Hardy spaces for semigroups with Gaussian bounds

Let Tt=e-tL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_t=e^{-tL}$$\end{document} be a semigroup of self-adjoint linear operators acting on L2(X,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(X,\mu )$$\end{document}, where (X,d,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,d,\mu )$$\end{document} is a space of homogeneous type. We assume that Tt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_t$$\end{document} has an integral kernel Tt(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_t(x,y)$$\end{document} which satisfies the upper and lower Gaussian bounds: C1μ(B(x,t))exp-c1d(x,y)2/t≤Tt(x,y)≤C2μ(B(x,t))exp-c2d(x,y)2/t.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{C_1}{\mu (B(x,\sqrt{t}))} \exp \left( {-\,c_1d(x,y)^2/ t} \right) \le T_t(x,y)\le \frac{C_2}{\mu (B(x,\sqrt{t}))} \exp \left( {-\,c_2 d(x,y)^2/ t} \right) . \end{aligned}$$\end{document}By definition, f belongs to H1(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1(L)$$\end{document} if ‖f‖H1(L)=‖supt>0|Ttf(x)|‖L1(X,μ)<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert f\Vert _{{H^1(L)}}=\Vert \sup _{t>0}|T_tf(x)|\Vert _{L^1(X,\mu )} <\infty $$\end{document}. We prove that there is a function ω(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (x)$$\end{document}, 0<c≤ω(x)≤C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<c\le \omega (x)\le C$$\end{document}, such that H1(L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1(L)$$\end{document} admits an atomic decomposition with atoms satisfying: suppa⊂B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{supp}}\, a\subset B$$\end{document}, ‖a‖L∞≤μ(B)-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert a\Vert _{L^\infty } \le \mu (B)^{-1}$$\end{document}, and the weighted cancelation condition ∫a(x)ω(x)dμ(x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int a(x)\omega (x)\hbox {d}\mu (x)=0$$\end{document}.


