Maximal characterisation of local Hardy spaces on locally doubling manifolds

We prove a radial maximal function characterisation of the local atomic Hardy space h1(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak {h}}}^1(M)$$\end{document} on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to h1(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak {h}}}^1(M)$$\end{document} if and only if either its local heat maximal function or its local Poisson maximal function is integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama.


Introduction
Goldberg [6] introduced a "local" Hardy space h 1 (R n ), which may be defined in several equivalent ways: (1.1) where I denotes the identity map, the standard Laplacian, H t the heat semigroup e t and P t the Poisson semigroup e −t √ − . Furthermore, h 1 (R n ) admits both an atomic and an ionic decomposition, and can be characterised in terms of a suitable "grand maximal function".
The main advantage of working with h 1 (R n ) rather than with the classical Hardy space H 1 (R n ) [5] is that h 1 (R n ) is preserved by multiplication by smooth functions with compact support. This makes h 1 (R n ) very effective in many situations in which localisation arguments are involved.
Analogues of h 1 (R n ) may be defined in a variety of settings. In particular, all the definitions mentioned above in the Euclidean case make sense on any (complete) Riemannian manifold M, the role of being played by the Laplace-Beltrami operator on M. It is then natural to speculate whether all such definitions give rise to the same space. Even a bare knowledge of the theory of h 1 (R n ) suggests that the key properties of this space depend mainly on the local structure of the Euclidean space. This leads to conjecture that a theory parallel to that in R n should hold on any Riemannian manifold where the local geometry is somewhat uniformly controlled.
A number of results in this direction are available in the case where the manifold is doubling. Indeed, an extensive theory of local Hardy spaces has been developed in the general context of doubling metric measure spaces (see, e.g., [3,8,9,21] and references therein), and includes both atomic and maximal characterisations. This theory is somewhat parallel to that of the "global" Hardy space H 1 à la Coifman-Weiss [2] on spaces of homogeneous type. However, due to the aforementioned local nature of h 1 , a global assumption such as the doubling condition does not appear entirely natural for its study, and one may expect that a richer theory could be developed, also encompassing non-doubling manifolds.
This problem has been considered by Taylor [18], who introduced a local Hardy space h 1 (M) on Riemannian n-manifolds M with strongly bounded geometry (positive injectivity radius and uniform control of all the derivatives of the metric tensor) via a grand maximal function characterisation; more precisely, Taylor defines where

