Pointwise multipliers for localized morrey-campanato spaces on RD-spaces

Acta Mathematica Scientia 2014,34B(6):1677–1694 http://actams.wipm.ac.cn

POINTWISE MULTIPLIERS FOR LOCALIZED MORREY-CAMPANATO SPACES ON RD-SPACES∗

°Å)

Haibo LIN (

College of Science, China Agricultural University, Beijing 100083, China E-mail : [email protected]

ŒS)

Dachun YANG (

†

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China E-mail : [email protected] Abstract In this article, the authors characterize pointwise multipliers for localized MorreyCampanato spaces, associated with some admissible functions on RD-spaces, which include localized BMO spaces as a special case. The results obtained are applied to Schr¨ odinger operators and some Laguerre operators. Key words

RD-space; pointwise multiplier; localized Morrey-Campanato space; BMO space

2010 MR Subject Classification

1

42B15; 42B30

Introduction

The pioneering work on the characterizations of pointwise multipliers on BMOφ (Tn ) and the Hardy space H 1 (Tn ) was due to Janson [1], where Tn denotes the n-dimensional torus and BMOφ (Tn ) is the function space defined by using the mean oscillation and some growth function φ. Specially, when n = 1, the same characterizations as in [1] for the pointwise multipliers on BMOφ (T) and H 1 (T) were obtained by Stegenga [2], which were further used to study the boundedness of the Toeplitz operator on Hardy spaces H 1 (T). Later, Nakai and Yabuta extended Janson’s results to the n-dimensional Euclidean space Rn in [3] and to spaces of homogeneous type in the sense of Coifman and Weiss in [4]. It should be mentioned that the class of pointwise multipliers for BMO(Rn ) was used by Lerner [5] to solve a conjecture of Diening in [6] on the boundedness of the Hardy-Littlewood maximal operators on the variable Lebesgue spaces. Also, by invoking the results of [3, 4] on pointwise multipliers, Lin et al. ∗ Received September 18, 2013; revised January 20, 2014. The first author is supported by the National Natural Science Foundation of China (11301534), Da Bei Nong Education Fund (1101-2413002) and Chinese Universities Scientific Fund (2013QJ003). The second (corresponding) author is supported by the National Natural Science Foundation of China (11171027 and 11361020), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20120003110003) and the Fundamental Research Funds for Central Universities of China (2012LYB26 and 2012CXQT09). † Corresponding author: Dachun YANG.

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[7] constructed a nonnegative function in BMO(Rn ) but not in BLO(Rn ), which showed that the results obtained in [7] on the boundedness of the Lusin area and the gλ∗ functions indeed improved the known corresponding results even on Rn . Recently, Ky further proved that the class of pointwise multipliers for BMO(Rn ) characterized by Nakai and Yabuta can be seen as the dual of the Hardy space of Musielak-Orlicz type; see [8] for details. On the other hand, in recent years, the study of function spaces, including Hardy spaces and BMO spaces, associated with Schr¨ odinger operators and Laguerre operators, inspired great interests; see, for example, [9–20] and their references. Let L := −∆ + V be the Schr¨odinger operator on Rn , where the potential V is a nonnegative locally integrable function. Let X be an RD-space (see Definition 1.1 below), which means that X is a space of homogenous type in the sense of Coifman and Weiss [21, 22] with the additional property that a reverse doubling property holds in X . Let ρL be a given auxiliary function modeled on the known auxiliary function determined by V . Yang et al. [16, 17] studied the localized Hardy space Hρ1L (X ) and the BMO-type space BMOρL (X ) and, especially, proved that the dual space of Hρ1L (X ) is BMOρL (X ). These extend the corresponding results on Rn established by Dziuba´ nski et al. [14, 15]; see Section 4 for details. The auxiliary function ρL in [16, 17] satisfies that there exist positive constants C0 and k0 such that, for all x, y ∈ X ,  k 1 1 d(x, y) 0 ≤ C0 1+ . (1.1) ρL (x) ρL (y) ρL (y) Moreover, let X be the space (0, ∞) equipped with the Euclidean distance | · | and the Lebesgue measure dx, which is a special RD-space. Let ρLa and ρLa be the auxiliary functions, respectively, associated with the Laguerre operators La with a ∈ (− 21 , ∞) and La with a ∈ (0, ∞) defined by setting, for all x ∈ (0, ∞), ρLa (x) :=

1 1 min{x, x−1 } and ρLa (x) := min{x, 1}. 8 8

(1.2)

Dziuba´ nski [18, 19] introduced the atomic Hardy spaces Hρ1La ((0, ∞)) and Hρ1La ((0, ∞)) and, moreover, Cha and Liu [20] studied the BMO-type space BMOρLa ((0, ∞)) and proved that the dual space of Hρ1La ((0, ∞)) is BMOρLa ((0, ∞)); see Section 4 for details. It is well known that the theory of Morrey-Campanato spaces (including BMO spaces as a special case) plays an important role in harmonic analysis and partial differential equations; see, for example, [23–31] and their references. Nakai [32] established a characterization of pointwise multipliers for Morrey spaces on Rn . Very recently, Liu and Yang [33] characterized pointwise multipliers for Campanato spaces on the Gauss measure space. Motivated by the work mentioned above, the main purpose of this article is to study characterizations for pointwise multipliers of the localized Morrey-Campanato spaces associated with some admissible functions ρ on RD-spaces. Since the admissible function ρ considered in this article unifies the auxiliary function ρL associated with Schr¨odinger operators and the auxiliary functions ρLa and ρLa associated with some Laguerre operators, the results obtained are then applied to Schr¨ odinger operators and some Laguerre operators. To state our main result, we first recall some necessary notions and notation. We begin with recalling the notions of spaces of homogeneous type in the sense of Coifman and Weiss [21, 22] and RD-spaces in [34].

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Definition 1.1 Let (X , d) be a metric space endowed with a regular Borel measure µ such that all balls defined by d have finite and positive measures. For any x ∈ X and r ∈ (0, ∞), let the ball B(x, r) := {y ∈ X : d(x, y) < r}. (i) The triple (X , d, µ) is called a space of homogeneous type if there exists a constant C1 ∈ [1, ∞) such that, for all x ∈ X and r ∈ (0, ∞), µ(B(x, 2r)) ≤ C1 µ(B(x, r)) (doubling property).

(1.3)

(ii) The triple (X , d, µ) is called an RD-spaces if there exist constants κ ∈ (0, n] and C2 ∈ [1, ∞) such that, for all x ∈ X , r ∈ (0, diam (X )/2) and λ ∈ [1, diam (X )/(2r)), (C2 )−1 λκ µ(B(x, r)) ≤ µ(B(x, λr)) ≤ C2 λn µ(B(x, r)),

(1.4)

where diam (X ) := sup d(x, y). x, y∈X

Remark 1.2 (i) Obviously, an RD-space is a space of homogeneous type. