Boundedness of Lusin-area and g ∗ λ functions on localized Morrey-Campanato spaces over doubling metric measure spaces

Let X be a doubling metric measure space and ρ an admissible function on X . In this paper, the authors establish some equivalent characterizations for the localized Morrey-Campanato spaces Eα, p ρ (X ) and Morrey-Campanato-BLO spaces  ̃ Eα, p ρ (X ) when α ∈ (−∞, 0) and p ∈ [1, ∞) . If X has the volume regularity Property (P ) , the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, from Eα, p ρ (X ) to  ̃ Eα, p ρ (X ) without invoking any regularity of considered kernels. The same is true for the g∗ λ function and, unlike the Lusin-area function, in this case, X is even not necessary to have Property (P ) . These results are also new even for R with the d -dimensional Lebesgue measure and have a wide applications.


Introduction
The theory of Morrey-Campanato spaces plays an important role in harmonic analysis and partial differential equations; see, for example, [1,5,16,22,23,25,27,29,31] and their references. It is well-known that λ functions the dual space of the Hardy space H p (R d ) with p ∈ (0, 1) is the Morrey-Campanato space E 1/p−1, 1 (R d ). Notice that the Morrey-Campanato spaces on R d are essentially related to the Laplacian Δ ≡ d j=1 On the other hand, there exists an increasing interest on the study of Schrödinger operators on R d and the sub-Laplace Schrödinger operators on connected and simply connected nilpotent Lie groups with nonnegative potentials satisfying the reverse Hölder inequality; see, for example, [6,7,8,9,17,18,26,35,37]. Let L ≡ −Δ + V be the Schrödinger operator on R d , where the potential V is a nonnegative locally integrable function. Denote by B q (R d ) the class of nonnegative functions satisfying the reverse Hölder inequality of order q . For V ∈ B d/2 (R d ) with d ≥ 3 , Dziubański et al. [6,7,8] studied the BMO-type space BMO L (R d ) and the Hardy space H p L (R d ) with p ∈ (d/(d + 1), 1] and, especially, proved that the dual space of H 1 L (R d ) is BMO L (R d ); moreover, they obtained the boundedness on these spaces of the Littlewood-Paley g -function associated to L.
Let X be a doubling metric measure space, which means that X is a space of homogeneous type in the sense of Coifman and Weiss [2,3], but X is endowed with a metric instead of a quasi-metric. Let ρ be a given admissible function modeled on the known auxiliary function determined by V ∈ B d/2 (R d ) (see [35] or (2.4) below). The localized atomic Hardy space H p, q ρ (X ) with p ∈ (0, 1] and q ∈ [1, ∞] ∩ (p, ∞], the localized Morrey-Campanato space E α, p ρ (X ) and localized Morrey-Campanato-BLO space E α, p ρ (X ) with α ∈ R and p ∈ (0, ∞) were introduced in [34]. Moreover, the boundedness from E α, p ρ (X ) to E α, p ρ (X ) of several maximal operators and the Littlewood-Paley g -function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator, was obtained in [34]. Meanwhile, the boundedness from localized BMO-type space BMO ρ (X ) to BLO-type space BLO ρ (X ) of the Lusin-area and g * λ functions was established in [19].
The purpose of this paper is to investigate behaviors of the Lusinarea and g * λ functions on Morrey-Campanato spaces over doubling metric measure spaces. Precisely, let X be a doubling metric measure space and ρ an admissible function on X . In this paper, we first establish some equivalent characterizations for E α, p ρ (X ) and E α, p ρ (X ) when α ∈ (−∞, 0) and p ∈ [1, ∞). To obtain the boundedness of the Lusin-area function on the Morrey-Campanato spaces, we need to assume that X has the volume regularity Property (P ), which was introduced in [19], motivated by Colding-Minicozzi II [4] and Tessera [30]. We remark that the volume regularity property is related to the Følner sequence of a compact generating set of a compactly generated locally compact group with polynomial growth in [30] and used to establish the generalized Liouville theorems for harmonic sections of Hermitian vector bundles over a complete metric space in [4].
