The atomic Hardy space for a general Bessel operator

We study Hardy spaces associated with a general multidimensional Bessel operator $\mathbb{B}_\nu$. This operator depends on a multiparameter of type $\nu$ that is usually restricted to a product of half-lines. Here we deal with the Bessel operator in the general context, with no restrictions on the type parameter. We define the Hardy space $H^1$ for $\mathbb{B}_\nu$ in terms of the maximal operator of the semigroup of operators $\exp(-t\mathbb{B}_\nu)$. Then we prove that, in general, $H^1$ admits an atomic decomposition of local type.


Introduction
Let ν = (ν 1 , . . . , ν d ) ∈ R d , d ≥ 1, and consider the multidimensional Bessel differential operator acting on functions on R d 1.1. Hardy spaces associated with the classical Bessel operator. When ν ∈ (−1, ∞) d there exists a classical self-adjoint extension of B ν (acting initially on C 2 c (R d + )), from now on denoted by B cls ν [15,18]. The operator B cls ν is the infinitesimal generator of the classical Bessel semigroup of operators W cls t,ν = exp(−tB cls ν ), which has the integral representation W cls t,ν f (x) = R d + W cls t,ν (x, y)f (y) dµ ν (y), t > 0. It is well known that W cls t,ν j (x j , y j ), for x, y ∈ R d + and t > 0, where I τ denotes the modified Bessel function of the first kind and order τ > −1, cf. [25]. The classical Bessel kernel satisfies the lower and upper Gaussian bounds, i.e.
with some constants c 1 , c 2 , C 1 , Recently, harmonic analysis related to the classical Bessel operator has been extensively developed, see e.g. [1-6, 8, 11, 13, 18, 19] and references therein. In particular, the Hardy space associated with B cls ν and its characterizations have been studied. An especially useful result is the characterization of H 1 (B cls ν ) by atomic decomposition, which was proved in [5] in the one-dimensional case and then extended to higher dimensions in [11]. This atomic characterization also follows from a more general result from [10]. This result states, in particular, that every function f ∈ H 1 (B cls ν ) can be represented as f = k λ k a k , where k |λ k | f H 1 (B cls ν ) and a k are the classical atoms on the space of homogeneous type (R d + , | · |, dµ ν ), that is there exist balls B k such that (1.4) supp a k ⊆ B k , a k ∞ ≤ µ ν (B k ) −1 , a k dµ ν = 0.
We say that the atoms a k satisfy localization, size and cancellation conditions [9].
The main goal of this paper is to prove an atomic decomposition theorem for the Hardy space associated with the multidimensional Bessel operator B ν in the general situation, admitting all ν ∈ R d . For a precise definition of the operator B ν see in Section 1.3. According to the best of the author's knowledge the theory of Hardy spaces for the Bessel operator for the full range of the parameter has never been studied. The most general results are known for ν j > −1, and a vast majority of them are restricted to ν j ≥ −1/2, j = 1, . . . , d. Our atomic decomposition theorem is a starting point for developing this theory in a general Bessel context and the first step towards proving other characterizations of H 1 (B ν ). Note that one of the difficulties in the general situation is that the measure is not locally finite, in particular it does not satisfy the standard doubling condition. Theory of Hardy spaces or Calderón-Zygmund operators for spaces with non-doubling measure is far more difficult and less known. It has been developed only quite recently, see e.g. [16,17,22,23,26]. Usually an essential assumption in this theory is the polynomial growth condition µ(B(x, r)) ≤ Cr n . But this condition fails for µ ν unless ν ∈ (−1, ∞) d , since otherwise there are balls of arbitrarily small radius and infinite measure. For the same reason, our measure does not satisfy the property known in the literature as a local doubling condition, considered for instance in [7]. Another obstacle is that, in general, operators considered in the literature are generators of semigroups of operators whose integral kernels satisfy the upper Gaussian bounds. This is not the case of B ν if ν / ∈ (−1, ∞) d . Nevertheless we overcome these difficulties and give an atomic decomposition of the Hardy space associated with the general multidimensional Bessel operator B ν for any ν ∈ R d .
