−k is bounded on H 1 (M ) by Theorem 4.9. Theorem 5.5. Assume that α and β are as in (1.3), and δ as in (4.1).
Suppose that b = β 2 and that k is an integer > max N + 2 + α/2 − δ, N + 1/2 , where N denotes the integer [n/2 + 1]] + 1. Then the first order Riesz transform ∇L−1/2 is bounded from X k (M ) to L1 (M ). Proof. Since Lk (β 2 I + L)−k is an isometry between H 1 (M ) and X k (M ), it suffices to prove that ∇Lk−1/2 (β 2 I + L)−k is bounded from H 1 (M ) to L1 (M ). Observe that ∇Lk−1/2 (β 2 I + L)−k = ∇(β 2 I + L)−1/2 Lk−1/2 (β 2 I + L)1/2−k .

The right hand side is the composition of the operators Lk−1/2 (β 2 I + L)1/2−k , which is bounded on H 1 (M ) by Theorem 4.9 and of the translated Riesz transform ∇(β 2 I + L)−1/2 , which is bounded from H 1 (M ) to L1 (M ) by [Ru]. The required result follows.  6. Atomic decomposition of X k (M ) For each ball B in M we denote by L20 (B) the subspace of L2 (M ) consisting of all R L2 (M ) functions f with support contained in the ball B, and satisfying B f dµ = 0. Atoms in X k (M ) will be L2 (M ) functions supported in a ball B that satisfy a size condition analogous to that for H 1 -atoms and certain cancellation conditions, which will be expressed as orthogonality to suitable subspaces of L2 (M ). Definition 6.1. Suppose that k is a positive integer, and that B is in B. Denote by QkB the space of all functions V in L2 (M ) such that Lk V is constant (in the sense of distributions) in a neighbourhood of B. Functions in QkB are sometimes called “k quasi-harmonic” on B. Remark 6.2. Observe that the following are equivalent: (i) V is in QkB ;

32

G. MAUCERI, S. MEDA AND M. VALLARINO

(ii) V is in L2 (M ) and is smooth in a neighbourhood of B, and Lk V is constant therein. Indeed, if V is in QkB , then V is in L2 (M ) by the definition of the space QkB , and k L V is a constant in the sense of distributions in a neighbourhood of B. Hence V is smooth on that neighbourhood by elliptic regularity. The converse is obvious. Proposition 6.3. Suppose that k is a positive integer, and that B is a ball in M . The following hold:  (i) (QkB )⊥ = F
∈ L2 (M ) : L−k F ∈ L20 (B) ; (ii) L−k (QkB )⊥ is contained in L20 (B) ∩ Dom(Lk ). Furthermore, functions in (QkB )⊥ have
support contained in B; (iii) Uβ−k (QkB )⊥ is contained in L20 (B). 2

Proof. We prove (i). First we show that (QkB )⊥ is contained in  2 −k 2 F ∈ L (M ) : L F ∈ L0 (B) . Suppose that F is in (QkB )⊥ . To show that the support of L−k F is contained in B it suffices to prove that (L−k F, 1B ′ ) = 0 for every ball B ′ contained in (B)c . Since L is self adjoint, (L−k F, 1B ′ ) = (F, L−k 1B ′ ).

Notice that L−k 1B ′ is in QkB , hence the last inner product vanishes, as required. Next we prove that the integral of L−k F is 0. Since the support of L−k F is contained in B and L is self adjoint, Z L−k F dµ = (L−k F, 12B ) = (F, L−k 12B ). M

Now, the last inner product vanishes, because F is in (QkB )⊥ by assumption and L−k 12B is in QkB , as required.  Next we prove that F ∈ L2 (M ) : L−k F ∈ L20 (B) is contained in (QkB )⊥ . Suppose that L−k F is in L20 (B). Observe that F is in Dom(Lk ) and that F = Lk L−k F . Suppose now that V is in QkB . Then V is smooth in a neighbourhood of B by Remark 6.2, and (F, V ) = (Lk L−k F, V ) = (L−k F, Lk V ) = 0.

