Schr\"odinger operators with reverse H\"older class potentials in the Dunkl setting and their Hardy spaces

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. Let $L=-\Delta +V$, $V\geq 0$, be the Dunkl--Schr\"odinger operator on $\mathbb R^N$. Assume that there exists $q>\max(1,\frac{\mathbf{N}}{2})$ such that $V$ belongs to the reverse H\"older class $\text{RH}^q(dw)$. We prove the Fefferman--Phong inequality for $L$. As an application, we conclude that the Hardy space $H^1_{L}$, which is originally defined by means of the maximal function associated with the semigroup $e^{tL}$, admits an atomic decomposition with local atoms in the sense of Goldberg, where their localization are adapted to $V$.

On R N , N ≥ 3, let us consider the Schrödinger differential operator where V ∈ L 2 loc (R N , dx) is a non-negative potential which V belongs to the reverse Hölder class B q with q > N 2 , i.e. the inequality holds for every ball B in R N . Define the auxiliary function m as follows: The integral defining the function m was introduced by Ch. Fefferman (see [20, p. 146, the assumption of the main lemma]). The function is then used in the well-known Fefferman-Phong inequality ( [20, p. 146], see also Shen [35], [36,Lemma 1.9]) which we state below. Theorem 1.1 (Fefferman-Phong inequality). There is a constant C > 0 such that for all f ∈ C 1 c (R N ) we have The proof of (1.4) is based on the usage of the fact that V ∈ A p for some p > 1 and the Poincaré inequality (1.5) 1 |B(x, r)| B(x, R) |f (y) − f B(x,r) | 2 dy ≤ Cr 2 |B(x, r)| B(x,r) |∇f (y)| 2 dx.
The Fefferman-Phong inequality and the function m itself are very useful tools which are used in analysis regarding the operator L , e.g., in investigating behavior of its eigenvalues [20], estimating of the fundamental solution of the equation L u = 0 ( [36,Theorem 2.7]) and studying L p -bounds of the operators ∇L iγ , ∇L −1/2 , ∇L −1 ∇, ∇ 2 L −1 (see Theorems 0.3, 0.4, 0.5, 0.8 in [36]). It was proved in [15] (see also [16,Theorem 2.11,Proposition 2.16]) that the integral kernel k t (x, y) of the Schrödinger semigroup e −tL behaves like the classical heat semigroup for 0 < t < m(x) −2 , while for t > m(x) −2 has essentially faster decay. These observations allowed Dziubaski and Zienkiewicz [15] to study the Hardy spaces associated with L and prove a local character of atoms (see also [17,18]).
The aim of this article is to prove the Fefferman-Phong inequality for Dunkl-Schrödinger operators and study its applications for describing behavior of the corresponding Dunkl-Schrödinger semigroups and their Hardy spaces H 1 .
The Dunkl theory is a generalization of the Euclidean Fourier analysis. It started with the seminal article [10] and developed extensively afterwards (see e.g. [8], [9], [11], [12], [21], [29], [30], [31], [38], [39]). We refer the reader to lecture notes [32] and [33] for more information and references. We fix a normalized root system R in R N and a multiplicity function k ≥ 0 (see Section 2). For ξ ∈ R N , N ≥ 1, the Dunkl operators T ξ are the following k-deformations of the directional derivatives ∂ ξ by a difference operator: where σ α is the reflection on R N with respect to the hyperspace orthogonal to α. The Dunkl operators are generalizations of the partial derivatives (in fact, they are ordinary partial derivatives for k ≡ 0), however they are non-local operators. Therefore, in order to obtain counterparts of classical Euclidean harmonic analysis results in the Dunkl setting, we have to deal with both: local and non-local parts of the operators under consideration. For instance, the question what would be a good counterpart of Poincare's inequality (1.5) is true in the rational Dunkl setting seems to be an interesting problem. Recently various different versions of (1.5) were proved (see [27], [40], [41]). The analysis is more complicated if we compose such operators. Furthermore, there are other technical problems and open questions in Dunkl theory. One of them is the lack of knowledge about boundendess of the so called Dunkl translations τ x on L p (dw)-spaces for p = 2. It makes analysis of convolution operators more complicated and delicate.
In the present paper we consider the Dunkl-Schrödinger operator where V ∈ L 2 loc (dw) is non-negative potential and ∆ = N j=1 T 2 e j is the Dunkl Laplacian. Such operators were recently studied by Amri and Hammi in [2] and [3]. An example of such operator is the so called Dunkl harmonic oscillator −∆ + x 2 , whose properties are better understood (see [1], [24], [28], [29], and [33]). Let N be the homogeneous dimension (see (2.2)). We shall assume that V satisfies an analogue of (1.2) with q > max(1, N 2 ) (see Subsection 2.3 for details). In the current paper we prove that a counterpart of the Fefferman-Phong inequality (1.4) is true in the Dunkl setting, which is one of our main results (see Theorem 5.1). The main difficulty which one faces trying to prove Theorem 5.1 is the lack of knowledge about the Poincare's inequality, which is the main ingredient of the proof in the classical case. Our idea of the proof is to mix the methods which are known from the theory of non-local operator (see [18, proof of Theorem 9.4]), a version of pseudo-Poincare's inequality (which is very close to that in [40,Section 5]), together with a careful analysis of properties of the counterpart of the function m compared to the structure of the Dunkl operator. The analysis of properties of the counterpart of the function m (see (4.1)) and the proof of Theorem 5.1 are the goals of Part 1 of the paper. Part 2 is devoted to the application of the Fefferman-Phong inequality to prove the characterization of the Hardy space H 1 L associated with the Dunkl-Schrödinger operator by the maximal function associated with the semigroup generated by −∆ + V and by a special atomic decomposition -see Section 6 for details. This application is inspired by [15] (see also [14] and [17]). The atoms for H 1 L have the structure of local atoms in the sense of Goldberg [23] with localization adapted to the behavior of the function m. So, in order to obtain our result, we need characterizations of a family local Hardy spaces in the Dunkl setting proved in [24, Section 5].