3)
μ is the Riemannian measure and, for every x ∈ M and r ∈ (0, 1], L(x, r ) is the collection of all C 1 functions on M with Lipschitz constant at most r −(n+1) supported in the ball of centre x and radius r . Further extensions of the theory are due to Volpi and the second-named author [16,20], who studied an atomic local Hardy space in the more general context of locally doubling metric measure spaces (see Sect. 2 below for additional details). The atomic space of [16] coincides with the space of [18] in the case of manifolds with strongly bounded geometry (see Remark 2.7 below), and also with that of [3,9,21] in the case of doubling spaces. These works, however, do not address the issue of whether the local Hardy space admits characterisations analogous to (1.1) in a non-doubling setting. Our research was motivated by the following simple question. Suppose that M is a Riemannian manifold of dimension n, denote by L the (positive) Laplace-Beltrami operator on M, consider the associated heat and Poisson semigroups, namely H t := e −tL and P t := e −t √ L , and the spaces What geometric assumptions are needed in order that where h 1 (M) denotes the atomic local Hardy space of [16,18]? Despite our efforts, we have not been able to find in the literature a proof of the equivalence of h 1 (M), h 1 H (M) and h 1 P (M) on a general class of noncompact manifolds extending beyond the doubling ones. As suggested above, some "uniformity" of the local geometry should be the essential feature of M in order that the desired equivalence hold. One of our main results states that, if M has bounded geometry, viz. positive injectivity radius and Ricci curvature bounded from below (a weaker assumption than that of [18]), then indeed (1.4) holds true.
As a matter of fact, for the same class of manifolds M, we prove a much more general characterisation of h 1 (M) in terms of an arbitrary "radial maximal function", associated to a family of integral operators x, y) f (y) dμ(y), t ∈ (0, 1], whose integral kernel K satisfies suitable assumptions. Roughly speaking (see Sect. 4 below for details), we require K to decompose as the sum K 0 + K ∞ , where the local part K 0 is supported in a t-independent neighbourhood of the diagonal and satisfies bounds of the form and a γ -Hölder condition for some γ ∈ (0, 1], while the global part K ∞ satisfies the "uniform integrability" condition Here d and μ denote the Riemannian distance and measure on M respectively. Under these assumptions on K , we prove the maximal characterisation for all manifolds M with bounded geometry.
for the semigroups e −tL α with α ∈ (0, 1], thus obtaining the characterisation which includes (1.4) as a special case. The present paper does not address the problem whether h 1 (M) also admits a local Riesz transform characterisation, analogous to the first identity in (1.1) for the case of h 1 (R n ). This deceptively simple question turns out to require a much more sophisticated analysis, and is solved in the affirmative in a recent work of Veronelli and the second-named author [15]. One of the ingredients used in [15] is the Poisson maximal characterisation h 1 (M) = h 1 P (M) that we prove here.
Another question that we do not address here is the investigation of spaces defined in terms of "global" maximal functions, such as in the context of a nondoubling manifold M. Nevertheless, the results in the present paper turn out to be instrumental in the analysis of such spaces and their relation to the Hardy-type spaces X γ (M) introduced in [13,20], which we plan to develop in a future work [14].
It is an interesting question whether the maximal characterisation (1.7) extends to larger classes of Riemannian manifolds, or even more general spaces. A particularly natural setting for this investigation would be that of the locally doubling metric measure spaces considered in [16]. Given that our core ingredient (the local variant of Uchiyama's result) is proved for general Ahlfors-regular spaces, it would seem natural to conjecture that a maximal characterisation of h 1 in terms of a single kernel holds at least on metric measure spaces satisfying a suitable "local Ahlfors" condition. Extensions to even broader classes of spaces may also be possible; however, even in the case of globally doubling spaces, radial maximal characterisations of H 1 and h 1 appear to be available only under additional assumptions, such as a reverse doubling condition on the underlying space or a normalisation condition on the kernel (see, e.g., [7,9,21,22]), so tackling the general case of locally doubling spaces may be a nontrivial problem.
We shall use the "variable constant convention", and denote by C 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. We shall also write 1 A for the characteristic function of a set A.

Background on Hardy-type spaces
Let M denote a connected, complete n-dimensional Riemannian manifold with Riemannian measure μ and Riemannian distance d. Throughout this paper we assume that M has bounded geometry, that is, (A) the injectivity radius ι M of M is positive; (B) the Ricci tensor of M is bounded from below.
For p in [1, ∞], f p denotes the L p (M) norm of f (with respect to the Riemannian measure).
We denote by B the family of all geodesic balls on M. For each B in B we denote by c B and r B the centre and the radius of B respectively. Furthermore, we denote by c B the ball with centre c B and radius c r B . For each scale parameter s in R + , we denote by B s the family of all balls B in B such that r B ≤ s. We also write B r (x) for the geodesic ball of centre x ∈ M and radius r > 0.
(a) M is uniformly locally n-Ahlfors, i.e. for every s > 0 there exists a positive constant D s such that in particular, M satisfies the local doubling property, i.e. for every s > 0 there exists a positive constant D s such that has at most exponential growth, i.e. there exist positive constants a and β such that (c) For all Q > 1 and α ∈ (0, 1), the (Q 2 , 0, α)-harmonic radius r H of M is strictly positive.
In particular, to each point x in M we can associate a harmonic co-ordinate system η x centred at x and defined on B r H (x) such that, in these coordinates, the metric tensor (g i j ) satisfies the estimate as quadratic forms (2.4) at every point of B r H (x). In particular, as a consequence, where Y = η x (y) and Z = η x (z). Moreover (2.4) implies that where g denotes the determinant of the metric tensor.
We point out that the key aspect of the estimates (a)-(c) is their uniformity with respect to the centre of the ball B or the point x. Indeed, if one does not care about uniformity, then estimates similar to those in (a) and (c) are easy consequences of the properties of normal coordinates (see, e.g., [11,Proposition 5.11]), and do not require the assumptions (A)-(B). This includes, for all x ∈ M, the volume asymptotics where ω n is the volume of the unit ball in Euclidean n-space. We now recall the definition of the atomic local Hardy space h 1 (M). This is a particular instance of the local Hardy space introduced by Volpi [20], who extended previous work of Taylor [18], and then further generalised in [16]. A global p-atom at scale s is a function a in L 1 (M) supported in a ball B of radius exactly equal to s satisfying the size condition above (but possibly not the cancellation condition). Standard and global p-atoms at scale s will be referred to simply as p-atoms at scale s. One can prove (see, for instance, [16]) that h 1, p s is independent of both s in (0, ∞) and p in (1, ∞] (in the sense that different choices of the parameters define equivalent norms); henceforth it will be denoted simply by h 1 (M), and f h 1 will denote the norm f h 1,2 1 . We will also say "h 1 (M)-atom at scale s" instead of "2-atom at scale s".
The following statement will be useful in proving boundedness properties of sublinear operators defined on h 1 (M). Proof Suppose that f is in h 1 (M), and write f = ∞ j=1 λ j a j , where a j are p-atoms at scale s, and ∞ j=1 |λ j | < ∞. For each positive integer N , write f N = N j=1 λ j a j and note that f N tends to f in On the other hand, by sublinearity, if N > N , then so from the uniform boundedness of T on atoms and the convergence of the series j |λ j | we deduce that T f N is a Cauchy sequence in L 1 (M). By the uniqueness of limits, we conclude that T f N converges to T f in L 1 (M), and By taking the infimum of both sides with respect to all the representations of f as sum of p-atoms, we obtain that T f 1 ≤ C f h 1, p s (M) , as required.
The space h 1 (M) can also be characterised in terms of ions as follows.