The last equality follows from the facts that Lk V is constant in a neighbourhood of B, and that L−k F is in L20 (B), so that its integral on B vanishes. Next we prove (ii). Clearly if F is in L2 (M ), then L−k F is in Dom(Lk ) by abstract set theory. Moreover L−k F is in L20 (B) by (i), and the first statement of (ii) follows. To prove the second statement of (ii), observe that the support of L−k F is contained in B, hence so is the support of Lk L−k F , i.e., of F .
2 k −k Finally, we prove (iii). Observe that Uβ−k . Since L−k (QkB )⊥ 2 = (I + β L) L 2 k is contained ) by (ii), it suffices to show that Lj L20 (B) ∩
in L0 (B) ∩ Dom(L k 2 Dom(L ) is contained in L0 (B) for all j in {0, 1, . . . , k}. Suppose that F is in L20 (B) ∩ Dom(Lk ). Denote by φ a function in Cc∞ (M ) such that φ = 1 on B. Since L is self adjoint and the support of F is contained in B, Z Lj F dµ = (Lj F, φ) = (F, Lj φ) = 0, M

HARDY SPACES ON NONCOMPACT MANIFOLDS

as required to conclude the proof of (iii) and of the proposition.

33



Definition 6.4. Suppose that k is a positive integer. A X k -atom is a function A in L2 (M ), supported in a ball B, such that (i) A is in (QkB )⊥ ; (ii) kAk2 ≤ µ(B)−1/2 . R Note that condition (i) implies that M A dµ = 0, because 12B is in QkB .

Remark 6.5. Note that if A is a X k -atom supported in B, then L−k A/|||L−k |||2 is a H 1 -atom with support contained in B. Indeed A is in (QkB )⊥ , so that L−k A is in L20 (B) by Proposition 6.3 (iii). Moreover kL−k Ak2 ≤ |||L−k |||2 kAk2

≤ |||L−k |||2 µ(B)−1/2 ,

so that L−k A/|||L−k |||2 is a H 1 -atom supported in B, as required. Note also that a X k -atom A is in X k (M ) and kAkX k (M) ≤ |||Uβ−k 2 |||2 .

(6.1)

2 Indeed, the function Uβ−k 2 A is in L0 (B) by Proposition 6.3 (iii) and −k kUβ−k 2 Ak2 ≤ |||Uβ 2 |||2 kAk2

−1/2 ≤ |||Uβ−k . 2 |||2 µ(B)

−k 1 Therefore Uβ−k 2 A/|||Uβ 2 |||2 is an H -atom, and the required estimate follows from the definition of X k (M ). k Definition 6.6. Suppose that k is a positive integer. The space Xat (M ) isP the space of all functions F in H 1 (M ) that admit a decomposition of the form F = j λj Aj , where {λj } is a sequence in ℓ1 and {Aj } is a sequence of X k -atoms supported in balls Bj in B1 . Atoms supported in balls in B1 will be called admissible. We endow k Xat (M ) with the norm X X kF kXat k (M) = inf |λj | : F = λj Aj , Aj admissible X k -atoms . j

j

We shall need the following variant of Lemma 4.6. Lemma 6.7. Suppose that k is a positive integer and that the covariant derivatives ∇j Ric of the Ricci tensor are uniformly bounded on M for all j in {0, 1, . . . , 2k−2}. k If a is a H 1 -atom in Dom(Lk ), then Lk a is in Xat (M ). Furthermore, if the support of a is contained in the ball B, then there exists a constant C such that 1/2 kLk akXat k (M) ≤ C (1 + rB ) µ(B) kLk ak2 .