In this paper, if X has Property (P ), we then establish the boundedness of the Lusin-area function from E α, p ρ (X ) to E α, p ρ (X ) without invoking any regularity of considered kernels. The corresponding boundedness of g * λ function from E α, p ρ (X ) to E α, p ρ (X ) is also established in this paper. Both the Lusin-area function and the g * λ function are defined via kernels modeled on the semigroup generated by the Schrödinger operator. Moreover, an interesting phenomena is that unlike the Lusin-area function, the boundedness of the g * λ function needs neither the regularity of the kernels nor Property (P ) of X , which reflects the speciality of the structure of the g * λ function. These results are new even on R d with the d-dimensional Lebesgue measure and the Heisenberg group, and apply in a wide range of settings, for instance, 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.
This paper is organized as follows. Let X be a doubling metric measure space and ρ an admissible function on X . In Section 2, we establish some equivalent characterizations for E α, p ρ (X ) and E α, p ρ (X ) when p ∈ [1, ∞) and α ∈ (−∞, −1/p) or α ∈ [−1/p, 0); see Theorems 2.1 and 2.2 below. Moreover, under the assumption that sup x∈X μ(B(x, ρ(x))) = ∞, we prove that the Morrey- In Section 3, assuming that X has Property (P ) and the Lusin-area where C is a positive constant independent of f ; see Theorem 3.1 below. As a corollary, we obtain the boundedness of the Lusin-area function from E α, p ρ (X ) to E α, p ρ (X ); see Corollary 3.1 below. If the g * λ function g * λ (f ) is bounded on L p (X ) with p ∈ (1, ∞), the corresponding results for g * λ (f ) are also established, and moreover, in this case, X is not necessary to have Property (P ); see Theorem 3.2 and Corollary 3.2 below. We point out that Theorems 3.1 and 3.2 and Corollaries 3.1 and 3.2 are true 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; see [34] for the detailed explanations. Notice that E 0, p ρ (X ) = BMO ρ (X ) and E 0, p ρ (X ) = BLO ρ (X ) when p ∈ [1, ∞). Thus, the results in this section when α = 0 were already obtained in [19]. λ functions We remark that the results obtained in Section 3 are also new even on R d with the d-dimensional Lebesgue measure and the Heisenberg group, since we do not need any regularity of involved kernels. However, to establish the boundedness of Lusin-area function on a doubling metric measure space X , we need certain regularity of X , namely, the volume regularity Property (P ), which reflects the speciality of the Lusin-area function, comparing with the corresponding results of the g * λ function. Moreover, R d with the Lebesgue measure and the Heisenberg group have the volume regularity Property (P ); see [19].
Finally, we make some conventions. Throughout this paper, we always use C 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 and K 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. We also use B to denote a ball of X , and for λ > 0, λB denotes the ball with the same center as B , but radius λ times the radius of B . Moreover, set B ≡ X \ B . Also, for any set E ⊂ X , χ E denotes its characteristic function. For all f ∈ L 1 loc (X ) and balls B , we always set