Preliminaries
2.1. The basic definitions of the Dunkl theory. In this section we present basic facts concerning the theory of the Dunkl operators. For details we refer the reader to [10], [32], and [33].
We consider the Euclidean space R N with the scalar product x, y = N j=1 x j y j , where x = (x 1 , ..., x N ), y = (y 1 , ..., y N ), and the norm x 2 = x, x . For a nonzero vector α ∈ R N , the reflection σ α with respect to the hyperplane α ⊥ orthogonal to α is given by In this paper we fix a normalized root system in R N , that is, a finite set R ⊂ R N \ {0} such that R ∩ αR = {±α}, σ α (R) = R, and α = √ 2 for all α ∈ R. The finite group G generated by the reflections σ α ∈ R is called the Weyl group (reflection group) of the root system. A multiplicity function is a G-invariant function k : R → C which will be fixed and ≥ 0 throughout this paper. Let be the associated measure in R N , where, here and subsequently, dx stands for the Lebesgue measure in R N . We denote by the homogeneous dimension of the system. Clearly, Observe that there is a constant C > 0 such that Moreover, there exists a constant C ≥ 1 such that, for every x ∈ R N and for every r 2 ≥ r 1 > 0, For a measurable subset A of R N we define Clearly, by (2.3), for all x ∈ R N and r > 0 we get For ξ ∈ R N , the Dunkl operators T ξ are the following k-deformations of the directional derivatives ∂ ξ by a difference operator: The Dunkl operators T ξ , which were introduced in [10], commute and are skew-symmetric with respect to the G-invariant measure dw.
For fixed y ∈ R N the Dunkl kernel E(x, y) is the unique analytic solution to the system The function E(x, y), which generalizes the exponential function e x,y , has the unique extension to a holomorphic function on C N × C N . Moreover, it satisfies E(x, y) = E(y, x) for all x, y ∈ C N .
Let {e j } 1≤j≤N denote the canonical orthonormal basis in R N and let T j = T e j . In our further consideration we shall need the following lemma.
There is a constant C > 0 such that for all x, ξ ∈ R N we have The Dunkl transform originally defined for f ∈ L 1 (dw), is an isometry on L 2 (dw), i.e., and preserves the Schwartz class of functions S(R N ) (see [7]). Its inverse F −1 has the form The Dunkl translation τ x f of a function f ∈ S(R N ) by x ∈ R N is defined by It is a contraction on L 2 (dw), however it is an open problem if the Dunkl translations are bounded operators on L p (dw) for p = 2.
The Dunkl convolution f * g of two reasonable functions (for instance Schwartz functions) is defined by or, equivalently, by where, here and subsequently, g(x, y) = τ x g(−y).