Definition 2.4
Suppose that s > 0, p is in (1, ∞] and let p be the index conjugate to p. A p-ion at scale s is a function g in L 1 (M) supported in a ball B in B s satisfying the following conditions: (2.10) The cancellation condition that standard atoms must satisfy is in general not preserved by changes of variables and localisations; however, as shown in the following lemma, performing such operations on an atom produces an ion (or a multiple thereof). This observation, together with the equivalence (2.10), confirms that h 1 (M) is amenable to localisations and changes of variables.
The following statement involves two Riemannian manifolds M and M , both satisfying the assumptions (A)-(B) above; correspondingly, we denote by d and d the respective Riemannian distances, and by μ and μ the Riemannian measures. The result is certainly known to experts, and implicit in the work of Taylor [18], under more restrictive assumptions on M and M .
then g is a p-ion on M at scale As.
Proof Let a be a p-atom at scale s on M, supported in a ball B ⊆ . Let H > 0 be a positive constant and define g as above. In the course of the proof, we will determine what conditions H must satisfy in order for the above statement to hold. Let B be the ball on M of centre −1 (c B ) and radius Ar B . Then clearly −1 (B) ⊆ B ∩ and g is supported in B . Moreover If a is a standard atom, then so the condition is satisfied provided H ≥ L/(As). As for the condition (i) of Definition 2.4, let us first notice that, for all x ∈ , the latter equality is a consequence of (2.7). Hence the size condition on the p-atom a implies that where p is the conjugate exponent to p. On the other hand, since M and M are both uniformly locally n-Ahlfors, there exists a constant κ ≥ 1, only depending on M, M and s, such that

Interlude: a result on metric measure spaces
In this section we prove a variation of a result of Uchiyama [19], which plays a fundamental role in our proof of the radial maximal characterisation for local Hardy spaces. Differently from the rest of the paper, here we do not work on a Riemannian manifold M, but on a metric measure space X . Due to the different setting, part of the notation used here differs from that used in other sections.
Let D ∈ (0, ∞). Let (X , d, m) be a metric measure space which is D-Ahlfors regular, i.e. there exists a constant A ≥ 1 such that, for all x ∈ X and r ∈ [0, ∞), here B(x, r ) denotes the ball of centre x and radius r in X . Lebesgue spaces L p (X ) on X are meant with respect to the measure m, and f p will denote the L p (X ) norm (or quasinorm, if p < 1) of f . The next definition closely follows [19,Eqs. (40)-(43)].
]. An approximation of the identity (AI in the sequel) of exponent γ on a D-Ahlfors regular space X is a measurable function K :

Remark 3.2 The bounds (3.2) and (3.4) can be equivalently rewritten as
in the case γ = 1 and X = R D , these bounds are clearly satisfied by , which is a constant multiple of the Poisson kernel.
In the course of this section the exponent γ ∈ (0, 1] will be thought of as fixed.
To a measurable kernel K : (0, 1] × X × X → C, we associate the corresponding integral operators K t for t ∈ (0, 1] and the (local) maximal operator K * defined by We also denote by M the (global) centred Hardy-Littlewood maximal function: As is well known, M is of weak type (1, 1) and bounded on L p (X ) for all p ∈ (1, ∞]. Finally, for all x ∈ X , r ∈ (0, ∞), let F γ (x, r ) be the family of γ -Hölder cutoffs on the ball B(x, r ), that is, the collection of all functions φ : X → R such that, for all y, z ∈ X , Then we define the (γ -Hölder) local grand maximal function G γ by The aim of the present section is the proof of the following result, which is a variation of [19, Theorem 1 ] for local maximal functions.
pointwisely. In particular, for all q

