Area Integral Characterization of Hardy space H 1 L related to Degenerate Schrödinger Operators

As a suitable substitute of Lebesgue spaces Lp(Rn), the classical Hardy space H1(Rn) plays an important role in various elds of analysis and partial di erential equations. Let ∆ be the Laplace operator onRn. It follows from [1] that H1(Rn) can be characterized by the maximal function supt>0 |e−t∆ f (x)|. In fact, H1(Rn) can be seen as a Hardy space associated with the operator −∆. We use L to denote a general di erential operators, such as Schrödinger operators with nonnegative potential or second order elliptic self-adjoint operators in divergence form and so on. The Hardy spaces associated with L become one of the most concerned problems of the harmonic analysis. Readers can refer to [2–10] and the references therein. In recent years, [3] and [10] study the Hardy spaces associated with the degenerate Schrödinger operators. As we know, the area integral is an important tool to characterize Hardy spaces. In [11], Fe erman and Stein obtain the area integral characterization of the classical Hardy spacesHp(Rn). From then on, such characterization was extended to other settings. We refer the reader to [4, 5, 12] and the references therein. Let L be a degenerate Schrödinger operator L onRn. In this paper, motivated by the above literatures, wewill prove that theHardy space associatedwith L also has such a characterization. The degenerate Schrödinger operator L on Rn is de ned as follows. Lf (x) = − 1 ω(x) ∑


Introduction
As a suitable substitute of Lebesgue spaces L p (R n ), the classical Hardy space H (R n ) plays an important role in various elds of analysis and partial di erential equations. Let ∆ be the Laplace operator on R n . It follows from [1] that H (R n ) can be characterized by the maximal function sup t> |e −t∆ f (x)|. In fact, H (R n ) can be seen as a Hardy space associated with the operator −∆. We use L to denote a general di erential operators, such as Schrödinger operators with nonnegative potential or second order elliptic self-adjoint operators in divergence form and so on. The Hardy spaces associated with L become one of the most concerned problems of the harmonic analysis. Readers can refer to [2][3][4][5][6][7][8][9][10] and the references therein. In recent years, [3] and [10] study the Hardy spaces associated with the degenerate Schrödinger operators.
As we know, the area integral is an important tool to characterize Hardy spaces. In [11], Fe erman and Stein obtain the area integral characterization of the classical Hardy spaces H p (R n ). From then on, such characterization was extended to other settings. We refer the reader to [4,5,12] and the references therein. Let L be a degenerate Schrödinger operator L on R n . In this paper, motivated by the above literatures, we will prove that the Hardy space associated with L also has such a characterization. The degenerate Schrödinger operator L on R n is de ned as follows. where (a ij (x)) i,j is a real symmetric matrix satisfying with ω being a nonnegative weight from the Muckenhoupt class A , and V ≥ belonging to a reverse Hölder class with respect to the measure dµ = ω(x)dx (see Section 2 for their de nitions). Denote by E(f , g) the Dirichlet form associated with L, that is, The operator −L is the in nitesimal generator of the heat semigroup {T t } t> of self-adjoint linear operators on L (dµ). Let K t (x, y) be the integral kernels of {T t }, i.e., In [3], Dziubański introduces the following Hardy space associated with the operator L.
De nition 1.1. We say a function f in L (dµ) belongs to H L (dµ) if the heat maximal function Mf is in L (dµ), where Mf (x) = sup The H L -norm of f is de ned by f H L = Mf L (dµ) .
Dziubański in [3] has given the following atomic decomposition of H L (dµ).

De nition 1.2. A function a is called an
where ρ(x) is the auxiliary function that de ned in ( . ). The atomic norm · H L −atom is de ned by where the in mum is taken over all decompositions f = j λ j a j , where {a j } is a sequence of H L -atoms and {λ j } is a sequence of scalars.
The rst main result of this paper can be stated as follows. Let S L h be the area integral associated with the heat semigroup generated by L, see (3.1) below. We have the following area integral characterization of H L (dµ): Assume that ω ∈ (RD)ν Dγ A , < ν ≤ γ, and V ∈ Bq,µ with q > γ/ . Then see Theorems 3.9 & 3.14 for the details. Following the classical case, we need a reproducing formula related to L in the distributional sense to divide the elements of H L (dµ) into atoms. As the dual space of H L (dµ) (cf. De nition 3.10), the BMO type spaces BMO L (dµ) are introduced by Yang-Yang-Zhou in [13,14]. Suppose f ∈ (BMO L (dµ)) * , we obtain the desired reproducing formula which can be seen from Theorem 3.13. Since (BMO L (dµ)) * is a subclass of the Schwartz temperate distribution space S , so we know that our reproducing formula is valid for the elements in (BMO L (dµ)) * due to the fact that for a general potential V, the kernel of e −tL only satis es some Lipschitz condition, see Proposition 3.3. Also, the reproducing formula can be extended to all temperate distributions under this assumption if the high order derivatives of the kernel of e −tL still have a Gaussian upper bound. Remark 1.4. (i) The Hardy space H L (dµ) in this paper is a special case of the localized Hardy space H ρ (X) associated with the admissible function ρ, which has been investigated by Yang and Zhou in [10], where X is a RD-space. However, the authors give several maximal function characterizations of H ρ (X) without the area integral characterization in [10]. We will focus on the latter in this paper. (ii) Our main results can be seen as the generalization of the classical case. In fact, if ω(x)dx = dx and L = ∆ = n i= ∂ ∂x i , the space H (dx) is exactly the classical space H (R n ). It is well-known that the Hardy space H (R n ) has the area integral characterization associated with heat semigroup e −t .
Throughout this article, we will use c and C to denote the positive constants, which are independent of main parameters and may be di erent at each occurrence. By B ∼ B , we mean that there exists a constant C > such that C ≤ B B ≤ C.

