Area Integral Characterization of Hardy space H¹_L related to Degenerate Schrödinger Operators

1 Introduction

As a suitable substitute of Lebesgue spaces L^p(ℝⁿ), the classical Hardy space H¹(ℝⁿ) plays an important role in various fields of analysis and partial differential equations. Let Δ be the Laplace operator on ℝⁿ. It follows from [1] that H¹(ℝⁿ) can be characterized by the maximal function sup_t>0|e^−tΔf(x)|. In fact, H¹(ℝⁿ) can be seen as a Hardy space associated with the operator –Δ. We use L to denote a general differential 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], Fefferman and Stein obtain the area integral characterization of the classical Hardy spaces H^p(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 on ℝⁿ. 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 ℝⁿ is defined as follows.

where (a_ij(x))_i,j is a real symmetric matrix satisfying

with ω being a nonnegative weight from the Muckenhoupt class A₂, and V ≥ 0 belonging to a reverse Hölder class with respect to the measure dμ = ω(x)dx (see Section 2 for their definitions). Denote by 𝓔(f, g) the Dirichlet form associated with L, that is,

The operator –L is the infinitesimal generator of the heat semigroup {T_t}_t>0 of self-adjoint linear operators on L²(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.

Dziubański in [3] has given the following atomic decomposition of HL1 (dμ).

The first main result of this paper can be stated as follows. Let ShL be the area integral associated with the heat semigroup generated by L, see (3.1) below. We have the following area integral characterization of HL1 (dμ):

Assume that ω ∈ (RD)_ν ⋂ D_y ⋂ A₂, 2 < ν ≤ y, and V ∈ B_q,μ with q > y/2. 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 HL1 (dμ) into atoms. As the dual space of HL1 (dμ) (cf. Definition 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 satisfies 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.

Throughout this article, we will use c and C to denote the positive constants, which are independent of main parameters and may be different at each occurrence. By B₁ ∼ B₂, we mean that there exists a constant C > 1 such that 1C≤B1B2≤C.

2 Preliminaries

A nonnegative function ω is an element of the Muckenhoupt class A₂ if there exists a constant C > 0 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 satisfies the doubling condition, that is, there exists a constant C₀ > 0 such that for every x ∈ ℝⁿ, r > 0,

Using the notation of [15], we say that ω ∈ D_y, y > 0, if there is a constant C > 0 such that for every t > 1,

Notice that (2.2) guarantees the existence of such a y.

Similarly, ω ∈ (RD)_ν if for every t > 1,

A nonnegative potential V belongs to the reverse Hölder class B_q,μ, q > 1, with respect to the measure dμ if there exists a constant C > 0 such that for every Euclidean ball B, one has

From now on we shall assume that ω ∈ A₂ ∩ D_y ∩ (RD)_ν, 2 < ν < y, dμ(x) = ω(x)dx and V ∈ B_q,μ, q > y/2. We set δ = 2 – y/q.

In Definitions 1.2 & 3.10, we have used the following auxiliary function m(x, V) which is defined 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>0 on L²(dμ) generated by –L₀, where

It is known that the kernels h_t(x, y) satisfy the Gaussian estimates:

which indicates that the kernels K_t(x, y) have a Gaussian upper bound. Furthermore, Dziubański in [3] proves that

In [16], Hebisch and Saloff-Coste have proved the following estimates for the heat kernels of L₀:

with constants α > 0, c > 0, C > 0, and

In the rest of this section, we state some properties of the function m(x, V) which will be used in the sequel.

3 Area integral characterization associated to the heat semigroup