Remark 3.5
As will be clear from the proof, the constants E and p in Theorem 3.4 only depend on the parameters A, D, γ, c in (3.1) and Definition 3.1, while E q only depends on those parameters and q. This fact will be crucial in the application of the above result in the following Sect. 4.
As in [19], the key ingredient in the proof of this result is the following decomposition of an arbitrary γ -Hölder cutoff φ supported in a ball of radius 1 as a superposition of kernels K (t, x, ·) at different times t and basepoints x. The main difference with respect to the decomposition obtained in [19, proof of Lemma 3 ] is that here we only use times t ≤ 1.

Proposition 3.6
Let K be an AI on X . There exist δ, η ∈ (0, 1) and κ, L ∈ (0, ∞) such that and the following hold. Let o ∈ X , and set d( Then, for all i ∈ N, there exists a finite index set J (i) and, for We postpone the proof of this decomposition to Sect. 3.1. Let us first show how to derive the main result from this decomposition. To this purpose the following lemma, which is a simple adaptation of [19, Lemma 1], will be useful.
We can now prove the main result of this section.

Proof of Theorem 3.4
Let us first observe that the estimate (3.14) actually holds (with the same constants) for all φ ∈ F γ (o, r ) and r ∈ (0, 1]. Indeed, it is sufficient to apply Corollary 3.8 to the rescaled metric d r , measure m r and kernel K r given by which satisfy the same assumptions as d, m, K (with the same constants). The pointwise estimate (3.7) then follows by taking the supremum for r ∈ (0, 1] and φ ∈ F γ (o, r ), for arbitrary o ∈ X . This estimate, together with the boundedness of the Hardy-Littlewood maximal function M on L s (X ) for s ∈ (1, ∞], immediately gives (3.8).

Proof of the decomposition
Here we prove the crucial Proposition 3.6. From now on we think of the AI K and the point o ∈ X as fixed. As in the statement of Proposition 3.6, we define d( Proof Immediate from the triangle inequality.

Lemma 3.10
For all a ∈ (0, 1], there exists C a ∈ (0, ∞) such that the following hold. Let t ∈ (0, 1/2] and let g ∈ L 1 loc (X ) be nonnegative. Then there exists a finite collection {x j } j of points of X such that Proof Let {y j } j be a collection of points of X t = {y ∈ X : td(y) ≤ 1/2} maximal with respect to the condition Finiteness of the collection is an immediate consequence of (3.1). Moreover, by maximality, By Lemma 3.9, whence y j ∈ B(x, 3td(x)); moreover, by (3.20) such points y j are at least at distance atd(x)/16 from each other, and therefore (3.17) follows from (3.1).