Proof. Suppose first that the support of a is contained in a ball B such that rB ≤ 1. Since Lk a is in (QkB )⊥ by Proposition 7.3, µ(B)−1/2 Lh a/kLk ak2 is a X k -atom supported in a ball in B1 and the lemma is proved. Next, suppose that rB > 1. We retain the notation introduced in the proof of Lemma 4.6. In particular, the functions ψj and φhj are smooth cut-offs, supported in balls Bj and Bjh , respectively. The existence of a uniform bound on the derivatives of the Ricci tensor implies that we can choose the functions ψj and φhj so that their

34

G. MAUCERI, S. MEDA AND M. VALLARINO

covariant derivatives of order up to 2k are uniformly bounded for all j and h (see [He, p. 14]). We recall that a=

j +1 N NX X

ahj ,

j=1 h=1

where

a0j

= aψj , Z a1j = a0j − φ0j a0j dµ, M

and ahj = (φh−2 − φh−1 ) j j

Thus a1j is in Dom(Lk ), and kLk a1j k2 ≤ kLk a0j k2 + kLk φ0j k2

Z

M

Z

M

a0j dµ,

2 ≤ h ≤ Nj + 1.

|a0j | dµ

≤ C kLk a0j k2 , where, in the last inequality, we have used the estimate ka0j k2 ≤ C kLk a0j k2 , which holds because L has spectral gap. Similarly, if h = 2, . . . , Nj + 1, then ahj is in Dom(Lk ), and kLk ahj k2 ≤ C kLk a0j k2 .

Hence Lk ahj /kLk a0j k2 is a multiple of a X k -atom supported in a ball of radius 1, with a constant C which does not depend on j and h by the uniform ball size property. Thus k 0 kLk ahj kXat k (M) ≤ C kL aj k2

(6.2)

∀j, h.

Adding up the inequalities in (6.2), we obtain kLk akXat k (M) ≤

j +1 N NX X

j=1 h=1

≤C

kLk ahj kXat k (M)

j +1 N NX X

j=1 h=1

kLk a0j k2 .

Remembering from Lemma 4.6 that Nj ≤ C rB and N ≤ C µ(B), and using Schwarz’s inequality, we see that the right-hand side is dominated by C rB

N X j=1

kLk a0j k2

≤ C rB N

N X

1/2

j=1

1/2

≤ C rB µ(B)

kLk a0j k2

 2 1/2

kLk ak2 .

In the last inequality we have used the fact that {ψj } is a partition of unity on B, subordinated to the uniformly locally finite covering {Bj }. This completes the proof of the lemma.  Remark 6.8. There exists a constant C such that 1/2 kf kXat k (M) ≤ C (1 + rB ) µ(B) kf k2

∀f ∈ (QkB )⊥ .

Indeed, if f is in (QkB )⊥ , then the function L−k f is a multiple of a H 1 -atom, by Proposition 6.3. The conclusion follows, by Lemma 6.7.

HARDY SPACES ON NONCOMPACT MANIFOLDS

35

k−1 Lemma 6.9. Suppose that k is a positive integer and that A is an p admissible X k 2 2 atom. Then U4β 2 +κ2 A is in Xat (M ), and for each η in (2β, 4β + κ ) and all sufficiently large integers J and N there exists a constant C, independent of A, such that   kU4β 2 +κ2 AkXat k (M) ≤ C kσN (·, D)b r k∞ + krkSη ;J ,

where r(λ) = (4β 2 + b + λ2 )−1 .

Proof. Suppose that the atom A is supported in the ball B(p, R). Then R ≤ 1, because A is admissible. Denote by N an integer > n/2 + 3. For notational convenience, in this proof we shall write J instead of JN −1/2 , R instead of R4β 2 +κ2 , p U instead of U4β 2 +κ2 and c instead of 4β 2 + b. Observe that R = r(D1 ). Fix η in (2β, c). Step I: splitting of the operator. As in Section 3, we denote by ω an even function in Cc∞ (R) which is supported in [−3/4, 3/4], is equal to 1 in [−1/4, 1/4], and satisfies X ω(t − j) = 1 ∀t ∈ R. j∈Z

Define the operators S and T spectrally by (6.3)

S = (b ω ∗ r)(D1 )

and

T = (r − ω b ∗ r)(D1 ).

