Localized BMO and BLO Spaces on RD-Spaces and Applications to Schr\"odinger Operators

An RD-space ${\mathcal X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in ${\mathcal X}$. Let $\rho$ be an admissible function on RD-space ${\mathcal X}$. The authors first introduce the localized spaces $\mathrm{BMO}_\rho({\mathcal X})$ and $\mathrm{BLO}_\rho({\mathcal X})$ and establish their basic properties, including the John-Nirenberg inequality for $\mathrm{BMO}_\rho({\mathcal X})$, several equivalent characterizations for $\mathrm{BLO}_\rho({\mathcal X})$, and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal functions and their localized versions associated to $\rho$, and the Littlewood-Paley $g$-function associated to $\rho$, where the Littlewood-Paley $g$-function and some of the radial maximal functions are defined via kernels which are modeled on the semigroup generated by the Schr\"odinger operator. These results apply in a wide range of settings, for instance, to the Schr\"odinger operator or the degenerate Schr\"odinger operator on ${{\mathbb R}}^d$, or the sub-Laplace Schr\"odinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

1. Introduction. Since the space, BMO(R d ), of functions with bounded mean oscillation on R d was introduced by John and Nirenberg [20], it then plays an important role in harmonic analysis and partial differential equations. For example, it is well known that BMO(R d ) is the dual space of the Hardy space H 1 (R d ) (see, for example, [27,13]), and also a good substitute of L ∞ (R d ). Recall that the Riesz transforms ∇(−∆) −1/2 are bounded on BMO(R d ) but not on L ∞ (R d ) (see again, for example, [27,13] is the Laplacian and ∇ is the gradient operator. However, the space BMO(R d ) is essentially related to the Laplacian ∆. Let L ≡ −∆+V be the Schrödinger operator on R d , where the potential V is a nonnegative locally integrable function. Recently, there is an increasing interest on the study of these operators. In particular, Fefferman [10], Shen [26] and Zhong [34] established some basic results, including some estimates of the fundamental solutions and the boundedness on Lebesgue spaces of Riesz transforms, for L on R d with d ≥ 3 and the nonnegative potential V satisfying the reverse Hölder inequality. Especially, the works of Shen [26] lay the foundation for developing harmonic analysis related to L on R d . Li [21] extended part of these results in [26] to the sub-Laplace Schrödinger operator on connected and simply connected nilpotent Lie groups. On the other hand, denote by B q (R d ) the class of functions satisfying the reverse Hölder inequality of order q. For V ∈ B d/2 (R d ) with d ≥ 3, Dziubański et al [9] introduced the BMO-type space BMO L (R d ) associated to the auxiliary function ρ determined by the potential V (see, for example, (4) below) and established the duality between H 1 L (R d ) and BMO L (R d ), as well as a characterization of BMO L (R d ) in terms of the Carleson measure and the BMO L (R d ) boundedness of the variants of some classical operators associated to L including semigroup maximal functions and the Hardy-Littlewood maximal function. These results were generalized to Heisenberg groups by Lin and Liu [22]. Also, it is now known that BMO L (R d ) in [9] is a special case of BMO-type spaces introduced by Duong and Yan [4,5]; see, in particular, [5,Proposition 6.11] and also [33].
Recently, a theory of Hardy spaces and their dual spaces on so-called RD-spaces was established in [15,16,14]. A space of homogenous type X in the sense of Coifman and Weiss ( [2,3]) is called to be an RD-space if X has the additional property that a reverse doubling condition holds in X (see [16]). It is well known that a connected space of homogeneous type is an RD-space. Typical examples of RD-spaces include Euclidean spaces, Euclidean spaces with weighted measures satisfying the doubling property, Heisenberg groups, connected and simply connected nilpotent Lie groups ( [29,30]) and the boundary of an unbounded model polynomial domain in C N ( [24]), or more generally, Carnot-Carathéodory spaces with doubling measures ( [25,16]). In [31], modeled on the known auxiliary function determined by V , a notion of admissible functions ρ was introduced and a theory of the localized Hardy space H 1 ρ (X ) associated with a given admissible function ρ was developed. In particular, the space H 1 ρ (X ) was characterized via several maximal functions modeled on the semigroup maximal operators generated by Schrödinger operators, including the localized radial maximal function S + ρ . One of the main purposes of this paper is to investigate behaviors of these maximal operators aforementioned on localized BMO spaces. Precisely, let ρ be an admissible function on RD-space X . We first introduce the localized BMO space BMO ρ (X ) and localized BLO space BLO ρ (X ), and establish their basic properties, including the John-Nirenberg inequality for BMO ρ (X ), several equivalent characterizations for BLO ρ (X ), and some relations between these spaces. Then we obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal functions and their localized versions associated to ρ, and the Littlewood-Paley g-function associated to ρ, where the Littlewood-Paley g-function and some of the radial maximal functions are defined via kernels which are modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings. Moreover, even when these results are applied, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on R d , or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, we also obtain some new results.
