End Point Estimate of Littlewood-Paley Operator Associated to the Generalized Schrödinger Operator

Let L = − Δ + μ be the generalized Schrödinger operator on ℝ d , d ≥ 3, where μ ≠ 0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. In this work, we give a new BMO space associated to the generalized Schrödinger operator L , BMO θ , L , which is bigger than the BMO spaces related to the classical Schrödinger operators A = − Δ + V , with V a potential satisfying a reverse Hölder inequality introduced by Dziuba ń ski et al. in 2005. Besides, the boundedness of the Littlewood-Paley operators associated to L in BMO θ , L also be proved.


Introduction
Consider the generalized Schrödinger operator where μ is a nonnegative Radon measure on ℝ d , d ≥ 3.
According to [1], there exist positive constants C 0 , C 1 , and δ such that μ satisfies the following conditions: for all x ∈ ℝ d and 0 < r < R, where Bðx, rÞ denotes the open ball centered at x with radius r: Condition (2) is regarded as scale-invariant Kato-condition, and from (3), we can see that the measure μ is doubling on balls satisfying μðBðx, rÞÞ ≥ cr d−2 : We will also assume that μ ≠ 0. If dμ = VðxÞdx and V ≥ 0 are in the reverse Hölder class, that is, there exists C = Cðd, VÞ > 0 such that then μ satisfies the conditions (2) and (3) for some δ > 0: However, in general, measures which satisfy (2) and (3) need not be absolutely continuous with respect to the Lebesgue measure on ℝ d , the counterexample is visible in ( [1], Remark 0.10). Let A = −Δ + V, it is easy to know that L is more general than A from the above. The boundedness of the operators associated to the classical operator A such as the Gauss-semigroup and Poissonsemigroup maximal functions, the Littlewood-Paley-square function and the fractional integral operator has attracted much interest [2][3][4]. It is worth noting that these operators fail to be bounded in BMO, even in the classic case (i.e.V = 0). In 2005, Dziubański et al. [2] identified the BMO-type space related to A, BMO A , namely, where ρðx, VÞ is the auxiliary function, defined by (see [2,[4][5][6][7]). They proved the above operators were bounded in this space. By the fact that BMO A is a subspace of BMO, we know that it is very meaningful to consider the boundedness of these operators if we can expand the BM O A space a little bit.
In this paper, we shall be interested in a new BMO space associated to the generalized Schrödinger operator L. To give the definition of the new BMO space, we first recall the auxiliary function ρðx, μÞ (see [8]), where C 1 is the constant in (3).
Here, we define the new BMO space, BMO θ,L , namely, where for θ > 0. The precise definition of the norm in the spaces BMO θ,L is given in Definition 1.
Definition 1. Let μ ≠ 0 be a nonnegative Radon measure in ℝ d , d ≥ 3: Assume that μ satisfies the conditions (2) and (3). For θ > 0, we shall say that a locally integrable function f belongs to BMO θ,L whenever there is a constant C ≥ 0 so that for all balls B R = Bðx, RÞ, B r = Bðx, rÞ such that R ≥ ρðx, μÞ ≥ r. The norm of f ∈ BMO θ,L , denoted by kf k BMO θ,L , is the smallest C in (10) above.
Here and subsequently, Remark 2. From the definition of BMO θ,L , we have BM O θ,L ⊆ BMO θ . Also, we emphasize that BMO θ,L is actually bigger than BMO A , which is defined by Dziubański et al. in [2], that is, BMO A ⊆ BMO θ,L . From the definition of BM O θ,L and the fact that L is more general than A, it is obvious that BMO A is the subset of BMO θ,L .
Since μ is nonnegative on ℝ d , the Feynman-Kac formula implies that the kernel k t ðx, yÞ of the semigro up e −tL of linear operators generated by −L satisfies also see in [9]. After Wang [10] considered the g-function defined on BMO functions, more and more scholars pay attention to the end point estimate of the Littlewood-Paley operator [3,8,[11][12][13]. In this paper, we will also consider the follows Littlewood-Paley operators associated to the generality Schrödinger operator are bounded in BMO θ,L .
where T t f ðxÞ = e −tL f ðxÞ = Ð ℝ d k t ðx, yÞf ðyÞdy. The main theorem is as follows.
The paper is organized as follows. In Section 2, we give some necessary lemmas. In Section 3, we consider sðf Þ and give the proof of Theorem 3.
Throughout this paper, give a ball B, we denote by B * the ball with the same center and twice radius. c and C will denote positive constants that may not be the same in each occurrence.

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Lemma 4.
Suppose μ satisfies (2) and (3). Then with Next, we recall some results about covering ℝ d by critical balls, which can be found in (see [14], Lemma 2.3).

Lemma 5. There exists a sequence of points
The kernel k t ðx, yÞ of the semigroup e −tL satisfies following upper bound in ( [15], Theorem 1.1).

Lemma 7.
There exist constants c, σ > 0 such that for k 0 in (14) and any N, there is a constant C so that Finally, following ( [16], Proposition 3), we recall some basic properties about the norm of BMO θ .

Proof of Theorem 3
Before we prove Theorem 3, we first recall some basic facts about the nonnegative Radon measure μ.

Journal of Function Spaces
Next, we give the following result, which is similar to the proof of ( [17], Corollary 1).

Lemma 10.
A function f belong to BMO θ,L with θ > 0 if and only if for every ball B = Bðx, rÞ with x ∈ ℝ d and r < ρðx, μÞ and for all x ∈ ℝ d .
Proof. From the Definition 1, it is easy to see that if f ∈ BM O θ,L , then f satisfies (21) and (22), furthermore, to prove f ∈ BMO θ,L , we need (22) and for every ball B = Bðx, rÞ with x ∈ ℝ d and r ≥ ρðx, μÞ.