Then UA = LRA = LSA + LT A. We shall prove that both LSA and LT A are in k Xat (M ) and that there exists a constant C, independent of A, such that (6.4)

kLSAkXat r k∞ k (M) ≤ C kσN (·, D)b

and

kLT AkXat k (M) ≤ C krkSη ;J .

The proof of estimates (6.4) will be given in Steps II and III. Step II: proof of the first inequality in (6.4). Note that ω rb has support in [−3/4, 3/4]. Define the functions Si as in (3.5) and (3.6), with k = N − 1, K = ω rb and ρi = 1/(4i R). Observe that, by Lemma 3.6 (iv), (6.5)

S=

d X

Si (D1 ),

i=0

where d = [log4 (3/R) + 1]]. Denote by Bi the ball with centre p and radius (4i+1 + 1)R. Since the support of the kernel of the operator Si (D1 ) is contained in {(x, y) : d(x, y) ≤ 4i+1 R} by the finite propagation speed, the function Si (D1 )A is supported in Bi .
k ⊥ . By Proposition 6.3 it suffices to Now we check that LSi (D1 )A is in HB i −k 2 show that L LSi (D1 )A is in L0 (Bi ). Now, L−k LSi (D1 )A = Si (D1 )L1−k A and L1−k A is a constant multiple of a H 1 -atom with support contained in B(p, R) by Remark 6.5. Thus the support of Si (D1 )L1−k A is contained in Bi and its integral over M vanishes by Proposition 4.4 (ii). Next, we claim that there exists a constant C, independent of A, such that for i in {0, . . . , d} (6.6)

kSi (D1 )Ak2 ≤ C kσN (·, D)b r k∞ µ(Bi )−1/2 4−i

and (6.7)

kLSi (D1 )Ak2 ≤ C kσN (·, D)b r k∞ µ(Bi )−1/2 4−i .

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G. MAUCERI, S. MEDA AND M. VALLARINO

Deferring momentarily the proof of the claim, we show that the first inequality in (6.4) follows from it. Indeed, by (6.5) and the triangle inequality, kLSAkXat k (M) ≤ C

d X i=0

kLSi (D1 )AkXat k (M) .

Now Remark 6.8 and (6.7) imply that 1/2 kLSi (D1 )AkXat kLSi (D1 )Ak2 k (M) ≤ C µ(Bi )

≤ C kσN (·, D)b r k∞ 4−i .

Hence kLSAkXat k (M) ≤ C kσN (·, D)b r k∞ , as required to prove the first inequality in (6.4). To conclude the proof of Step II it remains to prove (6.6) and (6.7). The proof of (6.6) is similar to the proof of the corresponding estimates kSi (D1 )ak2 ≤ C 4−i µ(Bi )−1/2 in the proof of Theorem 4.9, and is omitted. Now we prove (6.7). Recall that L = D2 + b I = D12 + (b − κ2 ) I. Therefore kLSi (D1 )Ak2 ≤ kD12 Si (D1 )Ak2 + |b − κ2 | kSi (D1 )Ak2 .