To be precise, this paper is organized as follows.
In Section 2, we first recall some notation and notions from [16,31], including the approximation of the identity, the admissible function ρ, the radial maximal function S + (f ) and the localized radial maximal function S + ρ (f ), where S + (f ) and S + ρ (f ) are defined via a given approximation of the identity.
In Section 3, letting ρ be an admissible function on X , we first introduce the localized BMO space BMO ρ (X ) and localized BLO space BLO ρ (X ); see Definitions 3.1 and 3.2 below. We also recall the notions of their global versions in Definitions 3.1 and 3.2 below. Then we establish some useful properties concerning these spaces, including the John-Nirenberg inequality for BMO ρ (X ) (see Theorem 3.1 below), several characterizations and inclusion relations of these spaces (see Lemma 3.1, Remarks 3.1 and 3.2, and Corollary 3.1 below). Then we prove that the function in BLO ρ (X ) has lower bound in Theorem 3.2, and establish several equivalent characterizations of BLO ρ (X ) in Theorems 3.2 and 3.3, Remark 3.3, and Corollaries 3.2 and 3.3 below.
In Section 4, we establish the boundedness of the natural maximal function, the Hardy-Littlewood maximal function and their localized versions from BMO ρ (X ) to BLO ρ (X ), and as an application, we obtain several equivalent characterizations for BLO ρ (X ) via the localized natural maximal function; see Theorems 4.1 and 4.2, Lemma 4.1 and Corollary 4.1 below. We point out that Corollary 4.1 improves the results of [9] and [22] even for the Schrödinger operators on R d or Heisenberg groups with the potentials satisfying certain reverse Hölder inequality; see Remark 4.1 below.
In Section 5, we establish the boundedness of some maximal operators from BMO ρ (X ) to BLO ρ (X ). To be precise, the boundedness of the radial maximal functions S + (f ), S + ρ (f ) and certain maximal operator T + from BMO ρ (X ) to BLO ρ (X ) are presented in Section 5.1; see Theorem 5.1, Corollaries 5.1 and 5.2 below. These operators were used, respectively in [14] and [31], to characterize the corresponding Hardy spaces H 1 (X ) and H 1 ρ (X ). Section 5.2 is devoted to the boundedness of P + from BMO ρ (X ) to BLO ρ (X ); see Theorem 5.2 below. Here, T + and P + are defined via kernels which are modeled on the semigroup generated by the Schrödinger operator, and were used in [31] to characterize the corresponding Hardy space H 1 ρ (X ). In Section 6, we obtain the boundedness on BMO ρ (X ) of the Littlewood-Paley gfunction which is also defined via kernels modeled on the semigroup generated by the Schrödinger operator. Assuming that g-function is bounded on L 2 (X ), we prove that if f ∈ BMO ρ (X ), then [g(f )] 2 ∈ BLO ρ (X ) with norm no more than C f 2 BMOρ(X ) , where C is a positive constant independent of f ; see Theorem 6.1 below. As a corollary, we obtain the boundedness of the Littlewood-Paley g-function from BMO ρ (X ) to BLO ρ (X ); see Corollary 6.1 below.
In Section 7, we apply results obtained in Sections 5 and 6, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on R d , the sub-Laplace Schrödinger operator on Heisenberg groups or on connected and simply connected nilpotent Lie groups. The nonnegative potentials of these Schrödinger operators are assumed to satisfy the reverse Hölder inequality. See Propositions 7.2, 7.3, 7.4 and 7.5 below. Even for these special cases, our results further improve and generalize the corresponding results in [9,22].