Some characterizations of localized Morrey-Campanato spaces
Let X be a doubling metric measure space and ρ an admissible function on X . In this section, we establish some equivalent characterizations for E α, p ρ (X ) and E α, p ρ (X ) when α ∈ (−∞, 0) and p ∈ [1, ∞). Moreover, under the assumption that sup x∈X μ(B(x, ρ(x))) = ∞, we prove that the Morrey-Campanato-BLO space E α, p ρ (X ) is a proper subspace of the Morrey-Campanato space E α, p ρ (X ) when p ∈ [1, ∞) and α ∈ [−1/p, 0). We begin with recalling the notion of doubling metric measure spaces [2,3]. Definition 2.1. Let (X , d) be a metric space endowed with a Borel regular measure μ such that all balls defined by d have finite and positive measures. For any x ∈ X and r ∈ (0, ∞), set the ball B(x, r) ≡ {y ∈ X : d(x, y) < r}. The triple (X , d, μ) is called a doubling metric measure space if there exists a constant C 1 ∈ [1, ∞) such that for all x ∈ X and r ∈ (0, ∞), From Definition 2.1, it is easy to see that there exist positive constants C 2 and n such that for all x ∈ X , r ∈ (0, ∞) and λ ∈ [1, ∞), (2.2) μ(B(x, λr)) ≤ C 2 λ n μ(B(x, r)).
Now we recall the notion of admissible functions introduced in [35].
. A positive function ρ on X is called admissible if there exist positive constants C 0 and k 0 such that for all x, y ∈ X , Obviously, if ρ is a constant function, then ρ is admissible. Moreover, let x 0 ∈ X be 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 written as B q (X ) for simplicity. 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+ (X ) for certain ∈ (0, ∞); see, for example, [27] and [28]. Thus B q (X ) = ∪ q1∈(q, ∞] B q1 (X ). For all V ∈ B q (X ) with q ∈ (1, ∞] and all x ∈ X , set see, for example, [26] and also [35]. It was also proved in [35] that ρ in (2.4) is an admissible function if n ∈ [1, ∞), q ∈ (max{1, n/2}, ∞] and V ∈ B q (X ).
The following localized Morrey-Campanato space and localized Morrey-Campanato-BLO space associated to the admissible function ρ were first introduced in [34]. x ∈ X , r ≥ ρ(x)} , p ∈ (0, ∞) and α ∈ R. Denote by B any ball of X . λ functions (i) A function f ∈ L p loc (X ) is said to be in the localized Morrey- . A function f on X is said to be in the localized Lipschitz space Lip ρ (α; X ) if there exists a nonnegative constant C such that for all x, y ∈ X and balls B containing x and y with B / ∈ D, The minimal nonnegative constant C as above is called the norm of f in Lip ρ (α; X ) and denoted by f Lip ρ (α; X ) . ( Remark 2.1. (i) For all α ∈ R and p ∈ (0, ∞), E α, p ρ (X ) ⊂ E α, p ρ (X ). (ii) When α = 0 and p ∈ [1, ∞), we denote E 0, p ρ (X ) by BMO p ρ (X ) and BMO 1 ρ (X ) by BMO ρ (X ). And we also denote E 0, p ρ (X ) by BLO p ρ (X ) and BLO 1 ρ (X ) by BLO ρ (X ). The localized BLO space was first introduced in [13] in the setting of R d endowed with a nondoubling measure.