Introduction
Let (X, d) be a metric space equipped with a nonnegative Borel measure μ. We shall assume that μ(X ) = ∞ and 0 < μ(B(x, r )) < ∞, r > 0, where B(x, r ) = {y ∈ X | d(x, y) ≤ r } denotes the closed ball centered at x and radius r . Suppose (X, d, μ) is a space of homoge- Jacek Dziubański jacek.dziubanski@math.uni.wroc.pl 1 Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland neous type in the sense of Coifman and Weiss [8], which means that the doubling condition holds, namely: There is C > 0 such that μ(B(x, 2r )) ≤ Cμ(B(x, r )) for r > 0 and x ∈ X . It is well known that the doubling condition implies that there are q > 0 and C d > 0 such that μ(B(x, sr)) ≤ C d s q μ(B(x, r )) for x ∈ X, r > 0, and s ≥ 1. (1.1) Suppose that L is a nonnegative densely defined self-adjoint linear operator on L 2 (X, μ). Let T t = e −t L , t > 0, denote the semigroup of linear operators generated by − L. We impose that there exists T t (x, y), such that (1.2) Clearly, T t (x, y) = T t (y, x) for t > 0 and a.e. x, y ∈ X . Moreover, we assume the following lower and upper Gaussian bounds, that is, there are constants c 1 ≥ c 2 > 0 and C 0 > 0 such that 3) for t > 0 and a.e. x, y ∈ X . It is well known that (1.3) implies that for every n ∈ N we have ∂ n ∂t n T t (x, y) ≤ for t > 0 and a.e. x, y ∈ X with some C n , c n > 0. For this fact see, e.g., [20, (7.1)], [10,27]. The Hardy space H 1 (L) related to L is defined by means of the maximal function of the semigroup T t , namely On the other hand, we define atomic Hardy spaces as follows. Suppose we have a space X with a doubling measure σ and a quasi-metric ρ. We call a function a an (ρ, σ )-atom if there exists a ball B = B ρ (x 0 , r ) := {x ∈ X | ρ(x, x 0 ) ≤ r } such that: supp a ⊆ B, a ∞ ≤ σ (B) − If such sequences exist, we define the norm f H 1 at (ρ,σ ) to be the infimum of ∞ k=1 |λ k | in the above presentations of f . Notice that in this paragraph we have changed the notation. This is because in the article we will use different metrics and measures.
Theorem 1 Suppose that (X, d, μ) is a space of homogeneous type and assume that for each x ∈ X the function Φ x is a bijection on (0, ∞). Let a nonnegative self-adjoint linear operator L on L 2 (X, μ) be given such that the semigroup T t = e −t L satisfies (1.3). Then there exist a constant C > 0 and a function ω on X, 0 < C −1 ≤ ω(x) ≤ C, such that the spaces H 1 (L) and H 1 at (d, ωμ) coincide and the corresponding norms are equivalent, Let us notice that the assumption on Φ x implies that μ(X ) = ∞ and μ is nonatomic. This will be used later on. By definition, we call a semigroup conservative if for t > 0 and x ∈ X .
It appears that Theorem 2 is equivalent to Theorem 1, see Sect. 3. Let us also emphasize that we do not require any regularity conditions on the kernels T t (x, y). However, it turns out that (1.3) implies Hölder-type estimates on T t (x, y), which are crucial for this paper. This will be discussed in Sect. 4 (see Theorem 5 and Corollary 14). In fact, Sect. 4 gives a short, independent, and self-contained proof of Hölder-type estimates of the heat kernel that satisfies (1.3), which can be interesting in its own right. Furthermore, although Theorems 1 and 2 are stated for the heat semigroup, we would like to emphasize that the same theorems can be proved for the Hardy space H 1 ( √ L) associated with the Poisson semigroup e −t √ L as well, see Theorem 4 and Remark 21.
The theory of the classical Hardy spaces on the Euclidean spaces R n has its origin in studying holomorphic functions of one variable in the upper half-plane. The reader is referred to the original works of Stein and Weiss [32], Burkholder et al. [6], Fefferman and Stein [15]. Very important contribution to the theory is the atomic decomposition of the elements of the H p spaces proved by Coifman [7] in the one-dimensional case and then by Latter [23] for H p (R n ) . The theory was then extended to the space of homogeneous type (see, e.g., [8,24,33]). For more information concerning the classical Hardy spaces, their characterizations and historical comments we refer the reader to Stein [31]. A very general approach to the theory of Hardy spaces associated with semigroups of linear operators satisfying the Davies-Gaffney estimates was introduced by Hofmann et al. [20] (see also [1,30]). Let us point out that the classical Hardy spaces can be thought as those associated with the classical heat semigroup e t . Finally we want to remark that the present paper takes motivation from [13,14], where the authors studied H 1 spaces associated with Schrödinger operators − + V on R n , n ≥ 3, with Green bounded potentials V ≥ 0.
In what follows C and c denote different constants that may depend on C d , q, C 0 , c 1 , c 2 .
The paper is organized as follows. In Sect. 2 we prove that estimate (1.3) implies the existence of an L-harmonic function ω such that 0 < C −1 ≤ ω(x) ≤ C. The equivalence of Theorems 1 and 2 is a consequence of the Doob transform, see Sect. 3. Then a proof of Theorem 2 is given in a few steps. First, in Sect. 4 we prove Hölder-type estimates for a conservative semigroup. Then, in Sect. 5 we introduce a new quasi-metric d and study its properties. In Sect. 6 we apply a theorem of Uchiyama on the space (X, d, μ) to complete the proof of Theorem 2. Finally, in Sect. 7 we provide some examples of semigroups that satisfy assumptions of Theorem 1.