We now make some conventions. Throughout this paper, we always use C or A to denote a positive constant that is independent of the main parameters involved but whose value may differ from line to line. Constants with subscripts, such as C 1 or A 1 , do not change in different occurrences. If f ≤ Cg, we then write f g or g f ; and if f g f , we then write f ∼ g. For any given "normed" spaces A and B, the symbol A ⊂ B means that for all f ∈ A, then f ∈ B and f B f A . We always use B to denote a ball of X , and for any ball B ⊂ X , we denote by x B the center of B, r B the radius of B, and B ∁ ≡ X \ B. Moreover, for any ball B ⊂ X and λ > 0, we denote by λB the ball centered at x B and having radius λr B . Also, χ E denotes the characteristic function of any set E ⊂ X . For all f ∈ L 1 loc (X ) and balls B, we always set f B ≡ 1 µ(B) B f (y) dµ(y).

Preliminaries.
We first recall the notions of spaces of homogeneous type in the sense of Coifman and Weiss [2,3] and RD-spaces in [16].
Definition 2.1. Let (X , d) be a metric space with a regular Borel measure µ such that all balls defined by d have finite and positive measure. For any x ∈ X and r ∈ (0, ∞), set the ball B(x, r) ≡ {y ∈ X : d(x, y) < r}.
Definition 2.2. ( [31]) A positive function ρ on X is said to be admissible if there exist positive constants C 0 and k 0 such that for all x, y ∈ X , We remark that the function ρ in Definition 2.2 does exist. Obviously, if ρ is a constant function, then ρ is admissible. Moreover, let x 0 ∈ X being fixed. The function ρ(y) ≡ (1+d(x 0 , y)) s for all y ∈ X with s ∈ (−∞, 1) also satisfies Definition 2.2 with k 0 = s/(1−s) when s ∈ [0, 1) and k 0 = −s when s ∈ (−∞, 0). Another non-trivial class of admissible functions is given by the well-known reverse Hölder class B q (X , d, µ), which is always written as B q (X ). Recall that a nonnegative potential V is said to be in B q (X ) with q ∈ (1, ∞] if there exists a positive constant C such that for all balls B of X , with the usual modification made when q = ∞. It is known that if V ∈ B q (X ) for certain q ∈ (1, ∞], then V is an A ∞ (X ) weight in the sense of Muckenhoupt, and also V ∈ B q+ǫ (X ) for some ǫ ∈ (0, ∞); see, for example, [27] and [28]. Thus B q (X ) = ∪ q 1 >q B q 1 (X ). For all V ∈ B q (X ) with certain q ∈ (1, ∞] and all x ∈ X , set see, for example, [26] and also [31]. It was also proved in [31] that ρ in (4) is an admissible function if n ≥ 1, q > max{1, n/2} and V ∈ B q (X ).
The following notion of approximations of the identity on RD-spaces was first introduced in [16], whose existence was given in Theorem 2.1 of [16]. Definition 2.3. Let ǫ 1 ∈ (0, 1] and ǫ 2 ∈ (0, ∞). A sequence {S k } k∈Z of bounded linear integral operators on L 2 (X ) is said to be an approximation of the identity of order (ǫ 1 , ǫ 2 ) (for short, (ǫ 1 , ǫ 2 )-AOTI), if there exists a positive constant A 4 such that for all k ∈ Z and all x, x ′ , y and y ′ ∈ X , S k (x, y), the integral kernel of S k is a measurable function from X × X into C satisfying (iii) Property (ii) also holds with x and y interchanged; (iv) X S k (x, z) dµ(z) = 1 = X S k (z, y) dµ(z) for all x, y ∈ X .
(i) For any f ∈ L 1 loc (X ) and x ∈ X , the radial maximal function S + (f ) is defined by (ii) For any f ∈ L 1 loc (X ) and x ∈ X , the radial maximal function S + ρ (f ) associated to ρ is defined by 3. Localized BMO and BLO spaces. This section is divided into two subsections. In Section 3.1, we introduce a localized BMO-type space BMO ρ (X ) and establish its several equivalent characterizations, John-Nirenberg inequality and some other properties; while Section 3.2 is devoted to the study of a corresponding localized BLO-type space BLO ρ (X ).
The following result follows from Definition 3.1.
Recall that the classical John-Nirenberg inequality (see [3]) says that there exist positive constants C 1 and C 2 such that for all f ∈ BMO(X ), balls B and λ > 0, From this, we deduce a variant of the John-Nirenberg inequality suitable for BMO ρ (X ) as follows.
