A maximal function characterisation of the Hardy space for the Gauss measure

In dimension one we give a maximal function characterisation of the Hardy space H^1(g) for the Gauss measure g, introduced by G. Mauceri and S. Meda. In arbitrary dimension, we give a description of the nonnegative functions in H^1(g) and use it to prove that L^p(g) is a contained in H^1(g) for 1

In [21] the first two authors defined an atomic Hardy-type space H 1 (γ) and a space BM O(γ) of functions of bounded mean oscillation, associated to γ. We briefly recall their definitions. A closed Euclidean ball B is called admissible at scale s > 0 if r B ≤ s min 1, 1/ |c B | ; here and in the sequel r B and c B denote the radius and the centre of B, respectively. We denote by B s the family of all balls admissible at scale s. For the sake of brevity, we shall refer to balls in B 1 simply as admissible balls. Further, B will be called maximal admissible if r B = min(1, 1/ |c B |). Now let r ∈ (1, ∞]. A Gaussian (1, r)-atom is either the constant function 1 or a function a in L r (γ) supported in an admissible ball B and such that (1.1) a dγ = 0 and a r ≤ γ(B) 1/r−1 ; here and in the whole paper, · r denotes the norm in L r (γ). In the latter case, we say that the atom a is associated to the ball B. The space H 1,r (γ) is then the vector space of all functions f in L 1 (γ) that admit a decomposition of the form j λ j a j where the a j are Gaussian (1, r)-atoms and the sequence of complex numbers {λ j } is summable. The norm of f in H 1,r (γ) is defined as the infimum of j |λ j | over all representations of f as above.
In [21] the spaces H 1,r (γ) were defined and proved to coincide for all 1 < r < ∞, with equivalent norms. In Section 2 we complement this by proving that they coincide also with the space H 1,∞ (γ). Once this is established, we shall denote the space by H 1 (γ) and use the H 1,∞ (γ) norm. Further, we shall frequently write atom for (1, ∞)-atom.
The space BM O(γ) consists of all functions f in L 1 (γ) such that where f B denotes the mean value of f on B, taken with respect to the Gauss measure. The norm of a function in BM O(γ) is If, in the definitions of H 1 (γ) and BM O(γ), we replace the family B 1 of admissible balls at scale 1 by B s for any fixed s > 0, we obtain the same spaces with equivalent norms, see [21]. We remark that a similar H 1 − BM O theory for more general measured metric spaces has been developed by A. Carbonaro and the first two authors in [1,2,3].
The main motivation for introducing these two spaces was to provide endpoint estimates for singular integrals associated to the Ornstein-Uhlenbeck operator L = −(1/2)∆ + x · ∇, a natural self-adjoint Laplacian on L 2 (γ). Indeed, in [21] the first two authors proved that the imaginary powers of L are bounded from H 1 (γ) to L 1 (γ) and from L ∞ (γ) to BM O(γ) and that Riesz transforms of the form ∇ α L −|α| and of any order are bounded from L ∞ (γ) to BM O(γ). In a recent paper [23], the authors proved that boundedness from H 1 (γ) to L 1 (γ) and from L ∞ (γ) to BM O(γ) holds for any first-order Riesz transform in dimension one, but not always in higher dimension.
The definition of the space H 1 (γ) closely resembles the atomic definition of the classical Hardy space H 1 (λ) on R n endowed with the Lebesgue measure λ, but there are two basic differences. First, the measured metric space (R n , d, γ) is non-doubling. Further, except for the constant atom, a Gaussian atoms must have "small support", i.e., support contained in an admissible ball. Despite these differences, H 1 (γ) shares many of the properties of H 1 (λ). In particular, the topological dual of H 1 (γ) is isomorphic to BM O(γ), an inequality of John-Nirenberg type holds for functions in BM O(γ) and the spaces L p (γ) are intermediate spaces between H 1 (γ) and BM O(γ) for the real and the complex interpolation methods.
It is well known that the classical Hardy space H 1 (λ) can be defined in at least three different ways: the atomic definition, the maximal definition and the definition based on Riesz transforms [6,31].