Gaussian estimates and bounded harmonic functions
In this section we assume that the semigroup T t satisfies (1.3). Clearly, for all t > 0 and a.e. x ∈ X . For a positive integer n define Then, Recall that a metric space with the doubling condition is separable, then so is L 1 (X, μ). Using the Banach-Alaoglu theorem for L ∞ (X, μ) there exists a subsequence n k and ω ∈ L ∞ (X, μ), such that ω n k → ω in * -weak topology. Obviously, Our goal is to prove that T t ω(x) = ω(x) for t > 0. To this end, we write Since the last two limits tend to zero, as k → ∞, we obtain

Doob transform
In this section we work in a slightly more general scheme. Let (X, μ) be a σ -finite measure space and L be a self-adjoint operator on L 2 (X, μ). We assume that the strongly continuous semigroup T t = exp(− t L) admits a nonnegative integral kernel T t (x, y), so that Moreover, assume that there exists ω satisfying 0 < C −1 ≤ ω(x) ≤ C, that is L-harmonic. Namely, for every t > 0 one has for a.e. x ∈ X . Obviously, this implies that sup x∈X,t>0 T t (x, y) dμ(y) ≤ C. Define a new measure dν(x) = ω 2 (x) dμ(x) and a new kernel The semigroup K t given by is a strongly continuous semigroup of self-adjoint integral operators on L 2 (X, ν). The mapping L 2 (X, μ) f → ω −1 f ∈ L 2 (X, ν) is an isometric isomorphism with the inverse L 2 (X, ν) g → ωg ∈ L 2 (X, μ). Clearly, Moreover, the positive integral kernel K t (x, y) is conservative, that is, Thus the above change of measure and operators, which is called Doob's transform (see, e.g., [16]), conjugates the semigroup T t with the conservative semigroup K t .
It is clear that the operators K t are contractions on L 1 (X, ν). Consequently, K t is a strongly continuous semigroup of linear operators on L 1 (X, ν). To see this, it suffices to show that lim t→0 K t χ A − χ A L 1 (X,ν) = 0 for any measurable set A of finite measure. We have that which completes the proof of the strong continuity of K t on L 1 (X, ν).

Further, we easily see that the semigroup T t is strongly continuous on
and Now we discuss the equivalence of Theorems 1 and 2. Let T t and K t be the semigroups related by (3.1) with generators − L and − R, respectively. It easily follows from the Doob is an isomorphism of the spaces.
Assume that the space H 1 (R) admits an atomic decomposition with atoms that satisfy the cancelation condition with respect to the measure ν, that is every g ∈ H 1 (R) can be written as g = λ j a j with j |λ j | g H 1 (R) and a j are atoms with the property a j dν = 0. Then every f ∈ H 1 (L) admits atomic decomposition f = λ j b j with atoms b j that satisfy cancelation condition b j ω dμ = 0.

Hölder-type estimates on the semigroups
In this section we consider a conservative semigroup T t having an integral kernel T t (x, y) that satisfies upper and lower Gaussian bounds (1.3). Let be the kernels of the subordinate semigroup P t = e −t √ L .
Theorem 4 Let L be a nonnegative self-adjoint linear operator on L 2 (X, μ) such that the semigroup T t = e −t L satisfies (1.3) and (1.6). Then there is a constant α > 0 such that Theorem 5 Under the assumptions of Theorem 4 there are constants α, c > 0 such that Hölder regularity of semigroups satisfying Gaussian bounds was considered in various settings by many authors. We refer the reader to Grigor'yan [16], Hebisch and Saloff-Coste [19], Saloff-Coste [28], Gyrya and Saloff-Coste [17], Bernicot, Coulhon, and Frey [2], and references therein. Here we present a short alternative proof of (4.3). To this end we shall first prove some auxiliary propositions and then Theorem 4. Finally, at the end of this section, we shall make use of functional calculi deducing Theorem 5 from Theorem 4.