Lemma 3.11
Let L ∈ (0, ∞) and a, b ∈ [0, ∞) be such that b ≥ a. Then there exists C a,b,L ∈ (0, ∞) such that the following hold. Let t ∈ (0, 1) and let {x j } j be a collection of points of X such that Then, for all x ∈ X and h ∈ [0, ∞), In addition, if td(x) ≥ 2, then Proof Note that, by the triangle inequality, Hence, ifh = (h − 1) + , then, by (3.24) and (3.1), where C may depend on b and L.
Let us also remark that, by (3.24) and (3.1), the number of j such that 2 k ≤ d(x j ) < 2 k+1 is bounded by a multiple of t −D . Hence, if d(x) < 2 k−1 , then d(x j , x) > d(x j )/2 by Lemma 3.9 and Summing over k ∈ N by exploiting the estimates (3.27), (3.28) and (3.29) immediately gives (3.25). On the other hand, if td(x) ≥ 2 and td(x j ) ≤ 1, then 2d(x j ) ≤ d(x) and d(x, x j ) ≥ d(x)/2 by Lemma 3.9, so (3.29) applies; however in this case the sum is restricted to 2 k ≤ t −1 , which leads to (3.26).
Assume that Let c 2 be the constant in (3.5). For all i ∈ N, by applying Lemma 3.10 with t = η 1+i , a = c 2 , and g = (K * f ) 1/2 , we construct a finite family {x i j } j∈J (i) of points of X satisfying sup x∈X j∈J (i) is independent of f , η and i. In particular, (3.10), (3.11) and (3.12) are certainly satisfied. Let φ ∈ F γ (o, 1). Up to rescaling it is not restrictive to assume that Let us now define recursively, for all i ∈ N, the function φ i : X → R by setting φ 0 = φ and and i j = sgn φ i (x i j ). We now want to prove, for all i ∈ N, that Clearly this implies that φ i → 0 locally uniformly as i → ∞, and consequently the representation (3.13) holds, provided we relabel κδ as κ. We will prove (3.36) by induction on i. Note that, because of (3.35), the estimate (3.36) trivially holds for i = 0. Before entering the proof of the induction step, we discuss a number of useful estimates.
Let us first obtain a few "a priori" estimates for the functions w i (that do not depend on the choices of the signs i j ). Let i ∈ N. By (3.2), (3.31) and Lemma 3.11, for all x ∈ X , Hence, if we assume that Similarly, for all x, y ∈ X such that d(x, y) ≤ η 1+i d(x)/4, by the triangle inequality we deduce that, for all j ∈ J (i), hence, by (3.4) and Lemma 3.11, Hence, if we assume that where we used that η γ /2 ≤ 1 − δ by (3.39) and (3.41). This shows that (3.36) is trivially satisfied whenever η i d(x) ≥ 2.
By summing the estimates (3.40), we can also derive, for all i ∈ N, an useful estimate for the difference of the values of φ i at different points.
Observe first that, since φ ∈ F γ (o, 1), for all x, y ∈ X such that d(x, y) ≤ d(x)/2, by Lemma 3.9 and the support condition, the difference |φ(x)−φ(y)| vanishes unless d(x) ≤ 4, so Consequently, if we define We now proceed with the proof of the inductive step; i.e., for a given i ∈ N, we assume the validity of (3.36) for φ i and prove the same estimate for φ i+1 .
Let x ∈ X . Consider first the case where In this case, since 3/4 ≤ 1 − δ by (3.39), from (3.38) we deduce that and we are done. Hence, to prove (3.36) for φ i+1 , it remains to consider the case where On the other hand, if we assume that η i+1 d(x) ≥ 1/2, then Note now that, by (3.48), (3.2) and Lemma 3.11, On the other hand, since K is nonnegative, by (3.5), Lemma 3.9, (3.47), and (3.32), Hence, if we assume that and consequently, by (3.38) and (3.45), which concludes the proof of the inductive step. By looking at all the above conditions, one sees that they are satisfied if we first fix the value of κ as in (3.

Maximal characterisation of h 1 (M) via approximations of the identity
We now return to the setting of a Riemannian manifold M satisfying the assumptions (A)-(B) of Sect. 2. In this section we shall prove a maximal characterisation of the atomic local Hardy space h 1 (M) in terms of a local maximal function associated to a single "approximation of the identity", in the sense defined below. the following hold: We denote by V γ the collection of all LAI of exponent γ on M.
We denote by V γ the collection of all AI of exponent γ on M.

Remark 4.3 Definition 4.
1 is analogous to Definition 3.1, but includes the additional constraint for K (t, ·, ·) to be supported in a t-independent neighbourhood of the diagonal. Definition 4.2 provides a relaxation of the support constraint, which is very convenient in applications. As a matter or fact, in the case M is globally n-Ahlfors, one can show that any kernel K satisfying the bounds (ii) to (iv) of Definition 4.1 is actually an AI in the sense of Definition 4.2. In these respects, Definition 4.2 can be considered as an appropriate extension of Definition 3.1 that applies also to spaces that are locally, but not globally Ahlfors.
In this section, the exponent γ ∈ (0, 1] will be thought of as fixed, and we will simply write "λ-LAI" in place of "λ-LAI of exponent γ ". We will also write V and V in place of V γ and V γ . K (t, x, y) is continuously differentiable in y, condition (iv) in Definition 4.1 is implied by the differential condition (iv') |∇ y K (t, x, y)| ≤ C 4 t −n−1 (1 + d(x, y)

Remark 4.4 In case
Indeed, by the fundamental theorem of calculus, ∈ (y, z), in the range (4.1) we deduce that 1 +

d(x, w)/t ≥ (3/4)(1 + d(x, y)/t) and
Clearly the restriction to the range (4.1) in Definition 4.1 is only relevant for condition (iv). As a matter of fact, the constant 4 in the range (4.1) could be replaced with any other constant greater than 1 without changing the class of λ-LAI, as shown by the following lemma and its proof.
Then K also satisfies condition (iv) in the range (4.1), with a constant C 3 (in place of C 3 ) only depending on κ, C 1 , C 3 .