To characterise H 1 (γ), we introduce the local grand maximal function defined on L 1 loc (R n , γ) by In Section 3 we shall prove that, in arbitrary dimension, f ∈ H 1 (γ) implies M loc f ∈ L 1 (γ). Moreover, in dimension one H 1 (γ) can be characterised as the space of all functions f in L 1 (γ) satisfying M loc f ∈ L 1 (γ) and the following additional global condition This is Theorem 3.3 below. Roughly speaking, if we interpret a function f as a density of electrical charge on the real line, this global condition says that the positive and negative charges nearly balance out, so that the net charges inside the intervals (−∞, −x) and (x, ∞) decay sufficiently fast as x approaches +∞. The condition is violated when the distance between the positive and the negative charges increases too much or the charges do not decay sufficiently fast at infinity. For instance, let (a n ) ∞ 1 and (a ′ n ) ∞ 1 be increasing sequences in (2, ∞) such that a n + 2/a n < a ′ n and a ′ n + 2/a ′ n < a n+1 < 2a n for all n. Then set c n χ (an,an+1/an) γ(a n , a n + 1/a n ) − for some c n > 0. One easily verifies that M loc f ∈ L 1 (γ) if and only if c n < ∞. But the global condition E(f ) < ∞ is equivalent to c n a n (a ′ n − a n ) < ∞, which is here a stronger condition.
We have not been able to find a similar characterisation of H 1 (γ) in higher dimension. However, in Section 4 we prove in all dimensions that if M loc f ∈ L 1 (γ) and the function f satisfies the stronger global condition then f ∈ H 1 (γ). Observe that for n = 1 and f ≥ 0, Fubini's theorem implies that the conditions E(f ) < ∞ and E + (f ) < ∞ are equivalent. In arbitrary dimension, E + (f ) can be used to characterise the nonnegative functions in H 1 (γ); see Theorem 4.2. This also leads to a simple proof of the inclusions L p (γ) ⊂ H 1 (γ) and We end the introduction with some technical observations and notation. In the following we use repeatedly the fact that on admissible balls at a fixed scale s, the Gauss and the Lebesgue measures are equivalent, i.e., there exists a constant C(s) such that for every measurable subset E of B ∈ B s In particular this implies that the Gauss measure is doubling on balls in B s , with a constant that depends on s (see [21,Prop. 2.1]). Further, it is straightforward to see that if B ′ ⊂ B are two balls and B ∈ B s then B ′ is also in B s . Given a ball B in R n and a positive number ρ, we shall denote by ρB the ball with the same centre and with radius ρr B .
In the following C denotes a constant whose value may change from occurrence to occurrence and which depends only on the dimension n, except when otherwise explicitly stated.

2.
Coincidence of H 1,∞ (γ) and H 1,r (γ) First we need a lemma which will play a role also in the maximal characterisation. It deals with the classical Hardy space H 1 (λ) with respect to the Lebesgue measure and the associated standard (1, ∞)-atoms, called Lebesgue atoms below. The result is probably well known, but we include a proof because we have not been able to find a reference in the literature. Some related results can be found in [4] and [5].
Lemma 2.1. If g ∈ H 1 (λ) and the support of g is contained in a ball B, then g has an atomic decomposition g = k λ k a k where the a k are Lebesgue (1, ∞)-atoms associated to balls contained in 2B and Proof. In this proof, all atoms are Lebesgue (1, ∞)-atoms. We claim that the grand maximal function of g satisfies To prove this, take x / ∈ 2B and observe that For each integer k, denote by Ω k the level set x : Mg(x) > 2 k . Then (2.2) implies that Let Ω k = i Q k i be a Whitney decomposition of Ω k into closed cubes Q k i , i ∈ N, whose interiors are disjoint, and whose diameters are comparable to δ times their distances from Ω c k , where δ is a (small) positive constant to be chosen later. Definẽ Q k i as the cube with the same centre as Q k i and side length expanded by a factor 2. Then iQ k i = Ω k and the family Q k i : i ∈ N will have the bounded overlap property, uniformly in k, provided that δ is small enough. Proceeding as in the proof of the atomic decomposition for H 1 (λ) in [31, p. 107-109], one shows that there exists a decomposition with the following properties.