Proposition 6
For x , y ∈ X and t > 0 we have .
Estimates (4.5)-(4.7) give upper estimate. For the lower estimate, recall that d(x, y) ≥ t and observe that P t (x, y) .
Proof The corollary is a simple consequence of (4.4). For the proof one may consider two Proposition 8 There exists a constant γ ∈ (0, 1) such that the following statement holds: and for all t ≥ 2t 0 and all x ∈ X one has Proof The proof, which takes some ideas from [21], is an adapted version of the proof of [13,Prop. 3.1]. For the reader convenience we present the details. Let m = (a 1 + b 1 )/2 and Here we shall assume that the latter holds. The proof in the opposite case is similar. Denote Notice that d(w, z) ≤ t 0 ≤ s and, by Corollary 7, Since t s, Proposition 6 implies where κ ∈ (0, 2). Moreover, from (4.9) and the semigroup property we easily get Now (4.10) together with (4.11) gives Proof of Thorem 4 Having Corollary 7 and Proposition 8 proved, we follow arguments of [13] to obtain the theorem. By Corollary 7 there are b 1 > a 1 > 0 such that for y, z ∈ X and t > 0 satisfying d(y, z) < t we have for all x ∈ X . From Proposition 8 we deduce that there exists ω(y, z) such that lim t→∞ P t (x, y) P t (x, z) = ω(y, z) uniformly in x ∈ X. (4.12) It follows from (4.1) that X P t (x, y) dμ(y) = 1. Recall that P t (x, y) = P t (y, x). Using (4.12), Thus ω(y, z) = 1. Assume that d(y, z) < t. Let n ∈ N be such that d(y, z) ≤ t2 −n < 2d(y, z). Set t 0 = t2 −n . Clearly, d(y, z) ≤ t 0 and Observe that n log(t/d(y, z)). Applying Proposition 8 n-times we arrive at with α > 0 and the proof of Theorem 4 is finished.
Finally, we devote the remaining part of this section for deducing Theorem 5 from Theorem 4. This is done by using a functional calculi. First, we need some preparatory facts. Recall that q is a fixed constant satisfying (1.1). By W 2,σ (R) we denote the Sobolev space with the norm defines a bounded linear operator on L 2 (X, μ). Further we shall use the following lemma, whose proof based on finite speed propagation of the wave equation (see [9]) can be found in [11].
Proposition 10 Let β > q and κ > 1/2. There is a constant C > 0 such that for every Proof For y ∈ X and j ∈ Z set U 0 = B(y, where a k = C 0 C q 2 kq exp(−c 2 2 2k−2 ) is a rapidly decreasing sequence.
For t > 0 set

Proof of Theorem 5 By the spectral theorem
Using Theorem 4 together with Lemma 11 and Proposition 6, for (4.18) We claim that for d(y, z) ≤ √ t one has To prove the claim we consider two cases.
Remark 12 Let us remark that Lemma 11, which is crucial in our proof of Theorem 5, can be proved by applying functional calculus of Hebisch [18]. Thus, Theorem 5 can be also obtained without using the finite propagation speed of the solution of the wave equation As a consequence of (1.4) and Theorem 5 we get what follows.

Corollary 13 The function T t (x, y) is continuous on
As a direct consequence of Theorem 5 and Doob transform (see (3.1)) we get the following corollary. Notice that in Corollary 14 we do not assume that T t (x, y) is conservative.

Measures and distances
To prove Theorem 2 we introduce a new quasi-metric d on X , which is related to d and μ.
To this end, set where the infimum is taken over all closed balls B containing x and y (see, e.g., [8,25]). Denote In the lemma below we state some properties of d, which are known among specialists, and which we shall need latter on. Since their proofs are very short and it is difficult for us to indicate one reference which contains all of them, we provide the details for the convenience of the reader.

Lemma 15
The function d has the following properties: 0 (a) there exists C b > 0 such that for x, y ∈ X we have , d(x, y))).  (z, y) .
Moreover, if the measure μ has no atoms and μ(X ) = ∞, then: (c) the measure μ is regular with respect to d, namely for x ∈ X and r > 0, μ( B(x, r )) r ; (d) for x ∈ X and r > 0 there exists R > 0 such that
On the other hand, if x and y belong to a ball B = B(z, r ), then R ≤ 2r ; hence, B(x, R) ⊂ B(z, 3r ) and μ (B(x, R)) ≤ μ (B(z, 3r )) μ(B(z, r )). By taking the infimum over all balls B containing both x and y, we conclude that μ (B(x, R) (z, y). Then x, y ∈ B(z, r ). By using (a), we deduce that (z, y) .
Let us recall that in Theorems 1 and 2 we assume that Φ x is a bijection on (0, ∞). This obviously implies that μ(X ) = ∞ and that μ is nonatomic. As a consequence of (d) and (e) of Lemma 15 we obtain the following corollary.