Proof We only need to check condition (iv) in the range
However in this range, by the triangle inequality, A simple example of λ-LAI on M is Proof Under our assumptions, it is immediately seen that K satisfies (i) and (ii) of Definition 4.1, by taking C 1 = C 1 L. As for (iv), note that, if x, y, z ∈ M and t ∈ (0, 1] satisfy 4d(y, z) ≤ t + d(x, y), then and from (4.4) we deduce that so the left-hand side of (4.5) vanishes and (4.5) is trivially true. Consequently K satisfies (iv) of Definition 4.1 by taking We now show that, in the decomposition of any given AI, the LAI part can be chosen so to be supported arbitrarily close to the diagonal.
To each measurable K : (0, 1] × M × M → C and t in (0, 1], we associate the integral operator K t and the maximal operator K * by the rules Let M R denotes the centred local Hardy-Littlewood maximal operator at scale R > 0, defined by It is well known (see, e.g., [ Proof By our assumptions on K , where we used the uniform local n-Ahlfors property of M. It is straightforward to check that the series above is uniformly bounded with respect to t, and the required estimate follows. Proof Assume that K is a λ-LAI. By Lemma 4.8 and the boundedness properties of the local Hardy-Littlewood maximal operator, we immediately deduce that K * is of weak-type (1, 1) and bounded on L p (M) for p ∈ (1, ∞]. Hence, in light of Lemma 2.3, to conclude that K * is bounded from h 1 (M) to L 1 (M), it is enough to show that K * is uniformly bounded on 2-atoms at scale λ.
Let a be a standard 2-atom supported in a ball B with centre c B and radius r B ≤ λ. Then by the L 2 -boundedness of K * and the local doubling property. Next, observe that d(x, y) ≥ 4d(y, c B ) for every x in M \ (5B) and y in B. By using the cancellation condition of the atom and Definition 4.1 (iv), we see that By optimising with respect to t in (0, 1], we find that Notice that, since K is a λ-LAI, K * a is supported in B 2λ (c B ). Therefore Assume now that a is a global 2-atom, with support contained in a ball B of radius λ. Then K * a is supported in 5B, and arguing as in (4.7) concludes the proof.
An immediate consequence of Proposition 4.9 is the following boundedness result. Proof By Definition 4.2, we can write K = K 1 + K 2 , where K 1 ∈ V , while K 2 satisfies (4.2). Since the latter bound implies the L 1 -boundedness of the maximal operator associated to K 2 , the desired boundedness result follows by applying Proposition 4.9 to K 1 .
For each λ > 0, define the local Riesz-type potential I λ by the rule From the uniform local n-Ahlfors condition, it readily follows that there exists a positive constant C, depending on λ, such that Proof Take f in L 1 (M), x ∈ M and t ∈ (0, 1]. Since φ is Lipschitz and K satisfies the estimate in Definition 4.1 (i), because sup α∈[0,∞) α n /(1 + α) n+γ < 1. By taking the supremum of both sides with respect to t in (0, 1], we obtain the required estimate. We now show that, for any AI K on M, the local Hardy space h 1 (M) can be characterised as the space of all f in L 1 (M) such that K * f is in L 1 (M). Our proof hinges on the fact that a similar characterisation is already known in the Euclidean case [6]. The main idea is to show that if φ is a suitable cutoff function in a neighbourhood of a point p in M, then the composition of φ f with an appropriate harmonic co-ordinate map η −1 p belongs to h 1 (R n ). The latter property is verified by showing that the maximal function of (φ f )•η −1 p with respect to an appropriate AI on R n is in L 1 (R n ). Part of the localisation argument is modelled over the proof of [18, Proposition 2.2], where however a grand maximal function is considered instead of an arbitrary AI, and much more restrictive conditions on M are assumed.
Let Q > 1 be fixed. Due to our assumptions on M, as discussed in Sect. 2, we can find R 0 ∈ (0, 1] such that, for each p ∈ M, there exist smooth coordinates η p : for all x, y ∈ B R 0 ( p), where X = η p (x) and Y = η p (y), and moreover the Riemannian volume element μ p in the coordinates η p satisfies Q −n ≤ μ p ≤ Q n (4.10) pointwise. Set U p := η p (B R 0 ( p)). A direct consequence of (4.9) is that be a smooth cutoff function on R n supported in B that is equal to 1 on (1/2)B, and set (X , Y ) = χ(X ) χ(Y ).
Let S be the R 0 /Q-LAI on R n defined (similarly to (4.3)) by