1.2. The exotic Bessel operator. Recall that dµ(x) = x 2ν+1 dx. When we take ν ∈ ((−∞, 1) \ {0}) d in (1.1), then there exists a self-adjoint extension of B ν , denoted by B exo ν and called the exotic Bessel operator [18]. It is quite remarkable that for the parameter ν ∈ ((−1, 1) \ {0}) d there exist both, the classical and the exotic, self-adjoint extensions of B ν , which differ significantly. We denote by W exo t,ν = exp(−tB exo ν ) the semigroup of operators generated by B exo ν . It is known that the exotic semigroup kernel can be expressed in a simple way in terms of the classical one, namely It turns out that if 0 < ν j < 1 then the so-called pencil phenomenon occurs. In the one-dimensional situation this means that for any fixed t > 0 the operator W exo t,ν j is well defined on L p (R + , dµ ν j ) and maps this space into itself if and only if ν j + 1 < p < (ν j + 1)/ν j . Therefore there is no reason in studying the theory of Hardy spaces for the exotic Bessel operator in the case ν ∈ (0, 1) d . For more details see the discussion in [18,Sec. 4] or in [4]. Thus, from now on we shall consider the operator For the sake of convenience we shall write B exo −ν , where ν ∈ (0, ∞) d . Hence, we will use such notation in the rest of the paper.
1.3. Main results. We consider a general multidimensional Bessel operator B ν = L 1 + . . . + L d , d ≥ 1, where each L i is either one-dimensional classical Bessel operator B cls ν i for ν i ∈ (−1, ∞) or one-dimensional exotic Bessel operator B exo −ν i for ν i ∈ (0, ∞) (acting on the ith coordinate variable, see Section 4.1 for details). Since B cls ν i and B exo −ν i are self-adjoint operators, B ν is well defined and essentially self-adjoint [20,Thm. 7.23]. To simplify the notation, by changing the coordinates, we shall consider B ν = B cls Then As we mentioned above, the case d 1 ≥ 1 and d 2 = 0, that is the classical case, is well known and slightly different from the context involving the exotic Bessel operator. Therefore we will consider d 1 ≥ 0 and d 2 ≥ 1.
Denote by W t,ν = exp(−tB ν ) the semigroup of operators generated by B ν . Clearly, the semigroup W t,ν has the integral representation We define the Hardy space for the operator B ν exactly in the same way as in (1.3), i.e. by means of the maximal operator associated with the semigroup of operators {W t,ν }, We will prove that elements of H 1 (B ν ) have an atomic decomposition, where atoms are either classical atoms (1.4) or some additional atoms of the form µ ν (Q) −1 1 Q for some cube Q ⊂ R d + . Such atoms µ ν (Q) −1 1 Q are called "local atoms" and notice that they do not satisfy cancellation condition. More precisely, we will show that every function f ∈H 1 (B ν ) belongs also to the local atomic Hardy space That means that f can be decomposed into a sum f = k λ k a k , where k |λ k | < ∞ and a k are either classical atoms described in (1.4) and supported in cubes Q ∈ Q or atoms of the form a k = µ ν (Q) −1 1 Q , Q ∈ Q.
For the general Bessel operator B ν let us define the family of cubes Q B which arises as follows. Let D = {[2 n , 2 n+1 ] : n ∈ Z} be the collection of all closed dyadic intervals in R + . Consider a family Then tile each cylinder R d 1 + × Q 1 × . . . × Q d 2 with countably many cubes having diameters equal to the smallest of the diameters of Q 1 , . . . , Q d 2 . The family Q B consists of all of these smaller cubes. For a rigorous definition of Q B see Section 4.
The main result of this paper is the following.
and H 1 at (Q B , µ ν ) are isomorphic as Banach spaces.