Corollary 16
Suppose that μ has no atoms and μ(X ) = ∞. Then, the atomic Hardy spaces H 1 at (d, μ) and H 1 at ( d, μ) coincide and the corresponding atomic norms are equivalent.
We finish this section by Lemma 17, which is used latter on. Define A 2 := C b (C d 2 q ) 3 , where C d , q, and C b are as in (1.1) and (5.1). (X, d, μ) such that the function Φ x defined in (1.5) is a bijection on (0, ∞). Assume that x ∈ X, r, t > 0 are related by r = μ(B(x, √ t)) and satisfy:

Lemma 17 Suppose that we have a space of homogeneous type
Proof Suppose, toward a contradiction, that d(x, y) < √ t. From (5.1) we get μ(B(y, d(y, z))) ≤ C d 2 q C b d(y, z) < r, so the first inequality is proved. y). Thus, using (5.1), √ t)) = A −1 2 r and we come to a contradiction.

Proof of Theorem 2
In order to prove Theorem 2 we shall use a result of Uchiyama [33], which we state below in Theorem 18. Denote by (X, d, μ) the space X equipped with a quasi-metric d and a nonnegative measure μ, where μ(X ) = ∞. Assume moreover that μ ( B(x, r )) r, (6.1) where x ∈ X , r > 0 and B(x, r ) ⊆ X is a ball in the quasi-metric d. Let A 1 be a constant in the quasi-triangle inequality, i.e., y)), x, y, z ∈ X. (6.2) Additionally, we impose that there exist constants γ 1 , γ 2 , γ 3 > 0, A ≥ A 1 and a continuous function T (r, x, y) of variables x, y ∈ X and r > 0 such that for all x, y, z ∈ X and r > 0. As in [33], we consider the maximal function In what follows t, r > 0 and x ∈ X are always related by (6.3). Let us notice that from Corollary 13 and by the assumption that Φ x is a continuous bijection on (0, ∞) we have that T is a continuous function on (0, ∞) × X × X.
follows from the upper estimates for T t (x, y), more precisely by combining The latter estimate holds for any δ > 0. Thus (U2) is proved with any γ 1 > 0. To finish the proof we need Hölder-type estimate (U3). This is proved in Proposition 20 below.

Proposition 20
Let α be a constant as in Theorem 5. There exists A ≥ A 1 such that for δ > 0 we have if d(y, z) ≤ (r + d(x, y))/(4 A).
Proof Set A = max(A 1 , A 2 ), see (6.2) and Lemma ( d,μ) . Using once again the assumption on Φ x and the definition of T we easily observe that which finishes the proof of Theorem 2.
Let us recall that Theorem 1 follows from Theorem 2. This is elaborated at the end of Sect. 3.

Remark 21
Under the assumptions of Theorem 1 one can prove, by the same methods, that the Hardy space coincides with H 1 at (d, ω μ) and the corresponding norms are equivalent. To this end, one uses: Proposition 6, Doob's transform, Theorem 4, and Theorem 18 applied to the kernel P(r, x, y) = P t (x, y), where t = t (x, r ) is defined by the relation μ(B(x, t)) = r .

Examples
In this section we give examples of self-adjoint semigroups with the two-sided Gaussian bounds.

Laplace-Beltrami operators
Let (X, g) be a complete Riemannian manifold with the Riemannian distance d(x, y) and the Riemannian measure μ satisfying the doubling property and the Poincaré inequality where ∇ denotes the gradient on X . It is well known that the kernel of the heat semigroup generated by the Laplace-Beltrami operator satisfies two-sided Gaussian bounds (1.3) and Hölder estimates (4.3). For details and more information concerning the heat equation on Riemannian manifolds we refer the reader to [28] and references therein.