2.2.
Dunkl Laplacian and Dunkl heat semigroup. The Dunkl Laplacian associated with R and k is the differential-difference operator ∆ = N j=1 T 2 j , which acts on C 2 (R N )functions by Obviously, F (∆f )(ξ) = − ξ 2 F f (ξ). The operator ∆ is essentially self-adjoint on L 2 (dw) (see for instance [2, Theorem 3.1]) and generates the semigroup H t of linear self-adjoint contractions on L 2 (dw). The semigroup has the form where the heat kernel is a C ∞ -function of all variables x, y ∈ R N , t > 0, and satisfies We shall need the following estimates for h t (x, y) -the proof can be found in [ Theorem 2.3 imply the following Lemma (see [13,Corollary 3.5]).
Lemma 2.4. Suppose that ϕ ∈ C ∞ c (R N ) is radial and supported by the unit ball. Then there is C > 0 such that for all x, y ∈ R N and t > 0 we have

Dunkl-Schrödinger operator and semigroup.
We present the main tools on Dunkl-Schrödinger operators, which are discussed in [2] (see also [3]) in details. Let V ≥ 0 be a measurable function such that V ∈ L 2 loc (dw). We consider the following operator on the Hilbert space L 2 (dw): We call this operator the Dunkl-Schrödinger operator. Let us define the quadratic form with domain The quadratic form is densely defined and closed (see [2,Lemma 4.1]), so there exists a unique positive self-adjoint operator L such that moreover, where L 1/2 is a unique self-adjoint operator such that (L 1/2 ) 2 = L. It was proved in [2,Theorem 4.6], that L is essentially self-adjoint on C ∞ 0 (R N ) and L is its closure. Consequently, L generates the semigroup of self-adjoint contractions on L 2 (dw). The semigroup has the form (see [2,Theorem 4.8]) where k t (x, y) is the integral kernel which satisfies Part 1. Fefferman-Phong inequality

Potential satisfying reverse Hlder inequality
In this part, we assume that q > max(1, N 2 ) and V belongs to the reverse Hölder class RH q (dw), that is, there is a constant C RH > 0 such that For any Lebesque measurable set A we define Our goal is to study the properties of the measure µ defined above. The proofs of the results in this section are standard and they are based on [ where, here and subsequently, 1 q + 1 q ′ = 1. Proof. Applying Hölder's inequality, then the reverse Hölder inequality (3.1), we get There is a constant 0 < γ < 1 such that for all x ∈ R N and r > 0 we have Proof. Thanks to (2.1) we obtain where v N is the Euclidean measure of the unit N-dimensional ball. Consequently, thanks to (2.3), we have where the constant C > 0 is independent of x and r. The claim follows easily.
Thanks to Lemma 3.2 for 1 − γ small enough we have There is n ∈ N such that γ n < 1/2. Applying (3.5) n times we get the claim.
As the consequence of the doubling property of µ, we obtain the following corollary.
Lemma 3.5. There are 0 < γ, δ < 1 such that for all cubes Q ⊂ R N and measurable sets E ⊆ Q the following implication is true: . Then by (3.7) we have the implication We will need the following classical result from theory of A p weights (see [22,Corollary 7

.2.4]).
Proposition 3.6. Let v be the weight and let ν be a doubling measure on R N . Suppose that there are 0 < γ, δ < 1 such that whenever E is a ν-measurable subset of a cube Q. Then there are constants C, η > 0 such that for every cube Q in R N we have Proposition 3.7. There is a constant C > 0 and p > 1 such that for every cube Q in R N we have Proof. Note that (3.8) is equivalent to Hence, applying Proposition 3.6 to v = V −1 and ν = µ (the assumption that ν is doubling is satisfied thanks to Lemma 3.3) we get that there are C, η > 0 such that (3.14) Finally, it can be checked that (3.14) is equivalent to (3.12) with p = 1 + 1 η . The reverse Hölder inequality (3.1) has the following consequence (see [36,Lemma 1.2]), which will be used in the next section many times.
Lemma 3.8. There is a constant C > 0 such that for all x ∈ R N and 0 < r 1 < r 2 < ∞ we have Proof. Thanks to Hölder's inequality and the reverse Hölder inequality (3.1), we get Finally, the claim follows by (2.5).