(6.8)

We first estimate kD12 Si (D1 )Ak2 when i is in {1, . . . , d}. Observe that Z 2N −1 ∞ φρi (t) 2 PN (O)(ω rb)(t) F (tD1 ) dt, D1 Si (D1 ) = √ 2π −∞ t2

where F (λ) = λ2 J (λ). Since the function λ 7→ F (tλ) is an even entire function of exponential type |t| and the support of φρi is contained in the set t ∈ R : 4i−1 R ≤ |t| ≤ 4i+1 R , the support of F (tD1 ) A is contained in Bi , by finite propagation speed. Thus Z   A(y) kF (tD1 ) (x, y) − kF (tD1 ) (x, p) dµ(y), F (tD1 ) A(x) = B(p,R)

and, by Minkowski’s integral inequality, kF (tD1 ) AkL2 (M) = kF (tD1 ) AkL2 (Bi ) Z |A(y)| Ii (y) dµ(y) ≤ B(p,R)

≤ kAk1

sup

Ii (y),

y∈B(p,R)

where Ii (y) = kkF (tD1 ) (·, y) − kF (tD1 ) (·, p)kL2 (Bi )

Observe that

∀y ∈ B(p, R).

Ii (y) ≤ d(y, p) sup d2 kF (tD1 ) (·, z) L2 (B ) . z∈M

i

Since supλ∈R+ (1 + λ)N −2 |F (λ)| < ∞ by the asymptotics of Bessel functions of the first kind and N − 2 > n/2 + 1 by assumption, we may use Proposition 4.3 (iii), and conclude that sup kd2 kF (tD1 ) (·, z)kL2 (Bi ) ≤ C |t|−n/2−1

z∈M

∀t ∈ [−1, 1].

HARDY SPACES ON NONCOMPACT MANIFOLDS

37

Therefore, since the support of φρi is contained in the set of all t in R such that 4i−1 R ≤ |t| ≤ 4i+1 R, Z ∞ ρi

φ (t) |PN (O)(ω rb)(t)| sup d2 kF (tD1 ) (·, z) L2 (Bi ) dt kD12 Si (D1 )Ak2 ≤ C R 2 t z∈M −∞ Z ∞ R ≤ C kσN (·, D)(ω rb)k∞ i n/2+3 φρi (t) |t| dt (4 R) −∞ ≤ C kσN (·, D)b r k∞ 4−i µ(Bi )−1/2

∀i ∈ {1, . . . , d}.

Now, (6.7) follows directly from this, (6.8) and (6.6). Next we consider LS0 (D1 )A. Observe that LS0 (D1 )A is supported in B(p, 5R), and that kL S0 (D1 )Ak2 ≤ L S0 (D1 ) 2 kAk2
−1/2 L S0 (D1 ) ≤ µ B(p, R) 2
−1/2 ≤ C µ B(p, 5R) L S0 (D1 ) 2 .

To prove that L S0 (D1 ) is bounded on L2 (M ), with norm independent of R in (0, 1] observe that, by the spectral theorem and Lemma 3.6 L S0 (D1 ) ≤ sup (λ2 + b) |S0 (λ)| 2 λ≥0

≤ C kσN (·, D)b r k∞ ,

where C is independent of R. This concludes the proof of in (6.7), and of Step II. Step III: proof of the second inequality in (6.4). By arguing much as in the proof of Theorem 4.9 (with D1 in place of D, and with r in place of m) we see that TA=

∞ X

Tj (D1 )A,

j=1

where Tj is defined as in (4.8), i.e. Z ∞ (6.9) Tj (λ) = Ωj,N (t) J (tλ) dt −∞

∀λ ∈ R,

and the function Ωj,N now denotes PN (O)(ωj rb). Recall that the support of Ωj,N is contained in the set {t ∈ R : j − 3/4 ≤ |t| ≤ j + 3/4} (see 4.7), and observe that r is in H ∞ (Sη ; J) for every integer J. From Lemma 4.8 we deduce that for every ε > 0 there exists a constant C such that (6.10)

kΩj,N k∞ ≤ C krkSη ;J e−(η−ε)j

∀j ∈ {1, 2, . . .}.

Now we estimate kTj (D1 )Ak2 . We argue as we did in the proof of Theorem 4.9 to estimate kTj (D1 )ak2 , and obtain that there exists a constant C such that kTj (D1 )Ak2 ≤ C krkSη ;J e−2β j

∀j ∈ {1, 2, . . .}.