(i) Each function A k i is supported in a ball B k i that contains the cubeQ k i as well as thoseQ k+1 m that intersectQ k i . Moreover, if δ is sufficiently small, the ball B k i is contained in Ω k and for each k the family B k i i has the bounded overlap property.
(iii) A k i ≤ C 2 k for each k and i. We split the sum in (2.4) in two parts Clearly, Σ 2 is an atomic decomposition with atoms associated to balls contained in 2B. Thus, it suffices to show that Σ 1 is a multiple of an atom associated to 2B. And indeed, Σ 1 is supported in 2B and has integral zero, because it is the difference of g and Σ 2 , both of which have these two properties. Moreover, by (i) and (iii) Hence by (2.3) Thus Σ 1 is a multiple of an atom associated to 2B. We thus have the desired atomic decomposition of g, and the norm estimate (2.1) also follows.
Proof. In this proof, the constants C may depend on r and n. Since any Gaussian (1, ∞)-atom is also a Gaussian (1, r)-atom, H 1,∞ (γ) is a subspace of H 1,r (γ) and f H 1,r (γ) ≤ f H 1,∞ (γ) . Conversely, suppose that a is a Gaussian (1, r)-atom associated to the ball B ∈ B 1 . Then the function a γ 0 is a multiple of a Lebesgue (1, r)-atom. Indeed, aγ 0 dλ = a dγ = 0 and, by the equivalence of the Gauss and Lebesgue measures on admissible balls, Hence, a 0 γ is in H 1 (λ) with norm at most C. By Lemma 2.1, it has a decomposition where each α j is a Lebesgue (1, ∞)-atom associated to a ball B j contained in 2B. Moreover j |λ j | ≤ C, and each B j is admissible at scale 2. Define a j = α j γ −1 0 . Then a j dγ = 0, and by the equivalence of the Gauss and Lebesgue measures on B j Thus the a j are multiples of Gaussian (1, ∞)-atoms. Since a = j λ j a j , we conclude that a ∈ H 1,∞ (γ) and 3. The characterisation of H 1 (γ) in R In this section, we shall prove that f ∈ H 1 (γ) implies M loc f ∈ L 1 (γ) and that, in dimension one, functions in H 1 (γ) can be characterised by the two conditions M loc f ∈ L 1 (γ) and E(f ) < ∞. We start with a simple but useful lemma dealing with the support of the local grand maximal function. Proof. Let x ∈ supp M loc f . We write ρ = |x| and c = |c B |, so that B ⊂ B(c B , min(1, 1/c)). The balls B and B(x, min(1, 1/ρ)) must intersect, and so To prove the lemma, it is enough to show that Considering the cases c ≤ 1 and c > 1, we conclude from this that  (1, ρ), which is trivially true. We have proved (3.2) and the lemma.
Proof. We shall prove that for any Gaussian atom a

(3.4)
M loc a 1 ≤ C, from which the lemma follows. Since (3.4) is obvious if a is the constant function 1, we assume that a is associated to an admissible ball B. By the preceding lemma, supp M loc f is contained in the ball denoted B ′ .
The integral of M loc a over 2B with respect to γ is no larger than C, since M loc a ≤ C sup |a| ≤ C/γ(B). To estimate M loc a at a point x in the remaining set B ′ \ 2B, we take φ ∈ Φ and 0 < t < min(1, 1/ |x|) and estimate a * φ t (x). We can assume that t > d(x, B) so that t > |x − c B |/2, since otherwise φ t * a(x) will vanish. Write (3.5) Here the first term to the right can be estimated in a standard way by To deal with the second term, we estimate a(y) dy, knowing that the integral of a against γ vanishes. Thus The fraction appearing here is Putting things together, we conclude that for x ∈ B ′ \ 2B An integration with respect to dγ, or equivalently γ 0 dλ, then leads to and (3.4) is proved. Proof. Suppose that f ∈ H 1 (γ). Then M loc f ∈ L 1 (γ) by Lemma 3.2. To prove the necessity of the condition E(f ) < ∞, it suffices to show that E(a) < C for all Gaussian atoms a. This is obvious for the exceptional atom 1. If a is associated to a ball B ∈ B 1 , it follows from the inequality Conversely, assume that f is a function in L 1 (γ) such that M loc f ∈ L 1 (γ) and E(f ) < ∞. We shall prove that f ∈ H 1 (γ), by constructing a Gaussian atomic Most of the following argument, up to the decomposition (3.15), works also in the n-dimensional setting. Since we shall need it in the next section, we carry out that part in R n .