Definition and growth properties of
Thanks to Lemma 3.8, for all x ∈ R N (and V ≡ 0) we have so the function m is well-defined. The next lemma is an adaptation of [36, Lemma 1.4].
Lemma 4.1. There are constants C, κ > 0 such that for all x, y ∈ R N we have Proof of (4.3). By the doubling property of w and µ we have w(B(x, r)) ∼ w(B(y, r)) and µ(B(x, r)) ∼ µ(B(y, r)) if r ≥ x − y . So, by Lemma 3.8, for any r < m(x) −1 we have where in the last inequality we have used the definition of m. Note that (4.6) implies that for so the inequality m(y) ≤ Cm(x) follows. Now we turn to the proof of m(x) ≤ Cm(y). For r > 2m(x) −1 , thanks to the doubling property of µ and w, then Lemma 3.8, we write where in the last inequality we have used the definition of m(x). Taking so, thanks to definition of m (see (4.1)), we are done.
Proof of (4.4). We may assume x − y m(x) ≥ 1, otherwise the claim follows by (4.3). Let r = m(x) −1 and let j ≥ 1, j ∈ Z, be such that Let 0 < r 1 < r. Thanks to Lemma 3.8, then the doubling property of µ and w together with (2.5), we have where C µ is the doubling constant for µ (see Lemma 3.3) and we have used (2.5) and the definition of m in the last line. Therefore, there is a constant C 1 > 1 independent of x, y ∈ R N and r > r 1 > 0 such that if r 1 ≤ rC −j 1 , then Consequently, by the definition of m(y) we have which lead us to Proof of (4.5). We may assume that x−y ≥ m(y) −1 , otherwise the claim follows by (4.3).
Associated collection of cubes Q. For a cube Q ⊂ R N , here and subsequently, let d(Q) denote the side-length of cube Q. We denote by Q * the cube with the same center as Q such that d(Q * ) = 2d(Q). We define a collection of dyadic cubes Q associated with the potential V by the following stopping-time condition: Thanks to the doubling property of w and µ together with (4.2) we see that the collection Q is well-defined and it forms a covering of R N by disjoint dyadic cubes. We list below simple facts about the collection Q, which are consequences of properties of w, µ and m(x).
Proof. It is an easy consequence of the doubling property of µ. Namely, let Q be the parent of cube Q ∈ Q. As the consequence of the stopping-time condition (4.7), we get There is a constant C > 0 such that for any Q ∈ Q and x ∈ Q * * * * we have Proof. The proof is essentially the same as the proof of (4.3). We provide details. Note that Q * * * * ⊆ B(x, 10 2 d(Q)) for x ∈ Q * * * * . Therefore, by the doubling property of µ and w together with (4.8) we have Consequently, for r < 10 2 d(Q), by Lemma 3.8 with r 1 = r and r 2 = 10 2 d(Q), we have By the same argument as in the proof of (4.3) we have m(x) ≤ Cd(Q) −1 . Similarly, for r > 10 2 d(Q), we have so repeating the argument from the proof of (4.3) we have Cm(x) ≥ d(Q) −1 .

Fefferman-Phong inequality
The goal of this section is the prove Fefferman-Phong inequality in the rational Dunkl setting. This result is crucial in the proof of condition (D) (see Section 6) for potential satisfying (3.1). The result for k ≡ 0 is due to C. Feffermann and D.H. Phong [20] (see also [36,Lemma 1.9]). The proof is inspired by the proof from [18,Theorem 9.4].
Theorem 5.1 (Fefferman-Phong type inequality). There is a constant C > 0 such that for all f ∈ D(Q) we have We need some lemmas before providing the proof of Theorem 5.1.