−1/2 −ε j Since η > 2β, there exists ε > 0 such that e−2β j ≤ C µ B(p, j +1) e , where we have used the uniform bound (1.3). Hence
−1/2 −ε j (6.11) kTj (D1 )Ak2 ≤ C krkSη ;J µ B(p, j + 1) e ∀j ∈ {1, 2, . . .}.

38

G. MAUCERI, S. MEDA AND M. VALLARINO

Observe that, at least formally, LT A =

∞ X j=1

L Tj (D1 )A.

We need to estimate kLTj (D1 )Ak2 . Note that (6.12)

kLTj (D1 )Ak2 ≤ kD12 Tj (D1 )Ak2 + |b − κ2 | kTj (D1 )Ak2 .

We have already estimated kTj (D1 )Ak2 in (6.11), so we concentrate on kD12 Tj (D1 )Ak2 . By (6.9) and the spectral theorem Z ∞ dt 2 D1 Tj (D1 ) = Ωj,N (t) F (tD1 ) 2 , t −∞ where F (λ) = λ2 J (λ). By using (6.10), Proposition 4.3 (ii) and the fact that the support of Ωj,N is contained in {t : j − 3/4 ≤ |t| ≤ j + 3/4}, we obtain that there exists constants C and ε > 0 such that Z ∞ dt 2 kD1 Tj (D1 )Ak2 ≤ C |Ωj,N (t)| kF (tD1 )Ak2 2 t −∞ Z ∞ dt ≤ C kAk1 |Ωj,N (t)| F (tD1 ) 1;2 2 t −∞ ≤ C krkSη ;J e−2β j


−1/2 −ε j ≤ C krkSη ;J µ B(p, j + 1) e

∀j ∈ {2, 3, . . .}.

This estimate, (6.11) and (6.12) then imply that (6.13)


−1/2 −εj kL Tj (D1 )Ak2 ≤ C krkSη ;J µ B(p, j + 1) e .

Now, by (6.9) we may write

LTj (D1 )A = L (6.14)

Z

= Lk

∞

Ωj,N (t) J (tD1 )A dt

−∞ ∞

Z

−∞

Ωj,N (t) J (tD1 )L1−k A dt

= Lk aj ,

R∞ where aj = −∞ Ωj,N (t) J (tD1 )L1−k A dt. Observe that aj is a multiple of a H 1 atom supported in B(p, j + 1). Indeed, by Remark 6.5, the function L1−k A is in
R∞ 2 L0 B(p, R) and the kernel of the operator −∞ Ωj,N (t) J (tD1 ) dt is supported in R {(x, y) : d(x, y) ≤ j}. Finally, M aj dµ = 0 by Proposition 4.4 (ii). Hence

εj


e aj ≤ C krkSη ;J µ B(p, j + 1) −1/2 . (6.15) 2

Then we may apply Lemma 6.7 to the function aj , and conclude that LTj (D1 )A = k (M ), and that, by (6.13), Lk aj is in Xat
1/2 kLTj (D1 )AkXat k (M) ≤ C j µ(B(p, j + 1) kLTj (D1 )Ak2 ≤ C krkSη ;J j e−εj .

HARDY SPACES ON NONCOMPACT MANIFOLDS

39

By summing over j, we see that kLT AkXat k (M) ≤ C krkSη ;J

∞ X

j e−εj ,

j=1

thereby concluding the proof of Step III and of the lemma.