In the proof we will use only general properties of B cls νc and B exo −νe , therefore some auxiliary results in Sections 3 and 4 will be formulated in a more general context. Moreover, we introduce universal conditions on a semigroup kernel so that the product case can be deduced from a lower-dimensional information. The methods we use have roots in [14], however in that paper the Lebesgue measure case is considered, and here we need to adapt them to our situation, which requires some effort.
1.4. Organization of the paper. Section 2 contains definitions and notation used in the paper. Also, some auxiliary results are proved there. In Section 3 we consider a general self-adjoint and nonnegative operator L, which generates a semigroup of operators possessing an integral representation. We present general assumptions on the semigroup integral kernel and some family of cubes that are sufficient to prove an atomic decomposition of local type for H 1 (L). Section 4 is devoted to a similar atomic characterization in a product situation. More precisely, we slightly generalize conditions from Section 3 and show that if (lower-dimensional) component operators L 1 , L 2 satisfy them, then the operator L = L 1 + L 2 also does. Hence, one may deduce the result for the sum of operators by checking "one-dimensional" conditions. Finally, in Section 5, we prove Theorem 1.5.
Throughout the paper we use standard notation. In particular, C and c at each occurrence denote some positive constants independent of relevant quantities. Values of C and c may change from line to line. Further, we write α β if there exists a positive constant C, independent of significant quantities, such that C −1 α ≤ β ≤ Cα.

Preliminaries
2.1. Notation and terminology. We consider the metric measure space (R d + , | · |, dµ ν ), where | · | stands for the standard Euclidean metric. It is well known that if ν ∈ (−1, ∞) d , then µ ν possesses the doubling property, i.e. there exists C > 0 such that Since both the space and the measure have product structure, it is convenient to use cubes and cuboids rather than balls. Thus denote by . . , d the cuboid centered at z ∈ R d + having axial radii r 1 , . . . , r d > 0. When r 1 = . . . = r d = r the cuboid becomes a cube which we denote briefly by Q(z, r). We denote by d Q the Euclidean diameter of a cuboid Q.
Definition 2.2. We call a family Q of cuboids in R d an admissible covering if there exist C 1 , C 2 > 0 such that: Observe that item 3 means that admissible cuboids are uniformly bounded deformations of cubes. In fact, we shall often use only cubes. From now on we always assume that Q is an admissible covering of R d + . Given a cuboid Q, by Q * we denote a (slight) enlargement of Q. More precisely, if Q = (z; r 1 , . . . , r d ), then Q * := Q(z; κr 1 , . . . , κr d ), where κ > 1 is a fixed constant. Let Q be a given admissible covering of R d + . We fix κ = κ(Q) sufficiently close to 1, so that for any Q 1 , Q 2 ∈ Q, The family {Q * * * } Q∈Q is a finite covering of R d + , For Q ∈ Q denote N (Q) = Q ∈ Q : Q * * * ∩ Q * * * = ∅, (all neighbors of the cuboid Q).
For the covering Q as above consider functions ψ Q ∈ C 1 (R d + ) satisfying (2.4) It is straightforward to see that such a family {ψ Q } Q∈Q exists. We call it a partition of unity related to Q.
We now define suitable atoms and a local atomic Hardy space H 1 at (Q, µ ν ) related to Q. Definition 2.5. A function a : R d + → C is a (Q, µ ν )-atom if: (i) either there is Q ∈ Q and a cube K ⊂ Q * , such that supp a ⊆ K, a ∞ ≤ µ ν (K) −1 , a dµ ν = 0; (ii) or there exists Q ∈ Q such that The atoms as in (ii) are called local atoms.
Having (Q, µ ν )-atoms at our disposal, we define the local atomic Hardy space H 1 at (Q, µ ν ) related to Q in the standard way. Namely, a function f belongs to H 1 at (Q, µ ν ) if it has an atomic decomposition where the infimum is taken over all possible representations of f as above. Note that H 1 at (Q, µ ν ) is a Banach space.

Auxiliary results.
Lemma 2.6. Let d = 1 and ν ∈ R. The following estimates hold.