Preliminaries
A nonnegative function ω is an element of the Muckenhoupt class A if there exists a constant C > such that for every ball B, Here and subsequently, |B| denotes the volume of the ball B with respect to the Lebesgue measure dx. It is well-known that (2.1) implies that the measure dµ(x) = ω(x)dx satis es the doubling condition, that is, there exists a constant C > such that for every x ∈ R n , r > , µ(B(x, r)) ≤ C µ(B(x, r)). (2.2) Using the notation of [15], we say that ω ∈ Dγ , γ > , if there is a constant C > such that for every t > , µ(B(x, tr)) ≤ Ct γ µ(B(x, r)).
Notice that (2.2) guarantees the existence of such a γ. Similarly, ω ∈ (RD)ν if for every t > , A nonnegative potential V belongs to the reverse Hölder class Bq,µ , q > , with respect to the measure dµ if there exists a constant C > such that for every Euclidean ball B, one has From now on we shall assume that ω ∈ A ∩ Dγ ∩ (RD)ν , < ν < γ, dµ(x) = ω(x)dx and V ∈ Bq,µ , q > γ/ . We set δ = − γ/q. In De nitions 1.2 & 3.10, we have used the following auxiliary function m(x, V) which is de ned by It is easy to see that, via a perturbation formula, Here h t (x, y) denotes the integral kernels of the semigroup {S t } t> on L (dµ) generated by −L , where It is known that the kernels h t (x, y) satisfy the Gaussian estimates: In [16], Hebisch and Salo -Coste have proved the following estimates for the heat kernels of L : with constants α > , c > , C > , and In the rest of this section, we state some properties of the function m(x, V) which will be used in the sequel.

Lemma 2.7. For V ∈ Bq,µ and l > , there exists a constant C > such that
For I , using Lemma 2.5, we have Similarly, for I , we have

Area integral characterization associated to the heat semigroup . Smoothness estimates associated with {T t }
In this section, by use of the area integral associated to the heat semigroup {T t } t> , we characterize the Hardy type space H L (dµ). The area integral associated to {T t } t> is de ned as follows. (3.1) Moreover, the Littlewood-Paley g-function associated to the heat semigroup {T t } t> is de ned by To prove our main results, we need some estimates for the integral kernels of the operators Q t : √ t))) e −c|x−y| /t . We divide the proof into two cases.
The above two estimates give Case II: |h| < ρ(x). We further divide Case II into two subcases.
(2) t > |h| . We only need to prove: we can use the change of variable to obtain We obtain, by Lemma 2.6 and (3.3), The term J can be further divided as follows: .
where l is the constant in Lemma 2.5. Using Lemma 2.5, we have In what follows, we consider J . In fact, For J , , by the symmetry, we have Similarly, we can deal with J , . It is easy to see that B(y, |h|) ⊂ B(z, |h|). Using < ν ≤ γ, we have For J , , we divide the estimate into two cases.
In what follows, we begin to estimate the smoothness of the kernel K t (·, ·).

Proposition 3.3.
For every < δ < δ = min{α, δ, ν}, there exists a constant C > such that for every M > and |h| < √ t, Proof. In the following proof, we denote by N the positive number, which may be di erent in di erent places. The proof is divided into four cases.
Using the triangle inequality, we have |x − y| / t ≤ + |x − y − h| /t. Hence, e −c|x−y−h| /t ≤ Ce −c|x−y| / t . Then Notice that ( + |h|m(y, V)) k /k + ≥ . We can use Lemma 2.4 to deduce that where we have used the fact |h| ≤ √ t in the last inequality. For any N, we can get Case 2: |h| < |x − y|/ . Similar to Case 1, for any N, we have Case 3: |h| < |x − y|/ and |h| ≤ ρ(y). On the one hand, we have Finally, we get Combining with the above two estimates, we get, for N large enough, For S , due to Lemma 2.1, we have For any u ∈ B(y, t/ ), we can see that B(y, t/ ) ⊂ B(z, t/ ). By the doubling property of the measure µ, we obtain Next, we consider S . For |z − y| > |h|, similar to Case 2, we have The above estimate gives Because |x − y|/ < |h| and |z − y| > |h|, then |x − y| < |z − y|. Using the estimate of K t/ (x, z) in Lemma 2.1, we know For I , we have So we have proved that I ≤ Cγ . Finally, we get This completes the proof of Proposition 3.3.