Theorem 3.1. Let ρ be an admissible function on X and D be as in Definition 3.1. If f ∈ BMO ρ (X ), then there exist positive constants C 3 and C 4 such that for all balls B and λ > 0, and, moreover, for all B ∈ D, Proof. The inequality (6) follows from (5) and Definition 3.1 directly. To show (7), let B ∈ D. If λ > 2 f BMO ρ (X ) , by the definition, we have λ > 2|f | B . Thus for all balls B in D, we obtain which together (6) yields (7); if 0 < λ ≤ 2 f BMOρ(X ) , we then have This finishes the proof of Theorem 3.1.
We now establish the relation between BMO(X ) and BMO ρ (X ) in terms of certain approximation of the identity. To begin with, let ρ be an admissible function on X . In [31], it was proved that there exist a nonnegative function K ρ on X × X and a positive constant C 5 such that It was proved in [31] [15,16,14], which coincides with the atomic Hardy space H 1 at (X ) of Coifman and Weiss in [3]. Moreover, there exists a positive constant C such that for all f ∈ H 1 ρ (X ), On the other hand, it was showed in [32] that the dual space of H 1 ρ (X ) is BMO ρ (X ). From these facts, we deduce the following corollary.
Corollary 3.1. Let ρ be an admissible function on X and K ρ be as in (8).

This shows (i).
To see (ii), by (K) 3 , (9) and (H 1 (X )) * = BMO(X ), we have that for all f ∈ BMO(X ) and the desired estimate. This finishes the proof of Corollary 3.1.

A localized BLO space.
Definition 3.2. Let ρ and D be as in Definition 3.1 and q ∈ [1, ∞).
with the Lebesgue measure was introduced by Coifman and Rochberg [1], and extended by Jiang [19] to the setting of R d with a nondoubling measure. Let q ∈ [1, ∞). Then the facts that BLO 1 (X ) ⊂ BMO(X ) = BMO q (X ) together with the Hölder inequality imply that BLO q (X ) = BLO 1 (X ) with equivalent norms. We denote BLO 1 (X ) simply by BLO(X ). Notice that BLO(X ) is not a linear space.
(ii) We also denote BLO 1 ρ (X ) simply by BLO ρ (X ). The localized BLO space was first introduced in [18] in the setting of R d with a non-doubling measure. For all q ∈ [1, ∞), a (X ) as in Definition 3.2 (ii) with D replaced by D a . If µ(X ) < ∞, then for all q ∈ [1, ∞) and admissible functions ρ, and any fixed a ∈ (0, ∞), BLO q ρ (X ) = BLO q a (X ) with equivalent norms. The proof is similar to that of Remark 3.1 (iii) and is omitted.
The following result follows from Definitions 3.1 and 3.2, whose proof is similar to that of Lemma 3.1 and is omitted.
for almost all x ∈ X . Moreover, the following statements are equivalent: (ii) f ∈ L 1 loc (X ) and there exists a nonnegative constant A such that (iii) f ∈ L 1 loc (X ) and there exists a nonnegative constant C such that Moreover, f BLO ρ (X ) , inf{A} and inf{ C} are mutually equivalent.
which together with X = ∪ x B(x, ρ(x)/2) and the Vitali-Wiener type covering lemma (see [3, p. 623]) implies that there exists certain positive constant C such that for µ-a. e.
x ∈ X , . From this, it is easy to see that (i) implies (ii). Obviously, (ii) implies (iii) and (iii) implies (i). Thus we complete the proof of Theorem 3.2.
(ii) During this paper being written, we learnt that when V ∈ B q (R d ) with q > d/2, and ρ is as in (4), Theorem 3.2 (iii) was used, independently, by Gao, Jiang and Tang [11] to introduce the space BLO L (R d ) corresponding to the Schrödinger operator L = −∆ + V .
As a consequence of Theorem 3.2, we have the following corollary.
As a consequence of Corollary 3.1 and Corollary 3.2, we have the following result. Corollary 3.3. Let ρ be an admissible function on X and K ρ be as in (8). Then Proof. Assume that f ∈ BLO ρ (X ) first. Then by Corollary 3.2, f ∈ BLO(X )∩BMO ρ (X ). From this and Corollary 3.