Lemma 5.2.
There are constants C, η > 0 such that for all Q ∈ Q and ε > 0 we have Proof. Let p > 1 be the number from (3.12). By the definition of E ε we write Thanks to (4.8) and the doubling property of w we have Consequently, applying (5.3) and (5.4) together with (3.12) we get Lemma 5.3. For all j ∈ {1, 2, . . . , N}, g ∈ C ∞ c (R N ), and f ∈ L 2 (dw) such that its weak Dunkl derivative T j f is in L 2 (dw) we have T j (f g) ∈ L 2 (dw). Moreover, Proof. It is a standard fact, but for the convenience of reader we provide the proof. Let us assume first that f ∈ C 1 (R N ). By the definition of T j (see (2.8)) we have In order to obtain the general case, let us take ψ ∈ C ∞ c (R N ). By the definition of T j (f g) and (5.6) we have Let {φ Q } Q∈Q be a smooth resolution of identity associated with Q, that means the collection of C ∞ -functions on R N such that supp φ Q ⊆ Q * , 0 ≤ φ Q (x) ≤ 1, Proof. This is the standard fact -we write so the claim is a consequence of (5.7).
Lemma 5.5. There is a constant C > 0 such that for all j ∈ {1, . . . , N}, f ∈ L 2 (dw) such that its weak Dunkl derivative T j f is in L 2 (dw), and Q ∈ Q we have (let us remind that O(Q * ) denotes the orbit of cube Q * , see (2.6)).
Proof. By Lemma 5.3 we have Thanks to the property that supp φ Q ⊆ Q * , (5.7), and Fact 4.3 we have . . . =: I 1 + I 2 for fixed α ∈ R. We consider I 1 first. Let us denote Consequently, by Lemma 5.4, we get In order to estimate I 2 , thanks to property 0 ≤ |φ Q (x) − φ Q (σ α (x))| ≤ 2, we write Note that, thanks to (4.5), for which ends the proof.
Proof of Theorem 5.1. Suppose first that Let ψ ∈ C ∞ c (R N ) be a radial non-negative function such that R N ψ dw = 1 and supp ψ ⊆ B(0, 1), and let A > 1 be a large constant (it will be chosen later). For Q ∈ Q we define the following scaled version of ψ:

It follows by Corollary 2.2 that
consequently, by Plancherel's theorem (see (2.11)) and Lemma 5.5, (5.10) The first inequality in (5.10) can be thought as a counterpart of the Poincaré inequality (cf. (1.5)). Furthermore, by Lemma 2.4 and the fact that by the doubling property of w we have w(B(x, d(Q))) ∼ w(Q) for all x ∈ Q * , we obtain Let ε > 0 (it will be chosen later) and let E ε be defined as in (5.2). We write By the Cauchy-Schwarz inequality and Lemma 5.2 we have Next, by the definition of E ε (see (5.2)) and Cauchy-Schwarz inequality we get (5.14) Combining (5.11), (5.12), (5.13), and (5.14) we get (5.15) Consequently, by (5.10) and (5.15) we get