By subtracting a multiple of the exceptional atom 1, we may without loss of generality assume that (3.7) f dγ = 0.
Let {B j } be a covering of R n by maximal admissible balls. We can choose this covering in such way that the family 1 2 B j is disjoint and {4B j } has bounded overlap [10,Lemma 2.4]. Fix a smooth nonnegative partition of unity {η j } in R n such that supp η j ⊂ B j and η j = 1 on 1 2 B j and verifying |∇η j | ≤ C/r Bj . Thus f = j f η j . We now need the following lemma.
Proof. Since the support of η j is contained in B j , the support of M loc (gη j γ 0 ) is contained in the ball 4B j , because of Lemma 3.1. Moreover, for φ ∈ Φ and x ∈ 4B j . Thus, to prove (3.8) it suffices to show that there exists a constant C such thatφ ∈ CΦ for x ∈ 4B j and 0 < t < min(1, 1/ |x|). The support ofφ is contained in B(0, 1) and Similarly ∇φ(z) ≤ C, because the gradients ∇ z η j (x−tz) and ∇ z γ 0 (x−tz)/γ 0 (c Bj ) give the factors t(1 + c Bj ) and t |x − tz| γ 0 (x − tz)/γ 0 (c Bj ), respectively, both of which are bounded. This concludes the proof of Lemma 3.4.
Continuing the proof of Theorem 3.3, we define b j ∈ C for each j ∈ N by Note that since η j = 1 on 1 2 B j , We now apply Lemma 3.4 with g = f − b j and use the subadditivity of M loc combined with (3.10), to get Proof. By the maximal characterisation of the classical space H 1 (λ), it suffices to show that Because of (3.11), all that needs to be verified is that (3.14) sup φ∈Φ sup t≥min(1,1/|x|) To prove (3.14), we split the integral in the left-hand side into the sum If x ∈ 4B j , then for φ ∈ Φ and t ≥ min(1, 1/ |x|) the last inequality because of (3.10). Hence If x ∈ (4B j ) c , we take φ and t as before and observe that we can assume that t > d(x, B j ), since otherwise the convolution in (3.14) will vanish. In view of (3.9) and (3.10), we then get

This implies that
We have proved (3.14) and the lemma.
We can now finish the proof of Theorem 3.3. By Lemmata 3.5 and 2.1, each function (f − b j )η j γ 0 has an atomic decomposition k λ jk α jk where the α jk are Lebesgue atoms with supports in 2B j and k |λ jk | ≤ C 4Bj M loc f dγ.
As we saw in the proof of Theorem 2.2, each a jk = γ −1 0 α jk is a multiple of a Gaussian atom, with a factor which is independent of j and k. Thus To complete the proof of Theorem 3.3, we need to find an atomic decomposition of j b j η j . It is here that we must restrict ourselves to the one-dimensional case and that the global condition E(f ) < ∞ plays a role.
Choose the intervals I 0 = (−1, 1), I j = ( √ j − 1, √ j + 1) for j ≥ 1 and I j = −I |j| for j ≤ −1. The intervals I j have essentially the same properties as the balls B j introduced above, and we can use them instead of the B j to construct η j and b j as before. To decompose now j b j η j , we first normalise the functions η j , letting Then b j η j = f η j dγη j , and clearly where µ k (x) = j≥k η j (x). Notice that f µ k dγ → 0 as k → ±∞, in view of (3.7). A summation by parts now yields Butη k −η k−1 is C times a Gaussian atom, if we use admissible balls at some scale s > 1 in the definition of atoms. Thus (3.16) is our desired atomic decomposition of j b j η j , provided we can estimate the coefficients by showing that To this end, observe that Since the support of µ ′ k is contained in I k and we obtain, using also the bounded overlap of the I j , here we used (3.7). This concludes the proof of Theorem 3.3.