Finally, if r ≤ x/2, then x + r x x − r. Therefore, applying the Mean Value Theorem, The proof of (b) is straightforward. We pass to proving (c). If x > r > x/2, then since now 2ν + 2 < 0. When r ≤ x/2, we proceed similarly as in the corresponding part of the proof of (a) above. The conclusion follows.
Proof. (a) Since r < x/2, we have that x + r x − r x and for ν = −1 the claim readily follows from (a) and (c) of Lemma 2.6. To cover ν = −1, use Lemma 2.6(b) and then observe that by the Mean Value Theorem log(x + r) − log(x − r) r/x. Part (b) is a simple reformulation of Lemma 2.6(a).
As a consequence of the above results we obtain the following.
Case 1: ν ≥ −1/2. Using Corollary 2.7(b) we see that the left hand side of (2.11) is controlled by The last inequality follows from the relation |x − y| ≥ cd Q , since x ∈ (Q * * ) c and y ∈ Q * .
In this case we again apply Corollary 2.7(b) to bound the left hand side of (2.11) by Step and write To estimate J 1 we apply Corollary 2.8(b), (2.14) and (2.10) with some small ε > 0 such that δ + ε < min(1/2, ν 1 + 1, . . . , ν d + 1). We get We deal with J 2 similarly, but this time using (2.10) for the integral over the cube Q and Step 1 for the integral over Q 1 . This gives Treating J 3 we apply (2.14) and Step 1 and obtain Q . This completes the proof.
2.3. Local Hardy space. In this subsection we consider ν ∈ (−1, ∞) d . Let τ > 0 be fixed. We are interested in decomposing into atoms a function f such that In the classical case of the Laplace operator on R d equipped with Lebesgue measure, if one restricts the supremum to 0 < t ≤ τ 2 in the maximal operator, then one obtains an atomic space with the classical atoms complemented with atoms of the form |B| −1 1 B , where the ball B has radius τ , c.f. [12].
It turns out that a similar phenomenon occurs in case of the classical Bessel operator. More precisely, (2.15) holds if and only if f = k λ k a k , where k |λ k | < ∞ and a k are either the classical atoms or local atoms at scale τ . The latter are atoms a supported in a cube Q of diameter comparable to τ such that a ∞ ≤ µ ν (Q) −1 but we do not impose cancellation condition. In other words one could say that these atoms built the space The next proposition states a local atomic decomposition theorem that will be suitable for the proof of our main result. This proposition can be obtained by known methods based on Uchiyama [24, Cor. 1']. We refer the reader also to [11,Sec. 4], where the authors check assumptions of Uchiyama's Theorem in the classical Bessel framework for the whole range of the parameter ν ∈ (−1, ∞) d . For the sake of completeness, below we present a sketch of the proof.
Let Q {τ } be a section of R d + consisting of cuboids, which have diameter uniformly comparable to τ , i.e. there exists a positive constant C such that C −1 τ ≤ d Q ≤ Cτ for every Q ∈ Q {τ } . Proposition 2.16. There exists C > 0 independent of τ such that: (a) For every classical atom a supported in K ⊂ Q * or atom of the form a = µ ν (Q) −1 1 Q , where then there exist a sequence λ k and (Q {τ } , µ ν )-atoms a k , such that f = k λ k a k , k |λ k | ≤ CM , and a k are either the classical atoms supported in Q * or a k = µ ν (Q) −1 1 Q .

Local atomic decomposition
3.1. Local atomic decomposition theorem. Let L be a self-adjoint and nonnegative operator defined on L 2 (R d + , dµ ν ). Denote by T t = exp (−tL) the semigroup generated by L and suppose there exists an integral kernel T t (x, y), such that T t f (x) = R d + T t (x, y)f (y)dµ ν (y), t > 0. Similarly as in (1.3) we consider the maximal Hardy space for the operator L For ν ∈ (−1, ∞) d , an admissible covering Q, and the semigroup {T t } we consider the following conditions.
Let us emphasize that the constants C, c in these above conditions are independent of Q ∈ Q.