(5.16)
If we divide both sides by d(Q) 2 and then use Fact 4.3, we get
For f ∈ D(Q) and n ∈ N we define f n (x) = f (x)η(x/n). Note that by Lemma 5.5 we have f n ∈ D(Q). Moreover, thanks to the fact that f ∈ L 2 (dw) and (4.4), the condition (5.9) is satisfied for f n . Therefore, by (5.1) for f n , we get Clearly, Moreover, thanks to the definition of η, the fact that f, T j f ∈ L 2 (dw), and Lemma 5.3, we have  [37]). In the seminal paper of Fefferman and Stein [19] the spaces H p were characterized by means of real analysis. One of the possible characterization assets that a tempered distribution f belongs to the H p (R N ), 0 < p < ∞, if and only if the maximal function sup t>0 |h t * f (x)| belongs to L p (R N ), where h t is the heat kernel of the semigroup e t∆ eucl . An important contribution to the theory is the atomic decomposition proved by Coifman [6] for N = 1 and Latter [26] in higher dimensions, which says that every element of H p can be written as an (infinite) combination of special simple functions called atoms. These characterizations led to generalizations of the Hardy spaces on spaces of homogeneous type, in particular, to H p spaces associated with semigroups of linear operators. In [5] (see also [4], [13]) a theory of Hardy spaces H 1 in the rational Dunkl setting parallel to the classical one was developed. The purpose of the remaining part of the paper is to study an H 1 L space related to L. Our starting definition is that by means of the maximal function for the semigroup e −tL . Then we shall prove that the space admits a special atomic decomposition. This result generalizes one of [24] where H 1 L for the Dunkl harmonic oscillator −∆ + x 2 was consider.
In [25] the authors provided a general approach to the theory of Hardy spaces associated with semigroups satisfying Davies-Gaffney estimates and in particular Gaussian bounds. We want to emphasize that the integral kernel for the Dunkl-Laplace semigroup does not satisfy the Gaussian bounds. Therefore the methods developed in [25] cannot be directly applied.
6.2. Hardy spaces associated with L. Let us introduce the notion of the Hardy space associated with the operator L.
We say that f belongs to the Hardy space H 1 L associated with operator L if and only if belongs to L 1 (dw). The norm in the space is given by . Let Q be a collection of closed cubes with parallel sides whose interiors are disjoint such that Q∈Q Q = R N . Let us remind that d(Q) denotes the side-length of cube Q and we denote by Q * the cube with the same center as Q such that d(Q * ) = 2d(Q). Assume that this family satisfies the following finite overlapping condition: . We define the atomic Hardy space associated with the collection Q (see [18]). The atomic Hardy space H 1,at Q associated with the collection Q is the space of functions f ∈ L 1 (dw) which admit a representation of the form where c j ∈ C and a j are atoms for the Hardy space c j a j (x) and a j are H 1,at Q atoms .
Inspired by [18], we consider the following two additional conditions on Q and V : where c > 0 is the constant from Theorem 2.3, The next theorem is one of the main result of the paper. We provide its proof in Section 9.
Theorem 6.3. Assume that the conditions (F), (D), and (K) hold for V and Q. There is a constant C > 0 such that for all f ∈ L 1 (dw) we have It can be checked that the conditions (F), (D), and (K) hold for potentials V satisfying the reverse Hölder inequality with q > N 2 and the associated collection of cubes (4.7), so we obtain the following corollary.
Corollary 6.4. Assume that the potential V satisfies the reverse Hölder inequality (3.1). There is a constant C > 0 such that for all f ∈ L 1 (dw) we have where Q is the collection of cubes defined in (4.7).
Corollary 6.4 is proved in Section 10, where the conditions (F), (D), and (K) are verified.

Local Hardy spaces
The following two definitions are inspired by [23] (see also [24]).
Definition 7.1. Let T > 0 and f ∈ L 1 (dw). We say that f belongs to the local Hardy space H 1 loc,T associated with the Dunkl Laplacian if and only if belongs to L 1 (dw). The norm in the space is given by Definition 7.2. Let T > 0. A function a(x) is called an atom for the local Hardy space H 1,at loc,T if (A) supp a ⊆ B(x, r) for some x ∈ R N and r > 0, (B) sup y∈R N |a(y)| ≤ w(B(x, r)) −1 , (C) If r < T , then R N a(x) dw(x) = 0.
A function f belongs to the local Hardy space H 1,at loc,T if there are c j ∈ C and atoms a j for H 1,at loc,T such that ∞ j=1 |c j | < ∞, In this case, set f H 1,at where the infimum is taken over all representations (7.3).
The following proposition was proved in [24] and its proof follows the pattern from [23].