Schrödinger operators
On X = R d with the Euclidean metric and the Lebesgue measure we consider the Schrödinger operator where is the standard Laplacian and V is a locally integrable function.
It is well known (see, e.g., [29]) that for V ≥ 0, d ≥ 3, the semigroup T t = e −t L admits kernels T t (x, y) with the upper and lower Gaussian bounds (1.3) if and only if V is a Green bounded potential, that is, Hardy spaces associated with Schrödinger operators on R d satisfying (7.1) were studied in [13]. Actually, as we have already mentioned, this work motivated us to study the problem of H 1 spaces with the Gaussian bounds in the generality as in Theorems 1 and 2.
Our second example concerns Schrödinger operators L = − + V with nonpositive potentials. For d ≥ 3 fix p 1 , p 2 > 1 satisfying p 1 < d/2 < p 2 . Then there is a constant c( p 1 , p 2 , d) > 0 such that if V ≤ 0 and then the integral kernel T t (x, y) of the semigroup T t = e −t L exists and satisfies two-sided Gaussian bounds The result can be found in Zhang [35]. A slightly different proof of (7.2), based on bridges of the Gauss-Weierstrass semigroup, can be obtained by using Lemma 1.2 together with Proposition 2.2 of [4].

Bessel-Schrödinger operator
Let α > − 1 and consider X = (0, ∞) and dμ(x) = x α dx. Notice that the space (X, μ) with the Euclidean metric d e (x, y) = |x − y| is a space of homogeneous type. We consider the classical Bessel operator which is self-adjoint positive on L 2 (X, μ), and the associated semigroup of linear operators S t = e −tB . It is well known that S t is conservative and has the integral kernel see, e.g., [5,Chapter 6]. Here I (α−1)/2 denotes the modified Bessel function of the first kind, see, e.g., [34]. The kernel S t (x, y) satisfies two-sided Gaussian estimates For a short proof of (7.4) see [12, proof of Lemma 4.2]. Therefore, using Theorem 2 we obtain atomic characterization of H 1 (B) that was previously proved in [3].
In this subsection we consider perturbations of B of the form where a potential V is nonnegative and locally integrable. More precisely, on L 2 (X, μ) we define the quadratic form with the domain where cl(A) stands for the closure of the set A in the norm f L 2 (X,μ) + f L 2 (X,μ) . The form Q is positive and closed. Thus, it corresponds to the unique self-adjoint operator L on L 2 (X, μ) with the domain Dom (L) = f ∈ Dom(Q) ∃h ∈ L 2 (X, μ) ∀g ∈ Dom(Q) Q( f, g) = h g dμ .
By definition, L f = h when f, h are related as above.
Let T t = exp (−t L) be the semigroup generated by − L. The Feynman-Kac formula states that where b s is the Bessel process on (0, ∞) associated with S t . Using (7.5) one gets that the semigroup T t has form (1.2), where 0 ≤ T t (x, y) ≤ S t (x, y). (7.6) Therefore the upper Gaussian estimates for T t (x, y) follows simply from (7.4) for any locally integrable V ≥ 0. On the other hand, the relation between S t (x, y) and T t (x, y) is given by the perturbation formula S t (x, y) = T t (x, y) + t 0 X S t−s (x, z)V (z)T s (z, y) dμ(z) ds (7.7) From now on we consider only α > 1. We are interested in proving the lower Gaussian estimates, but this can be done only for some potentials V . For other potentials Hardy spaces may have a local character, (see, e.g., [22]). Let This can be easily obtained from (7.3) and well-known asymptotics for the Bessel function I (α−1)/2 , see also [26,Sect. 2]. In Lemmas 22 and 23 we prove that under assumption (7.8) lower Gaussian estimates (1.3) hold for T t (x, y). The estimates are proved in a similar way as in the case of the Schrödinger operator on the Euclidean space. For the convenience of the reader we provide the details. Obviously, the space (X, d e , μ) satisfies the assumptions of Theorem 1. Since we have the two-sided Gaussian estimates for T t (see Lemma 22,(7.6), and (7.4)) we obtain the following corollary.