The main result of this section is the following. To prove Theorem 3.1 we need one more lemma.
Proof. First, we prove (3.3). Let Q ∈ Q and take y ∈ Q * . Notice that if x ∈ Q * * then in particular x ∈ N (Q), hence we may use (A 0 ). We have since µ ν (B(x, d Q )) µ ν (Q), x ∈ Q * , by the doubling property of µ ν .
3.2. Proof of Theorem 3.1. We shall prove the two inclusions.
First inclusion. Let f ∈ H 1 at (Q, µ ν ). We need to show sup t>0 |T t f | L 1 (R d + ,dµν ) ≤ C f H 1 at (Q,µν ) . By the standard density argument it is enough to check that sup t>0 |T t a| L 1 (R d + ,dµν ) ≤ C for every (Q, µ ν )-atom a, where C does not depend on a.
Take an atom a associated with a cuboid Q ∈ Q, see Definition 2.5. Using (A 1 ), (3.3), (A 2 ) and Proposition 2.16(a) with τ = d Q we get sup t>0 |T t a| . Let ψ Q be a partition of unity related to Q, see (2.4). Clearly, .
Here we have used the fact that . Now we apply Proposition 2.16(b) for each f Q separately with τ = d Q . For each f Q we obtain a sequence λ Q k and atoms a Q k such that .
Finally, we observe that for each Q the atoms a Q k obtained by Proposition 2.16 are either local atoms of the form µ ν (Q) −1 1 Q or classical atoms supported in Q * , therefore they are indeed (Q, µ ν )-atoms.

4.
Local atomic decomposition in the product case 4.1. Product of local atomic Hardy spaces. In this section we consider operators of the form L = L 1 + . . . + L N , where each L i acts nontrivially only on the variable x i ∈ X i = R d i + . We introduce conditions on the kernels of the semigroups generated by L i , and admissible coverings Q i of X i , which are sufficient to prove the local atomic decomposition theorem for the Hardy space H 1 (L).

More precisely, we consider
We equip the space X with the Euclidean metric and the measure dµ ν (x) = x 2ν+1 dx, where ν = (ν 1 , . . . , ν N ) and ν i = (ν i,1 , . . . , ν i,d i ) ∈ (−1, ∞) d i for i = 1, . . . , N . Assume that L i is an operator on L 2 (X i , dµ ν i ), as in Section 3.1. Slightly abusing the notation we keep the symbol for the operator on L 2 (X, dµ ν ), where I denotes the identity operator on the corresponding subspace, and we define We denote by T Given two admissible coverings Q 1 and Q 2 of X 1 and X 2 , respectively, we would like to produce an admissible covering on X 1 × X 2 . However, the products {Q 1 × Q 2 : Q 1 ∈ Q 1 , Q 2 ∈ Q 2 } do not form an admissible covering (in general item 3 of Definition 2.2 fails). Therefore, we need a finer construction. We split each Q = Q 1 × Q 2 (without loss of generality let us assume that d Q 1 > d Q 2 ) into cuboids Q j , j = 1, . . . , M , such that all of them have diameters comparable to d Q 2 . Then the cuboids Q j = Q j 1 × Q 2 , j = 1, . . . , M, satisfy: • each Q j satisfies condition 3 of Definition 2.2.
Definition 4.2. The admissible covering of X 1 ×X 2 described in the above construction will be called the product admissible covering and denoted by Q 1 Q 2 .
As an example, consider the family D = {[2 n , 2 n+1 ] : n ∈ Z}, which is an admissible covering of R + . The product admissible covering D D of R 2 + is illustrated by Figure 1 below. For a product admissible covering Q 1 Q 2 we also fix a new κ such that (2.3) is fulfilled. The covering Q 1 . . . Q N of X 1 × . . . × X N is constructed as above by induction. We shall assume that each T t (x i , y i ) , i = 1, . . . , N , and the corresponding admissible covering Q i satisfy condition (A 0 ). Furthermore, we consider certain modifications of conditions (A 1 ), (A 2 ). Namely, for i = 1, . . . , N , there exists γ ∈ (0, 1/3) such that: and • for each δ ∈ [0, γ) Notice that if {T t } and Q satisfy (A 1 ) and (A 2 ), then (A 1 ) and (A 2 ) also hold.
t (x i , y i ) satisfy the upper Gaussian estimates, that is the bound of condition (A 0 ) for all x i , y i ∈ X i . This is a direct consequence of (2.11).