Auxiliary lemmas
Lemmas in this section are inspired by [18]. It turns out that the presence of the factor " in the estimate from Theorem 2.3 is crucial in the proof of Theorem 6.3 and its proper usage is the main difficulty and difference between the proofs here and in [18]. Let {φ Q } Q∈Q be the resolution of identity associated with the collection Q, which satisfies the analogous properties to that from Section 5 (see e.g. (5.7)).
Lemma 8.1. There is a constant C > 0 such that for all Q ∈ Q and f ∈ L 1 (dw) we have Proof. We will prove just (8.1), thanks to (2.19) the proof of (8.2) is the same. We have Thanks to Theorem 2.3 and the fact that for x ∈ R N \ Q * * and y ∈ Q * we have x − y ≥ d(Q), so we obtain The latest estimate together with (8.3) implies the claim.
There is a constant C > 0 such that for every Q ∈ Q and f ∈ L 1 (dw) we have For Q ∈ Q we define There is a constant C > 0 such that for every Q ∈ Q and f ∈ L 1 (R N ) we have Consequently, by the Fubini theorem, Lemma 8.4. Assume that Q and V satisfy condition (D). Then there is a constant C > 0 such that for all f ∈ L 1 (dw) we have Proof. Let us denote the left-hand side of (8.8) by S. Then by property (F) we get Therefore, integrating over the x-variable we obtain Consequently, by assumption (D), we get Lemma 8.5. For all f ∈ L 1 (dw) we have Proof. The lemma is well-known. We provide the proof for the sake of completeness. By perturbation formula we have Integrating (8.10) with respect to the x-variable, using the Fubini theorem and the fact that for all v > 0 we have R N h v (x, y) dw(x) = 1 (see (2.13)), we get Letting t → ∞ we obtain the lemma.
Lemma 8.6. Assume that Q and V satisfy (K). There is a constant C > 0 such that for all Q ∈ Q and f ∈ L 1 (dw) we have Proof. Thanks to (8.1) and (8.2) it is enough to estimate By perturbation formula we write where V 1 + V 2 = V and V 1 = V χ Q * * * . In order to estimate the term with V 2 , we use Theorem 2.3 and the fact that for y ∈ R N \ Q * * * and x ∈ Q * * we have x − y ≥ d(Q), so, for x ∈ Q * * we get Therefore, by the Fubini theorem and (8.9) we obtain In order to estimate the term containing V 1 in (8.12), we write Clearly, by Theorem 2.3 and the Fubini theorem, we get Similarly, we write then by changing of variables we have  ≤ e −cd(x,y) 2 /(2s) e −cd(x,z) 2 /(2 −j+1 d(Q) 2 ) , so (8.14) and the doubling property of w lead us to Furthermore, by assumption (K), we get (8.15) Then, by Lemma 8.3 and (8.8) we get Hence, we have obtained therefore, by Proposition 7.3 we get where a j,Q are atoms of local we have a H 1 L ≤ C. Suppose that a(x) is associated with a cube Q ∈ Q. We write Thanks to (F) and the fact that supp a ⊆ Q * * * * , there is a number M > 0 independent of Q such that in (9.1) there are at most M nonzero summands with d(Q ′ ) ∼ d(Q). Let ℓ ≥ 0 be the smallest positive integer such that d(Q ′ ) ≥ 2 −ℓ/2 d(Q) for all such a cubes in (9.1). Clearly, thanks to (F), ℓ is independent of a and Q ∈ Q. We write |K t a| L 1 (dw) =: I 1 + I 2 . Further, Thanks to the fact that atom a is, by definition, an atom for H 1 loc,d(Q) , we have sup Thanks to (8.11) and (9.1), we get In order to estimate I 2 , we repeat the argument presented in the proof of (8.8). We provide details. We write By the semigroup property and Theorem 2.3 together with (2.19) for e −cd(x,z) 2 /(2 j+1 d(Q) 2 ) k 2 j−1 d(Q) 2 (z, y) dw(z)|a(y)| dw(y).
Therefore, integrating over the x-variable, we obtain Consequently, by condition (D) and (9.2), we get Lemma 10.1. There is a constant C > 0 such that for all y ∈ R N and t > 0 we have (10.1) Lk t (·, y), k t (·, y) ≤ C tw(B(y, √ t)) .
Proof. Thanks to the fact that operator L is positive and self-adjoint, we have that the semigroup {K t } t≥0 is analytic on L 2 (dw), so the operator LK t/2 is bounded on L 2 (dw) for all t > 0. Therefore, by the semigroup property and the definition of L (here L x denotes the action of L with respect to x-variable) we have (10.2) L x k t (x, y) = L x R N k t/2 (x, z)k t/2 (z, y) dw(z) = ((LK t/2 )k t/2 (·, y))(x).
The proof is finished (we set δ = 1 − N 2q ). Acknowledgment. The author would like to thank Jacek Dziubański for careful reading of the text and his helpful comments and suggestions.