and K ρ (f ) ∈ L ∞ (X ), then the obvious fact BLO(X ) ⊂ BMO(X ) together with another application of Corollary 3.1 (i) implies that f ∈ BMO ρ (X ), which together with Corollary 3.2 yields that f ∈ BLO ρ (X ). This finishes the proof of Corollary 3.3. Theorem 3.3. Let ρ be an admissible function on X and K ρ be as in (8). Then there Proof. Let f ∈ BLO(X ). By the homogeneity of · BLO(X ) and · BLO ρ (X ) , we may assume that f BLO(X ) = 1. Let B ≡ B(x 0 , r) ∈ D. Observe that by (3), for any a ∈ (0, ∞), there exists a constant C a ∈ [1, ∞) such that for all x, y ∈ X with d(x, y) ≤ aρ(x), By this and r ≥ ρ(x 0 ), we obtain that for all x ∈ B, ρ(x) r. Then there exists a positive constant C such that for all x ∈ B, B(x, C 5 ρ(x)) ⊂ CB. By (K) 1 through (K) 4 , (1) and the Tonelli theorem, we obtain On the other hand, let B ≡ B(x 0 , r) / ∈ D. Using r < ρ(x 0 ) and (10) with a = 1, we obtain that there exists a constant A 1 ∈ [1, ∞) such that for all x, y ∈ B, B(x, ρ(x)) ⊂ B(y, A 1 ρ(y)). From this together with (1), it follows that for all x, y ∈ B, By this together with (K) 1 through (K) 4 and (1), we have that for all x, y ∈ B, From this fact, we deduce that 1.
This together with (11) gives the desired estimate and hence, finishes the proof of Theorem 3.3.

4.
Boundedness of the natural and the Hardy-Littlewood maximal functions. In this section, we first obtain the boundedness of the natural maximal function, the Hardy-Littlewood maximal function and their localized versions from BMO ρ (X ) to BLO ρ (X ); as an application, we then establish several equivalent characterizations for BLO ρ (X ) via the localized natural maximal function.
Theorem 4.1. Let ρ be an admissible function on X . Then M ρ is bounded from BMO ρ (X ) to BLO ρ (X ), namely, there exists a positive constant C such that for all f ∈ BMO ρ (X ), and Proof. Let f ∈ BMO ρ (X ). By the homogeneity of · BMOρ(X ) and · BLOρ(X ) , we may assume that f BMOρ(X ) = 1. We first prove that for all balls B ≡ B(x 0 , r) ∈ D, From this, it follows that for all balls B ∈ D, To prove (12), for all balls B ∈ D, write The Hölder inequality together with the L 2 (X )-boundedness of HL (see [2]) and Lemma 3.1 gives us that We now claim that for all y ∈ B, 1. The claim then follows from the two estimates above, which together with (13) leads to (12).
We now prove that there exists a positive constant C such that for all balls Write Using the Hölder inequality, the L 2 (X )-boundedness of HL, (1) and Lemma 3.1, we obtain that Now we show that for all Now we assume that B ⊂ 3 B. If 3 B / ∈ D, then the fact y ∈ B ⊂ 3 B gives us that f 3 e B ≤ M ρ (f )(y), which together with (1) implies that 1.
Combining the two inequalities above and (16) leads to that which together with (15) further implies (14). This finishes the proof of Theorem 4.1.
Proof. Assuming that f ∈ BLO ρ (X ), we then see that (17) holds. Since µ is regular, for µ-a. e. x ∈ X , there exists a sequence of balls Let x be any point satisfying (18) Conversely, assume that f satisfies (17) and which together with (17) implies that f ∈ BLO ρ (X ) and f BLO ρ (X ) M ρ (f )−f L ∞ (X ) . This finishes the proof of Lemma 4.1.
As another corollary of Theorem 4.1, we obtain the boundedness of HL, HL ρ and M from BMO ρ (X ) to BLO ρ (X ). To this end, we first establish the following useful lemma.
Lemma 4.2. Let ρ be an admissible function on X and Y be one of the spaces BMO(X ), BMO ρ (X ), BLO(X ) and BLO ρ (X ).
Proof. We only consider the case that Y = BLO ρ (X ) by similarity. For all balls B ∈ D, we have that On the other hand, for all balls B / ∈ D, Combining the two inequalities above finishes the proof of Lemma 4.2. and Proof. Since for all locally integrable functions f , This finishes the proof of Corollary 4.1.
Remark 4.1. Let d ≥ 3, V be a nonnegative integrable function on R d and L = −∆ + V . If q > d/2, V ∈ B q (R d ) and ρ is as in (4), then BMO ρ (R d ) is just the space BMO L (R d ) introduced in [9]. It was proved in [9] that HL is bounded on BMO ρ (R d ). Recall that BLO ρ (R d ) BMO ρ (R d ). Thus, Corollary 4.1 improves the result of [9]. Similar claim is also true for HL on Heisenberg groups; see [22].