Our main result in this section is the following.   Proof. We will show the following claim. If conditions (A 0 ) -(A 2 ) hold for T [i] t (x i , y i ) together with admissible coverings Q i for i = 1, 2, then (A 0 ) -(A 2 ) also hold for T t (x, y) = T [1] t (x 1 , y 1 )T [2] t (x 2 , y 2 ), together with Q = Q 1 Q 2 . This is enough, since by simple induction we shall get that in the general case T t (x, y) = T [1] t (x 1 , y 1 ) . . . T [N ] t (x N , y N ) with Q 1 . . . Q N satisfy (A 0 ) -(A 2 ), and consequently, the assumptions of Theorem 3.1 will be fulfilled.
To prove the claim let T Without loss of generality we may assume that where K ⊆ Q 1 is a cuboid emerging from the very construction of Q 1 Q 2 , see the description preceding Definition 4.2 and Figure 2 below. Denote by z = (z 1 , z 2 ) the center of Q = K × Q 2 . Verification of (A 0 ). Notice that if x = (x 1 , x 2 ) ∈ N (Q), then in particular x 1 ∈ N (Q 1 ) and x 2 ∈ N (Q 2 ). Similarly, if y = (y 1 , y 2 ) ∈ Q * , then y 1 ∈ Q * 1 and y 2 ∈ Q * 2 . Therefore, for such x, y we may use (A 0 ) for each of T [1] t (x 1 , y 1 ) and T [2] t (x 2 , y 2 ) obtaining (A 0 ) for T t (x, y) = T [1] t (x 1 , y 1 )T [2] t (x 2 , y 2 ). Verification of (A 1 ). Fix δ ∈ (−γ, γ). Let y ∈ Q * . Recall that We start with estimating the integral over S 1 . We shall consider two cases.
To estimate the integral over S 3 notice that since x 1 ∈ Q * * 1 we may use (A 0 ) for T [1] t (x 1 , y 1 ). Let ε > 0 be such that δ + ε < γ. Then we use (A 1 ) for T [2] t (x 2 , y 2 ) and (2.11) and arrive at Treating the integral over S 4 we make use of (A 1 ) for both T [1] t (x 1 , y 1 ) and T [2] t (x 2 , y 2 ). We get Turning to the last integral over S 5 we consider two cases.
The proof of Theorem 4.5 is complete.

4.2.
Product of local and nonlocal atomic Hardy space. As we have mentioned, all atoms of the Hardy space H 1 (B cls ν 1 ) satisfy cancellation condition, i.e. they are nonlocal atoms. However, if we consider the product R d + = R d 1 + × R d 2 + with the measure µ ν = µ ν 1 ⊗ µ ν 2 and the operator L = B cls ν 1 + L 2 , where the semigroup K t := exp(−tL 2 ) generated by L 2 , together with an admissible covering Q 2 satisfy conditions (A 0 ) -(A 2 ) on R d 2 + , then the resulting Hardy space H 1 (L) shall have local character.
In the proof we will need the following standard asymptotics of the modified Bessel function I τ , see e.g. [25, p. 203-204], for x ∼ ∞. Proof. To verify (A 0 ) -(A 2 ) for K t,ν with D, let D Q = [2 n , 2 n+1 ] for some n ∈ Z and take y ∈ Q * .
To treat I 1 we use (5.3) and (5.4). Thus it suffices to show that functions of the form a k (y) = a k (y)y 4νe 2 are (Q B , µ (νc,−νe) )-atoms after a modification by multiplying by some function which is comparable to a constant.
This finishes the proof.