Lemma 4.12 Suppose that K is a LAI on M.
Then, for all p ∈ M, the function K p defined in (4.12) is a R 0 /Q-LAI on R n , with constants C 1 , C 2 , C 3 in Definition 4.1 independent of p ∈ M.
Proof For the sake of notational simplicity, in this proof we write x, y and z in place of and we are done because S is a R 0 /Q-LAI. If not, then X , Y ∈ U p and we can use that together with (4.9) to deduce conditions (i) and (ii) for K p from the corresponding conditions for K and S and the fact that |X − Y | < R 0 /Q if both X , Y ∈ B; similarly, since both K and S satisfy (iii), we deduce that where C 2 > 0 is the minimum of the corresponding constants for K and S.
As for condition (iv), by Lemma 4.5 it is enough to check it for all t ∈ (0, 1] and X , Y , Z ∈ R n such that If both (X , Y ), (X , Z ) / ∈ B × B, then again we are done, since S satisfies condition (iv). If not, then X and at least one of Y , Z belong to B, but then (4.13) implies that and consequently all of X , Y , Z ∈ 2B ⊆ U p . Hence we can write and consider the three summands in the right-hand side separately. Since S is a LAI on R n , by Lemma 4.6 both S and (t, X , Y ) → (X , Y )S(t, X , Y ) satisfy condition (iv), and the last two summands are dealt with. As for the first summand, in light of (4.9), the condition (4.13) implies that and we are reduced to checking that the function (t, x, y) → (η p (x), η p (y))K (t, x, y) satisfies condition (iv) on M. This however follows again from Lemma 4.6, since K is a LAI on M and x → χ(η p (x)) is bounded and Lipschitz on M (uniformly in p ∈ M) by (4.9). Then for every p ∈ P define ψ p = ψ d(·, p) and φ p = ψ p / q∈P ψ q . From (4.14) it is clear that the locally finite sum q∈P ψ q is bounded above and below by positive constants and is Lipschitz. This readily implies that the φ p have the desired properties, and part (i) is proved.
Finally, for part (iii), take as P 1 any subset P of P which is maximal with respect to the property min d( p, p ) : p, p ∈ P , p = p ≥ 4κ. (4.16) Recursively, define P k+1 as any subset P of P \(P 1 ∪· · ·∪P k ) which is maximal with respect to (4.16). In order to conclude, we need to show that this procedure terminates after finitely many steps, that is, P k = ∅ for some integer k ≥ 1. Indeed, if P k = ∅, then, given any p ∈ P k , by maximality of P 1 , . . . , P k−1 , the ball B 4κ ( p) intersects each of P 1 , . . . , P k−1 , and therefore B 4κ ( p) contains k distinct points of P (including p). However by (4.15) the cardinality of the intersection P ∩ B 4κ (x) is bounded by a constant independent of x ∈ M. Hence P k = ∅ only for finitely many integers k.