Remark 2.2. (i) It turns out that Theorem 2.2(i), (ii) and (iii) hold for
are already obtained in [34], which are used to prove Theorem 2.2(i) & (ii). Also we show Theorem 2.2(iii) by first assuming that it is true for RD-space X , which is proved in Proposition 2.1 below. Recall that the space X is said to have the reverse doubling property if there exist constants κ ∈ (0, n] and K 1 ∈ (0, 1] such that for all x ∈ X , r ∈ (0, 2 diam (X )) and λ ∈ (1, 2 diam (X )/r), If (X , d, μ) satisfies the conditions (2.2) and (2.5), then (X , d, μ) is called an RD-space, which was first introduced in [12] (see also [12,36] for some equivalent characterizations of RD-spaces).
(iii) By an argument similar to that used in the proof of Theorem 2.2(i) & (ii) when (X , d, μ) is an RD-space with d being a metric in [34], it is easy to see that if (X , d, μ) is an RD-space with d being a quasi-metric, Theorem 2.2(i) & (ii) are also true. Moreover, a slight modification of the proof below shows that the whole Theorem 2.2 holds for X with d being a quasi-metric.
To prove Theorem 2.2, we need some technical lemmas. Following Macías and Segovia [20], we call a doubling metric measure space to be normal if there exist positive constants K 2 and K 3 such that for all x ∈ X and μ({x}) < r < μ(X ), For a doubling metric measure space (X , d, μ), let Macías and Segovia [20] showed that (X , δ, μ) is a normal space of homogeneous type, namely, δ is a quasi-metric and μ satisfies (2.1) and (2.6). Moreover, the topologies induced on X by d and δ coincide.
Lemma 2.1. Let X be a doubling metric measure space and ρ an admissible function on X , and let ρ δ be as in (2.8).
Lemma 2.2. Let X be a doubling metric measure space and ρ an admissible function on X , and let ρ δ be as in (2.8).
To prove Lemmas 2.1, 2.2 and 2.3, we first state some basic facts. For x, r)). By the definition of δ , we have that Moreover, Conversely, by [22,Lemma 3.9] or [14, Proposition 2.1], for any δ -ball B δ (x, r), there exists a positive constantr , which may depend on x and r , such that That is, Proof of Lemma 2.1. By (2.9) and (2.11), it is easy to see . In this case, by (2.10),r ≥ ρ δ (x), which together with (2.9) leads to that . In this case, by (2.12),r < ρ(x). By an argument similar to the estimate of (2.13), we have In this case, by (2.12),r > ρ(x). By an argument similar to the estimate of (2.14), we have .
and we are done.
. In this case, by (2.10),r ≥ ρ δ (x). By an argument similar to the estimate of (2.14), we have .
By the same way as in the proof of Lemma 2.2, we divide the proof into (Part 1) and (Part 2). Then we have the same conclusions as in Case 2 of (Part 1) and in Cases 1 and 2 of (Part 2) of the proof of Lemma 2.2. So we only need to consider Case 1 of (Part 1) and Case 3 of (Part 2) therein.
(ii) If f ≥ 0 , then, by Remark 2.2(iii), we obtain that f E α, p ρ δ (X , δ, μ) ∼ f E α, p ρ δ (X , δ, μ) , which together with Lemmas 2.1 and 2.2 yields that (iii) In the case M ≡ sup x∈X μ(B(x, ρ(x))) < ∞, if −∞ < essinf X f < 0 , then, by Proposition 2.1 below, we obtain that which together with Lemmas 2.1 and 2.3 yields that (iv) Since sup x∈X μ(B(x, ρ(x))) = ∞, we choose B j ≡ B(z j , ρ(z j )/2), j ∈ N, so that μ(B j ) → ∞ as j → ∞. Then, we have two situations that Then essinf X f = −b and by Lemma 2.4, we have On the other hand, Then essinf X f = −b and by Lemma 2.4, we have On the other hand, for j > j 0 , Combining the estimates for Cases (I) and (II) yields (vi), which completes the proof of Theorem 2.2.
In the proof of Theorem 2.2(iii) above, we used the following proposition.