Proposition 3.4. Let Q t (x, y) = t ∂ ∂s Ks(x, y)| s=t . (a) For N > , there exists a constant C N > such that
(b) Let < δ ≤ δ and |h| < t, where δ appears in Proposition 3.3. For any N > , there exists a constant C N > such that (c) For any N > , there exists a constant C N > such that Remark 3.5. In fact, we could assume < δ < in (c) of Proposition 3.4. Because for δ > , by use of the arbitrariness of N, we can choose a < δ < < δ such that In order to prove Proposition 3.4, we need the following lemmas. Similar to [17, Corollary 6.2], we can use (2.4) to obtain Proof. Let t = Reξ , we have K ξ (x, y) = [T ξ −t/ K t/ (·, y)](x). Using Lemma 2.1 we have We get Lemma 3.8. There exists a constant c > such that for every M > there is a constant C M > such that for any ξ ∈ π/ , Similarly, we can prove This completes the proof of Lemma 3.8.
(b) By the de nition of Q t (x, y), we have It can be deduced from Proposition 3.3 that The integral in the last inequality is divided as follows: Next, we consider I and I separately. For I , because |x − η| + |η − y| ≥ |x − y| / , we have For I , we obtain The estimates for I and I imply that (c) It is easy to see that If t ≤ c ρ(x), a direct computation gives If t > c ρ(x), as we have proved, This completes the proof of Proposition 3.4.

. The area integral characterization of H L (dµ)
Now we give the area integral characterization for Hardy spaces associated with the degenerate Schrödinger operator L. We will divide the proof into two steps. At rst, we prove that for any f ∈ H L , S L h f belongs to L (dµ). Proof. At rst, we can show that the Littlewood-Paley g-function g h L is bounded on L (dµ), where g h L is de ned in (3.2). In fact, using the reproducing formula on L (dµ) and the spectral theorem, g L h f L (dµ) = f L (dµ) . To prove Theorem 3.9, we only need to verify that S L h (a) is uniformly in L (dµ) for any H L -atom a. For y ∈ Γ(x), we have for z ∈ B(y, t), |x − z| ≤ |x − y| + |y − z| < t, that is, B(y, t) ⊂ B(x, t). So we can get where in the last inequality we have used the condition: a L ∞ ≤ µ (B(x, r) For I, it is easy to see that For the estimate of II, the following two cases are considered. Case I: r < ρ(x )/ . By the canceling property of the atom a, we have x ∈ B c (x , r), we have |x − z| < r ≤ c|x − y|/ . Using (b) of Proposition 3.4, for |h| < t and symmetry, we have By the fact that |x − x | ∼ |y − x |, we obtain We can choose l large enough such that Similarly, we have Case II: ρ(x )/ ≤ r < ρ(x ). In this case, the atom a has no canceling property. So Q t (y, z)a(z)dµ(z) dµ(y)dt tµ (B(x, t)) .
Because |x − x | > r, |x − z| < r and |x − y| < t < r/ , we have |y − x | > r/ . Set s = t . By where the last equality holds, since we have in the sense of (BMO L (dµ)) * and I is the identity operator in (BMO L (dµ)) * .
For the converse of Theorem 3.9, we assume that f ∈ (BMO L (dµ)) * ∩ L (dµ). On the one hand, if f ∈ H (dµ), it is obvious that f ∈ (BMO L (dµ)) * ∩ L (dµ). Theorem 3.9 guarantees that S h L (f ) ∈ L (dµ). Conversely, if f ∈ (BMO L (dµ)) * ∩ L (dµ) and S h L (f ) ∈ L (dµ), we will use the reproducing formula (3.4) to derive that f can be represented as the linear combination of H -atoms and the scalars. By Proposition 1.3, this means that f ∈ H L (dµ). Precisely, we have the following theorem. Theorem 3.14. Suppose V ∈ Bq,µ , q > . Let L = − ω(x) i,j ∂ i (a ij (·)∂ j )(x)+V(x) be the degenerate Schrödinger operator. For every f ∈ (BMO L (dµ)) * ∩L (dµ) and equals to zero weakly at ∞ associated with L. If S L h f ∈ L (dµ), we have f ∈ H L (dµ). We need to prove that the H L -norm of α i is bounded uniformly. For simplicity, we denote by α the function α i . In fact, we have