Theorem 4.14 Suppose that K is an AI on M. If f is in L
where the constant C only depends on M and K .
Proof Let κ = R 0 /(8Q 2 ). In view of Lemma 4.7 and the fact that the bound (4.2) implies boundedness on L 1 (M) of the maximal operator associated to K 2 , it is enough to prove the result when K is a κ-LAI.
Let the collection of points P, the sets P 1 , . . . , P N , and the families of Lipschitz cutoffs {φ p } p∈P and { φ p } p∈P be defined as in Lemma 4.13 corresponding to the above choice of κ. Recall that supp φ p ⊆ B κ ( p), and moreover d( p, p ) ≥ 4κ for all p, p ∈ P and ∈ {1, . . . , N }. For every ∈ {1, . . . , N }, define ψ = p∈P φ p ; note that the supports of the summands are pairwise disjoint, and in particular each ψ is bounded and Lipschitz. Let Since ψ is a Lipschitz function, by Lemma 4.11, Since K is a κ-LAI and supp(φ p f ) ⊆ B κ ( p) for all p ∈ P, we deduce that supp K * (φ p f ) ⊆ B 2κ ( p). Consequently, for every ∈ {1, . . . , N }, the supports of the functions K * (φ p f ), p ∈ P , are mutually disjoint. Therefore Recall that μ p is the Riemannian volume element in the coordinates η p , and set g p := μ p (φ p f ) • η −1 p . Notice that the support of g p is contained in η p B κ ( p) , which by (4.9) is contained in B R n κ Q (0) = (1/4)B. Consequently, if K p is defined as in (4.12), then Since the function 1− vanishes in (1/2)B×(1/2)B, and S is a 8κ Q-LAI on R n , we deduce that sup t∈(0,1] where κ Q is the difference of the radii of (1/2)B and (1/4)B. Therefore, if K p, * denotes the maximal operator associated to K p , then from (4.10) we deduce that Recall that K p is a LAI on R n by Lemma 4.12, with constants independent of p ∈ P. Consequently, there exists a constant A > 0 (independent of p ∈ P) such that A K p satisfies Definition 3.1 on the n-Ahlfors space R n , with constant c independent of p ∈ P. Hence, by Theorem 3.4, Proof It is immediately seen that, if K is a LAI constructed as in (4.3) with ψ supported in [0, 1], bounded by 1, and satisfying |ψ(u) − ψ(v)| ≤ |u − v| γ for all u, v ∈ [0, ∞), then K (t, x, ·) ∈ F γ (x, t) for all t ∈ (0, 1], and therefore pointwise for all f ∈ L 1 loc (M). Moreover, for all x ∈ M, by using normal coordinates centred at x, one easily deduces that lim t→0 + M K (t, x, y) dμ(y) = c, where c := R n ψ(|y|) dy > 0. From this it readily follows that if f ∈ C c (M), then K t f → c f pointwise as t → 0 + ; a density argument then shows that, if f ∈ L 1 loc (M),

Characterisation of h 1 (M) via heat and Poisson maximal functions
In this section we prove that h 1 (M) can be characterised in terms of the maximal operators associated either to the heat semigroup H t or to the Poisson semigroup P t , that is, . In light of Theorem 4.15, it will be enough to show that the heat and Poisson kernels on M are AI in the sense of Definition 4.2. We will actually show that a similar result holds for all semigroups e −tL α with α ∈ (0, 1]. Denote by h t (x, y) the integral kernel of the heat semigroup H t . We call the function (t, x, y) → h t (x, y) the heat kernel. It is well known (see, for instance, [17,Theorem 5.5.3 and Sect. 5.6.3]) that, under our assumptions on M, there exist positive constants C 1 and C 2 such that for every x and y in M and t in (0, 1]. Furthermore (see, e.g., [4,Theorem 6]), there exists positive constants C and c such that for every x and y in M and t in (0, 1]; in the discussion below, we will actually use the gradient bound (5.2) only for d(x, y) small.  (x, y)). Then (x, ·) is smooth for all x ∈ M, and |∇ y (x, y)| ≤ L for all x, y ∈ M and some L > 0. Set now K 1 (t, x, y) = (x, y)h t 2 (x, y) and K 2 (t, x, y) = (1 − (x, y))h t 2 (x, y). Then clearly K 1 (t, x, y) = 0 whenever d(x, y) > r /2, and moreover from (5.1) we deduce that so K 1 satisfies conditions (i), (ii) and (iii) of Definition 4.1 with λ = r /2. In addition and from (5.1) and (5.2) we deduce that Hence by Remark 4.4 we deduce that K 1 also satisfies condition (iv) of Definition 4.1, and consequently is a r /2-LAI on M.
Note now that K 2 (t, x, y) = 0 whenever d(x, y) ≤ r /4; hence, by ( and from the volume bound (2.3) it readily follows that K 2 satisfies condition (4.2). This proves that K = K 1 + K 2 is an AI on M.
Now we consider the semigroups e −tL α with α ∈ (0, 1), and denote by p α t their integral kernels; the Poisson semigroup P t corresponds to α = 1/2. As it is well known (see, e.g., [23,Sect. IX.11]), these semigroups can be subordinated to the heat semigroup. for all z ∈ C with z > 0.
As a consequence of Propositions 5.1 and 5.3 and Theorem 4.15, we deduce the following characterisation of h 1 (M). and there exists a positive constant C such that In particular, with equivalent norms.