A characterisation of nonnegative functions in H 1 (γ)
The dimension n is now arbitrary. The following lemma will be needed.
Proof. We shall construct atoms whose supports form a chain connecting B(0, 1) to B. First we define a finite sequence of maximal admissible balls and ρ 0 = 0, ρ 1 = 1. Finally, N is defined so thatB N −1 is the first ball of the sequence that contains c B , andB N = B. Squaring (4.1), we get N ≤ |c B | 2 + 1.
Next, we denote by B j , j = 1, . . . , N , the largest ball contained inB j ∩B j−1 . Notice that the three ballsB j ,B j−1 and B j have comparable radii and comparable Gaussian measures. Define now functions φ j and g j by setting Clearly, Each function g j is a multiple of an atom. Indeed, its integral against γ vanishes. Moreover, if 1 ≤ j ≤ N , the support of g j is contained inB j−1 and The support of φ N +1 is contained in B and Thus g j H 1 (γ) ≤ C γ(B) g L ∞ , j = 1, . . . , N + 1.
Summing the coefficients in the atomic decomposition (4.3), we then obtain via (4.2) The proof of the lemma is complete.
then f is in in H 1 (γ) and If f is nonnegative, the conditions M loc f ∈ L 1 (γ) and E + (f ) < ∞ are also necessary for f to be in H 1 (γ).
Proof. Let f be a function in L 1 (γ) such that M loc f ∈ L 1 (γ) and E + (f ) < ∞.
Since c(f ) is a multiple of the exceptional atom, it suffices to find an atomic decomposition of f 0 . Note that f 0 satisfies M loc f 0 ∈ L 1 (γ) and Let {B j } be the covering of R n by maximal admissible balls and {η j } the corresponding partition of unity introduced in the proof of Theorem 3.3. As there, we Then the argument leading to (3.15) shows that where the a jk are Gaussian atoms supported in 4B j and j,k It remains only to prove that j b j η j is in H 1 (γ). We write g j = b j η j and observe that where φ 0 is as in Lemma 4.1. Since (3.10) remains valid for f 0 , we have (4.6) g j ∞ ≤ C 1 γ(B j ) Bj |f 0 | dγ. Lemma 4.1 thus applies to each g j , and using also the bounded overlap of the B j we conclude This concludes the proof of the sufficiency and the norm estimate. The necessity of the condition M loc f ∈ L 1 (γ) was obtained in Lemma 3.2.
To prove the necessity of (4.4), let 0 ≤ f ∈ H 1 (γ). We first observe that the function x → |x| 2 is in BM O(γ). Indeed, its oscillation on any admissible ball is bounded. Since BM O(γ) is a lattice, the functions g k (x) = min |x| 2 , k are in BM O(γ), uniformly for k ≥ 1. By the monotone convergence theorem and the duality between H 1 (γ) and BM O(γ), The theorem is proved.
The following result is a noteworthy consequence of Theorem 4.2.
Proof. We claim that the operator M loc is bounded on L p (γ) for 1 < p ≤ ∞. Deferring momentarily the proof of this claim, we complete the proof of the corollary. Suppose that f is in L p (γ). Then M loc f is in L 1 (γ), because since γ(R n ) = 1. Moreover, E + (f ) ≤ |x| 2 p ′ f p < ∞, by Hölder's inequality. Thus f ∈ H 1 (γ) by Theorem 4.2. It also follows that the inclusion L p (γ) ⊂ H 1 (γ) is continuous, and by duality we get the continuous inclusion BM O(γ) ⊂ L p ′ (γ).
It remains to prove the claim. We shall use again the covering {B j } from the proof of Theorem 3.3. First we observe that the inequality (4.7) M loc g p ≤ C g p holds when supp g ⊂ B j , with a constant C independent of j. Indeed, M loc is bounded on L p (λ), and M loc g is supported in the ball 4B j , where the Gaussian measure is essentially proportional to dλ.
Given a function f ∈ L p (γ), we write it as a sum f = f j with supp f j ⊂ B j and with the sets {f j = 0} pairwise disjoint. We can then apply (4.7) to each f j and sum.