Abstract

Let be the generalized Schrödinger operator on where is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. In this work, we give a new space associated to the generalized Schrödinger operator , which is bigger than the spaces related to the classical Schrödinger operators , with a potential satisfying a reverse Hölder inequality introduced by Dziubański et al. in 2005. Besides, the boundedness of the Littlewood-Paley operators associated to in also be proved.

1. Introduction

Consider the generalized Schrödinger operator where is a nonnegative Radon measure on . According to [1], there exist positive constants , , and such that satisfies the following conditions: for all and where denotes the open ball centered at with radius Condition (2) is regarded as scale-invariant Kato-condition, and from (3), we can see that the measure is doubling on balls satisfying We will also assume that . If and are in the reverse Hölder class, that is, there exists such that then satisfies the conditions (2) and (3) for some However, in general, measures which satisfy (2) and (3) need not be absolutely continuous with respect to the Lebesgue measure on the counterexample is visible in ([1], Remark 0.10). Let , it is easy to know that is more general than from the above.

The boundedness of the operators associated to the classical operator such as the Gauss-semigroup and Poisson-semigroup maximal functions, the Littlewood-Paley-square function and the fractional integral operator has attracted much interest [2–4]. It is worth noting that these operators fail to be bounded in , even in the classic case (i.e.). In 2005, Dziubański et al. [2] identified the -type space related to , , namely, where is the auxiliary function, defined by

(see [2, 4–7]). They proved the above operators were bounded in this space. By the fact that is a subspace of , we know that it is very meaningful to consider the boundedness of these operators if we can expand the space a little bit.

In this paper, we shall be interested in a new space associated to the generalized Schrödinger operator . To give the definition of the new space, we first recall the auxiliary function (see [8]), where is the constant in (3).

Here, we define the new space, , namely, where for . The precise definition of the norm in the spaces is given in Definition 1.

Definition 1. Let be a nonnegative Radon measure in Assume that satisfies the conditions (2) and (3). For , we shall say that a locally integrable function belongs to whenever there is a constant so that for all balls , such that . The norm of , denoted by , is the smallest in (10) above. Here and subsequently, .

Remark 2. From the definition of , we have . Also, we emphasize that is actually bigger than , which is defined by Dziubański et al. in [2], that is, . From the definition of and the fact that is more general than , it is obvious that is the subset of .

Since is nonnegative on the Feynman-Kac formula implies that the kernel of the semigro up of linear operators generated by satisfies also see in [9].

After Wang [10] considered the -function defined on functions, more and more scholars pay attention to the end point estimate of the Littlewood-Paley operator [3, 8, 11–13]. In this paper, we will also consider the follows Littlewood-Paley operators associated to the generality Schrödinger operator are bounded in . where .

The main theorem is as follows.

Theorem 3. Let and defined as in (12), then is bounded in . That is, if , there exist a constant such that

The paper is organized as follows. In Section 2, we give some necessary lemmas. In Section 3, we consider and give the proof of Theorem 3.

Throughout this paper, give a ball , we denote by the ball with the same center and twice radius. and will denote positive constants that may not be the same in each occurrence.

2. Preliminaries

We first recall some basic properties of the auxiliary function in (7) (see [1], Proposition 1.8). In the sequel, , and are positive constants in (2) and (3).

Lemma 4. Suppose satisfies (2) and (3). Then (i)If then (ii)If then (iii)There exist constants such that for with and .

Next, we recall some results about covering by critical balls, which can be found in (see [14], Lemma 2.3).

Lemma 5. There exists a sequence of points in , so that the family of critical balls , satisfies (i)(ii)There exists such that for every ,

The kernel of the semigroup satisfies following upper bound in ([15], Theorem 1.1).

Lemma 6. Let be a nonnegative Radon measure in Assume that satisfies the conditions (2) and (3). There exist constants , , the constant in (14), the heat kernel of satisfies

Using the inequality (15) and repeating the proof process in ([2], Proposition 4), we can obtain the estimates for the integral kernel .

Lemma 7. There exist constants such that for in (14) and any , there is a constant so that (i)(ii), for all (iii)

Finally, following ([16], Proposition 3), we recall some basic properties about the norm of .

Lemma 8. For , let , , and , then (i) and for all (ii) where and the constant appearing in (14).

3. Proof of Theorem 3

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

Lemma 9. (i)For all balls in , let as in (3), then (ii)For , let and as in (3), then (iii)If , then

Proof. The proof of (i) and (ii) can be obtained directly from (2) and (3), respectively. From Lemma 4(i), the proof of (iii) can be obtained.

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

Lemma 10. A function belong to with if and only if for every ball with and and for all .

Proof. From the Definition 1, it is easy to see that if , then satisfies (21) and (22), furthermore, to prove , we need (22) and for every ball with and .
Obviously, we just have to prove that if satisfies (22) then satisfies (23). Let be a sequence as in Lemma 5, it follows from (22) that where for all . According to Lemma 5, we have the set is finite.
For every , and , by Lemma 4(iii) we get then for every . Thus That is for Now, we turn to estimate the boundedness of the operator from to . According to Lemma 10, we only need to check that (1)(2)

Proof of Theorem 3. Without loss of generality, fix with , we first to show that We split For , we set Using the self-adjointness of , we get Then, combining Hölder’s inequality, (31), Lemma 8(i), and Lemma 4(ii) with for , we have where the last inequality follows from Remark 2.
Next, for and , it follows from Lemma 7(i) and Lemma 8(ii) that Then, we obtain Now, we turn to estimate the term for . Using for and Lemma 7(iii), we have For , one can easily check using Lemma 7(i), Minkowski’s inequality and for that Next, we deal with the following case In this section, inspired by the methods in [2], we decompose as follows Set It is easy to see that is a finite constant and To complete the proof, it suffices to show that Note that for any , by (36), we get for It remains to estimate . Let . We claim for . It follows from the perturbation formula (see ([18], Chapter 9, (2.3))) that By Lemma 9, we have for , By the same argument as estimating , combining Lemma 7, we also obtain The same estimate as , we have The proof of (43) is finished. For , , it follows from Definition 1 and Lemma 8(ii) that For , since the formula . It is known that is bounded on ([19]). Hence, combining with Lemma 8(i), we obtain We set the kernel of . We recall this kernel satisfies the estimates and For and , choosing such that , it follows from Minkowski’s inequality and Lemma 8(ii) that Therefore, By (42), (48), and (54), we finish the proof of (37). Then, combining (28) and (37), we finish the proof of Theorem 3.