5.
Boundedness of several maximal operators. This section consists of two subsections. Subsection 5.1 is devoted to the boundedness of several radial maximal operators from BMO ρ (X ) to BLO ρ (X ); while in Subsection 5.2, we obtain the boundedness of the Poisson semigroup maximal operator from BMO ρ (X ) to BLO ρ (X ).
Observe that by Definition 2.3 (i), S + (f ) HL(f ). From this and Corollary 4.1, it follows that for all balls B ≡ B(x 0 , r) ∈ D, This also implies that S + (f )(x) < ∞ for µ-a. e. x ∈ X . Moreover, by the inequality above, to finish the proof Theorem 5.1, it suffices to show that for all balls B ≡ B(x 0 , r) / ∈ D and y ∈ B, Let By using Definition 2.3 (iv), we have that for all y ∈ B, By the Hölder inequality, S + (f ) HL(f ), the L 2 (X )-boundedness of HL, (1) and Lemma 3.1, we obtain Recall that {S t } t>0 is a continuous (ǫ 1 , ǫ 2 )-AOTI. By Remark 2.1, Definition 2.3 (i), (1) and the fact that for all x ∈ B and j ∈ N, 2 j+1 B ⊂ B(x, 2 j+2 r), we have that for all t ∈ (0, r), which implies that L 2 1. By Definition 2.3 (iv) and (i) together with (1) and the fact that for all y ∈ B and j ∈ N ∪ {0}, 2 j+1 B ⊂ B(y, 2 j+2 r), we have On the other hand, for all x, y ∈ B and t ∈ [r, ∞), B(x, t) ⊂ B(y, 2t) ⊂ B(x, 3t). It follows from this fact and (1) that By this and an argument similar to the estimate for L 3 , we have that for all x, y ∈ B and t ∈ [r, ∞), which implies that Combining the estimates for L 1 through L 4 yields (20), which completes the proof of Theorem 5.1.

By Definition 2.3 (i), we have that for all
x ∈ X and t ∈ [ρ(x), ∞), This implies that there exists a positive constant C such that for all f ∈ BMO ρ (X ), . From this, Lemma 4.2 and Theorem 5.1, we deduce the following corollary.
Then by Theorem 5.1, Corollaries 5.1 and 5.2, we obtain the following result.
As a consequence of Theorem 6.1, we have the following conclusion. Corollary 6.1. With the assumptions same as in Theorem 6.1, then there exists a positive constant C such that for all f ∈ BMO ρ (X ), g(f ) ∈ BLO ρ (X ) and g(f ) BLOρ(X ) ≤ C f BMOρ(X ) .

Proof.
Since , by the Hölder inequality and Theorem 6.1, we have that for all balls B / ∈ D, On the other hand, by (31) and the Hölder inequality, we obtain that for all balls B ∈ D, Combining the two inequalities above finishes the proof of Corollary 6.1.

7.
Applications. This section is divided into Subsections 7.1 through 7.4, which are devoted to the applications of results obtained in Sections 5 and 6, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on R d , the sub-Laplace Schrödinger operator on Heisenberg groups or on connected and simply connected nilpotent Lie groups. Let q ∈ (d/2, d], V ∈ B q (R d , | · |, dx) and ρ be as in (4). Then we have the following estimates; see [8,6,7].
Proposition 7.1. Let q ∈ (d/2, d], β ∈ (0, 2 − d/q) and N ∈ N. Then there exist positive constants C and C, where C is independent of N , such that for all t ∈ (0, ∞) and and for all t ∈ (0, ∞) and x, x ′ , y ∈ X with d(x, Observe that { T t 2 } t>0 is a continuous (1, N )-AOTI for all positive constants N . Thus {T t 2 } t>0 and { T t 2 } t>0 satisfy the assumption (22). Moreover, the L 2 (R d )-boundedness of g-function was obtained in [8]. Using these facts and Proposition 7.1 and applying Theorems 5.1, 5.2 and 6.1, and Corollaries 5.1, 5.2 and 6.1, we have the following result.
Proposition 7.2. Let q ∈ (d/2, ∞], V ∈ B q (R d , | · |, dx) and ρ be as in (4). There exists a positive constant C such that for all f ∈ BMO ρ (R d ), and We also point out that when ρ is as in (4), Dziubański et al [9] obtained the boundedness of T + , P + and g on BMO ρ (R d ). Proposition 7.2 improves their results.

Degenerate Schrödinger operators on R d . Let d ≥ 3 and R d be the d-dimensional
Euclidean space endowed with the Euclidean norm |·| and the Lebesgue measure dx. Recall that a nonnegative locally integrable function w is said to be an where the supremum is taken over all the balls in R d . Observe that if we set w(E) ≡ E w(x)dx for any measurable set E, then there exist positive constants C, Q and κ such that for all x ∈ R d , λ > 1 and r > 0, C −1 λ κ w(B(x, r)) ≤ w(B(x, λr)) ≤ Cλ Q w(B(x, r)), namely, the measure w(x) dx satisfies (1). Thus (R d , | · |, w(x) dx) is an RD-space.
Let w ∈ A 2 (R d ) and {a i, j } 1≤i, j≤d be a real symmetric matrix function satisfying that for all x, ξ ∈ R d , Then the degenerate elliptic operator L 0 is defined by We also denote the kernel of T t by T t (x, y) for all x, y ∈ R d and t ∈ (0, ∞). Then it is known that there exist positive constants C, C 6 , C 6 and α ∈ (0, 1] such that for all t ∈ (0, ∞) and x, y ∈ R d , that for all t ∈ (0, ∞) and x, y, y ′ ∈ R d with |y − y ′ | < |x − y|/4, and that for all t ∈ (0, ∞) and x, y ∈ R d , Let q ∈ (Q/2, Q], V ∈ B q (R d , | · |, w(x) dx) and ρ be as in (4).
, and d by Q. In fact, the corresponding Proposition 7.1 (i) and (iii) here were given in [8]. The proof of (ii) here is similar to that of Proposition 7.1; see [7] and also Lemma 7.4 below. The proofs of the corresponding Proposition 7.1 (iv), (v) and (vi) here are similar to that of Proposition 4 of [9]. We omit the details here.
Observe that { T t 2 } t>0 is a continuous (1, N )-AOTI for all positive constants N . Thus {T t 2 } t>0 and { T t 2 } t>0 satisfy the assumption (22). Moreover, the L 2 (R d )-boundedness of g-function can be obtained by the same argument as in Lemma 3 of [8]. Using these facts and applying Theorems 5.1, 5.2 and 6.1, and Corollaries 5.1, 5.2 and 6.1, we have the following conclusions. Proposition 7.3. Let w ∈ A 2 (R d ). Let q ∈ (Q/2, ∞], V ∈ B q (R d , | · |, w(x) dx) and ρ be as in (4) with dµ = w(x) dx. Then there exists a positive constant C such that for The homogeneous norm on H n is defined by |(x, t)| = (|x| 4 + |t| 2 ) 1/4 for all (x, t) ∈ H n , which induces a left-invariant metric d((x, t), (y, s)) = |(−x, −t)(y, s)|. Moreover, there exists a positive constant C such that |B((x, t), r)| = Cr Q , where Q = 2n + 2 is the homogeneous dimension of H n and |B((x, t), r)| is the Lebesgue measure of the ball B ((x, t), r). The triplet (H n , d, dx) is an RD-space.
A basis for the Lie algebra of left invariant vector fields on H n is given by All non-trivial commutators are [X j , X n+j ] = −4X 2n+1 , j = 1, · · · , n. The sub-Laplacian has the form ∆ H n = 2n j=1 X 2 j . Let V be a nonnegative locally integrable function on H n . Define the sub-Laplacian Schrödinger operator by L ≡ −∆ H n + V. Denote by {T t } t>0 ≡ {e −tL } t>0 the semigroup generated by L and by { T t } t>0 ≡ {e t∆ H n } t>0 the semigroup generated by −∆ H n .
Observe that { T t 2 } t>0 is a continuous (1, N )-AOTI for all positive constants N . Thus {T t 2 } t>0 and { T t 2 } t>0 satisfy the assumption (22). Moreover, the L 2 (H n )-boundedness of g-function was obtained in [22]. Using these facts and applying Theorems 5.1, 5.2 and 6.1, and Corollaries 5.1, 5.2 and 6.1, we have the following conclusions.