Theorem 6.10. Suppose that k is a positive integer and that the covariant derivatives ∇j Ric of the Ricci tensor are uniformly bounded on M for 0 ≤ j ≤ 2k − 2. k Then X k (M ) and Xat (M ) agree as vector spaces and there exists a constant C such that −k U 2 kF kX k (M) (6.16) C kF kXat k (M) ≤ kF kX k (M) ≤ ∀F ∈ X k (M ). β 2 at

k Proof. First we prove that Xat (M ) is contained in X k (M ), and that the right-hand inequality in (6.16) holds. P Suppose that F = j λj Aj , where Aj is an admissible X k -atom. By Remark 6.5 −k 1 1 (see (6.1)), the function Uβ−k The series 2 Aj /|||Uβ 2 |||2 is an H -atom in H (M ). P −k 1 k j λj Uβ 2 Aj is then convergent in H (M ). Denote by f its sum. Since Uβ 2 is 1 bounded on H (M ) by Theorem 4.9, X
= F. λj Uβk2 Uβ−k Uβk2 f = 2 Aj j

Thus, F is in X k (M ), and X |λj |. kF kX k (M) = kf kH 1 (M) ≤ Uβ−k 2 2 j

The right-hand inequality in (6.16) followsP from this by taking the infimum over all the decompositions of F of the form F = j λj Aj , as required. k (M ) and that the left-hand inNext we prove that X k (M ) is contained in Xat equality in (6.16) holds. For notational convenience, in the rest of this proof we denote H 1 (M ) also by X 0 (M ), and write R instead of R4β 2 +κ2 and U instead of U4β 2 +κ2 . We argue inductively. The result is trivial in the case where k = 0, because U 0 = I. Suppose that the result holds for k − 1. Recall the factorisation U = LR and the diagram R
−→ X k−1 (M ) R X k−1 (M ) Uց

X k (M )

ւL

Suppose that F is in X k (M ). Then U −1 F is a function in X k−1 (M ), which we denote by f . Furthermore kf kX k−1 (M) = kF kX k (M) . By the induction hypothesis for each ε in R+ there exist a sequence {Aj } of admissible X k−1 -atoms and a summable sequence {cj } of complex numbers such that X X |cj | − ε. cj Aj and kf kX k−1 (M) ≥ (6.17) f= j

j

40

G. MAUCERI, S. MEDA AND M. VALLARINO

Observe that we may write (6.18)

F = LRf =

X j

cj LRAj .

The first equality in (6.18) is just the factorisation U = LR. The second is a P consequence of the facts that the series j cj Aj converges in H 1 (M ) to f , and that LR is bounded on H 1 (M ) by Theorem 4.9. From (6.18) and Lemma 6.9 we see that if N and J are suitably large integers, then X |cj | kLRAj kXat k (M) kF kXat k (M) ≤ C j

X  |cj | ≤ C kσN (·, D)b r k∞ + krkSη ;J j


 ≤ C kσN (·, D)b r k∞ + krkSη ;J kf kX k−1 (M) + ε 
 = C kσN (·, D)b r k∞ + krkSη ;J kF kX k (M) + ε .

k Therefore F is in Xat (M ), and kF kXat k (M) ≤ C kF kX k (M) , as required. This concludes the proof of the theorem.



Remark 6.11. Suppose that k is a positive integer and that s is a scale parameter in 1 R+ . The admit a decomposition of the form P space of all functions F in H (M ) that F = j λj Aj , where {λj } is a sequence in ℓ1 and {Aj } is a sequence of X k -atoms k supported in balls Bj in Bs agrees with Xat (M ) (hence with X k (M )). The norm k on Xat (M ) defined by X X λj Aj , Aj are X k -atoms supported in balls of Bs |λj | : F = inf j

j

k (M ). is an equivalent norm on Xat To prove this, it suffices to observe that minor modifications in the proof of Theorem 6.10 and Lemma 6.9 show that F is in X k (M ) if and only if it admits a decomposition in terms of X k -atoms supported in balls in Bs .