Proposition 2.1. Theorem 2.2 (iii) holds if X is an RD-space.
To prove Proposition 2.1, we begin with some technical lemmas. A straightforward computation via (2.5) leads to the following technical lemma.

Lemma 2.5.
Let X be an RD-space and θ ∈ (0, ∞). Then, there exists a positive constant C such that for all z ∈ X and 0 < r < s < ∞, Then, by Lemma 2.4 in [24], there exists a positive constant C such that for all z ∈ X and 0 < r < s < ∞, Lemma 2.6. Let X be an RD-space and α ∈ [−1/p, 0). Then there exists a positive constant C such that for all f ∈ E α, p ρ (X ), z ∈ X and 0 < r < ρ(z), Proof. Case 1. ρ(z)/2 ≤ r < ρ(z). By (2.1) and the Hölder inequality, we have Case 2. 0 < r < ρ(z)/2 . Using (2.17) and Lemma 2.5, we have Combining the estimates for Case 1 and Case 2 completes the proof of Lemma 2.6.

Proof of Proposition
which completes the proof of Proposition 2.1.

Boundedness of Lusin-area and g * λ functions
Let X be a doubling metric measure space and ρ an admissible function. In this section, we consider the boundedness of certain variants of Lusinarea and g * λ functions from E α, p ρ (X ) to E α, p ρ (X ). The boundedness from BMO ρ (X ) to BLO ρ (X ) of these operators were obtained in [19]. We remark that unlike the boundedness of the g * λ function, to obtain the boundedness of the Lusin-area function, we need to assume that X has the following volume regularity Property (P ), which was introduced in [19]; see also [4,30].  , μ), the nilpotent Lie group G with a Carnot-Carathéodory (control) distance d and a left invariant Haar measure μ and so on; see [19] for more details.
To prove Theorem 3.1, we begin with the following two technical lemmas, which were obtained in [34].
Case I. B ≡ B(x 0 , r) ∈ D . In this case, r ≥ ρ(x 0 ). We need to prove that For all x ∈ X , write By the L p (X )-boundedness of S(f ) and (2.1), we have From (Q) i , (3.7), (2.2), the Hölder inequality and γ > αn, it follows that for all t < 8r and y ∈ X with d(x, y) < t, we have Notice that for all x, y ∈ X satisfying d(x, y) < t, we have It then follows from (3.8) and (3.9) together with γ ∈ (0, ∞) that (3.10) which together with (3.6) tells us that Observe that for all y ∈ X with d(x, y) < t, by (2.3), we have and that for all x ∈ B with r ≥ ρ(x 0 ), by (2.3), we also have that ρ(x) r . Combining these two observations yields that for all x ∈ B and y ∈ X with d(x, y) < t, It then follows from Lemma 3.2, (3.13) and (2.2) that for all x ∈ B , t ≥ 8r and y ∈ X with d(x, y) < t, λ functions which together with the assumption that δ 1 > (1 + k 0 )αn implies that By this and (3.11), we obtain (3.5). Moreover, it follows from (3.5) that In this case, r < ρ(x 0 ). We need to prove that To this end, for all x ∈ X , write Write . By the L p (X )-boundedness of S(f ) and (2.2), we have It follows from (Q) i , (3.7), (2.2), the Hölder inequality, Lemma 3.1(ii) and γ > αn that for all x ∈ B and y ∈ X with d(x, y) < t ≤ 8r , which together with (3.9) leads to that (3.17) Observe that by (2.3), 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, we obtain that for all x ∈ B and y ∈ X satisfying d(x, y) < t with t ∈ (0, 8r) and r < ρ(x 0 ), ρ(y) ∼ ρ(x 0 ). Hence, by (Q) ii and Lemma 3.1(i), we have which together with δ 2 > αn, r < ρ(x 0 ) and (3.9) implies that Combining this, (3.16) and (3.17) yields I 1 [μ(B)] 1+αp . λ functions Now we turn our attention to prove that for all x, x ∈ B , . Since X has the volume regularity Property (P ), we obtain , t)).
As a consequence of Theorem 3.1, we have the following conclusion, which can be proved by an argument similar to that used in the proof of [34,Corollary 4.1]. We omit the details.
By similarity, we only prove the case when α > 0. Let f ∈ E α, p ρ (X ), by the homogeneity of · E α, p ρ (X ) and · E 2α, p/2 ρ (X ) , we may assume that f E α, p ρ (X ) = 1. Let B ≡ B(x 0 , r). For any nonnegative integer k , let For any f ∈ E α, p ρ (X ) and x ∈ X , write We now consider the following two cases.
As a consequence of Theorem 3.2, we have the following conclusion.

Corollary 3.2.
With the assumptions same as in Theorem 3.2, then there exists a positive constant C such that for all f ∈ E α, p ρ (X ), g * λ (f ) ∈ E α, p ρ (X ) and g * λ (f ) E α, p ρ (X ) ≤ C f E α, p ρ (X ) .
We point out that Remark 3.2 is also suitable to Theorem 3.2 and Corollary 3.2.
The following is a simple corollary of Theorems 3.1 and 3.2, and Corollaries 3.1 and 3.2. We omit the details here; see [34,Section 5].