Proposition 7.4. Let q ∈ (n + 1, ∞], V ∈ B q (H n , d, dx) and ρ be as in (4). Then there exists a positive constant C such that for all f ∈ BMO ρ (H n ), T + (f ), T + (f ), T + ρ (f ), P + (f ), P + (f ), P + ρ (f ), g(f ), [g(f )] 2 ∈ BLO ρ (H n ) and We also point out that when ρ is as in (4), Lin and Liu [22] introduced BMO ρ (H n ) and established the boundedness of T + , P + and g on BMO ρ (H n ). The results in this subsection improve their corresponding results. 7.4. Schrödinger operators on connected and simply connected nilpotent Lie groups. Let G be a connected and simply connected nilpotent Lie group. Let X ≡ {X 1 , · · · , X k } be left invariant vector fields on G satisfying the Hörmander condition that {X 1 , · · · , X k } together with their commutators of order ≤ m generates the tangent space of G at each point of G. Let d be the Carnot-Carathéodory (control) distance on G associated to {X 1 , · · · , X k }. Fix a left invariant Haar measure µ on G. Then for all x ∈ G, V r (x) = V r (e); moreover, there exist κ, D ∈ (0, ∞) with κ ≤ D such that for all x ∈ G, when r ∈ (0, 1], and C −1 r D ≤ V r (x) ≤ Cr D when r ∈ (1, ∞); see [25] and [29]. Thus (G, d, µ) is an RD-space. The sub-Laplacian is given by ∆ G ≡ k j=1 X 2 j . Denote by { T t } t>0 ≡ {e t∆ G } t>0 the semigroup generated by −∆ G . Then there exist positive constants C, C 7 and C 7 such that for all t ∈ (0, ∞) and x, y ∈ G, that for all t ∈ (0, ∞) and x, y, y ′ ∈ G with d(y, y ′ ) ≤ d(x, y)/4, and that for all t ∈ (0, ∞) and x, y ∈ G, see, for example, [29]. Define the radial maximal operator T + by T + (f )(x) ≡ sup t>0 | T t (f )(x)| for all x ∈ G. Then by (46), it is easy to see that T + is bounded on L p (G) for p ∈ (1, ∞].
Let V be a nonnegative locally integrable function on G. Then the sub-Laplace Schrödinger operator L is defined by L ≡ −∆ G + V. The operator L generates a semigroup of operators {T t } t>0 ≡ {e −tL } t>0 , whose kernels are denoted by {T t (x, y)} t>0 . Define the radial maximal operator T + by T + (f )(x) ≡ sup t>0 |e −tL (f )(x)| for all x ∈ G. Then from Lemma 7.1 below, it is easy to see that T + is bounded on L p (G) for p ∈ (1, ∞]. Let q > D/2, V ∈ B q (G, d, µ) and ρ be as in (4). Then Li [21] established some basic results concerning L, which include estimates for fundamental solutions of L and the boundedness on Lebesgue spaces of some operators associated to L. To apply the results obtained in Sections 5 and 6 to L, we need the following estimate, which is a consequence of Proposition 5.2 and (5.12) in [31] together with the symmetry of T t and the fact that for all x, y ∈ G and t ∈ (0, ∞), V t (x) ∼ V t (y). We omit the details.  (G, d, µ). Then for all N ∈ (0, ∞), there exist positive constants C and C 8 , where C 8 is independent of N , such that for all t ∈ (0, ∞) and x, y ∈ G, For t ∈ [0, ∞), set E t ≡ T t − T t . Denote also by E t the kernel of E t . The following estimate for E t was established in [31].  (G, d, µ), then for all N ∈ (0, ∞), there exist positive constants C and C 9 , where C 9 is independent of N , such that for all t ∈ (0, ∞) and x, y ∈ G, Moreover, to estimate the regularity of T t , we need the regularity of E t . To this end, we recall the following lemma.  (G, d, µ), then for all positive constants C and C, there exists positive constant A 6 such that for all x ∈ G and t > 0, when where ℓ 0 is a positive constant independent of C, C and A 6 .
We write To estimate F 1 , we consider the following two cases. Case (i) t < 2[ρ(y)] 2 . For s ∈ (0, t/2), we have t − s ∼ t. By (47), Lemma 7.1, Lemma 7.3, the symmetry of T t , the assumption that D/2 < q ≤ D and the fact that V r (x) ∼ V r (y) for all x, y ∈ G and r ∈ (0, ∞), we have .
For all x, y ∈ G and t ∈ (0, ∞), define Q t (x, y) ≡ t 2 d ds s=t 2 T s (x, y).
Following the proof of Proposition 4 in [9], we have the following result. We omit the details.
Acknowledgements. The authors would like to thank the referees for their several valuable remarks which improve the presentation of this article.