Remark 6.12. Suppose that p is in (1, ∞) and denote by p′ the index conjugate k to p. Assume that k is a positive integer and that B is in B. Define HB,p ′ to be ′ p k the space of all functions V in L (M ) such that L V is constant (in the sense of k ⊥ distributions) in a neighbourhood of B. Then denote by (HB,p the annihilator of ′) k p k p k ⊥ HB,p′ in L (M ). Then a X -atom in L (M ) is an element A of (HB,p ′ ) , satisfying the size condition ′ kAkp ≤ µ(B)−1/p . It is straightforward to modify the theory of this section to show that X k (M ) admits an atomic characterisation in terms of X k -atoms in Lp (M ). The fact that U is an isomorphism of Lp (M ) for all p in (1, ∞) plays an important rˆole here. As a consequence of the atomic decomposition of the space X k (M ), we may describe explicitly the action of elements of the dual X k (M )∗ on finite linear combinations of X k -atoms.

k Definition 6.13. Suppose that k is a positive integer. We denote by Xfin (M ) the k 1 vector space of all finite linear combination of X -atoms and by Hfin (M ) the vector space of all finite linear combination of H 1 -atoms.

HARDY SPACES ON NONCOMPACT MANIFOLDS

41

k Suppose that ℓ is a continuous linear functional on Xat (M ). Since Uβk2 is an k isomorphism between H 1 (M ) and X k (M ) and X k (M ) and Xat (M ) are isomorphic by Theorem 6.10, ℓ ◦ Uβk2 is a continuous linear functional on H 1 (M ). By [CMM1, Thm 5.1], there exists a function f in BM O(M ) such that Z 1 g f dµ ∀g ∈ Hfin (M ). (ℓ ◦ Uβk2 )(g) = M

Clearly

|||ℓ|||X k (M)∗ = ℓ ◦ Uβk2 H 1 (M)∗ = kf kBMO(M) .

It may be worth describing how the functional ℓ acts on X k -atoms, or, more k generally, on functions in Xfin (M ). Suppose that A is a X k -atom with support
2 −1 k contained in an arbitrary ball B. Since Uβ−k , there exist constants 2 = I +β L cj such that k X cj L−j A. Uβ−k A = 2 j=0

Then

Uβ−k 2 A

is a finite linear combination of H 1 -atoms by Remark 6.5. Therefore ℓ(A) = (ℓ ◦ Uβk2 )(Uβ−k 2 A) Z = (Uβ−k 2 A) f dµ. M

Uβ−k 2 A

Observe that is supported in B, so that the last integral is just the inner −k 2 product in L (M ) between Uβ−k 2 A and 1B f . Since Uβ 2 is self adjoint, we may write Z ℓ(A) = A Uβ−k 2 (1B f ) dµ. M

k (M ) and its support is contained in A similar argument shows that if F is in Xfin the ball B, then Z ℓ(F ) = F Uβ−k 2 (1B f ) dµ. M

It may be worth observing that a consequence of this representation formula for ℓ, and of the fact that |||ℓ|||X k (M)∗ = kf kBMO(M) , is that N ′ (f ) ≤ |||ℓ|||X k (M)∗ ≤ |||Uβk2 |||2 N ′ (f ), where

 1 Z 2 1/2 −k U −k dµ 2 (1B f ) − fB Uβ 2 1B β B∈B1 µ(B) B and fB denotes the average of f over B. The proof of this fact is straightforward and is omitted. N ′ (f ) = sup

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` di Genova, via DodeGiancarlo Mauceri: Dipartimento di Matematica, Universita caneso 35, 16146 Genova, Italy – [email protected] ` di MilanoStefano Meda: Dipartimento di Matematica e Applicazioni, Universita Bicocca, via R. Cozzi 53, I-20125 Milano, Italy – [email protected] ´matiques et Applications, Physique Mathe ´maMaria Vallarino: Laboratoire de Mathe ˆ timent de Math´ tiques d’Orl´ eans, Universit´ e d’Orl´ eans, UFR Sciences, Ba ematiqueRoute de Chartres, B.P. 6759, 45067 Orl´ eans cedex 2, France – [email protected]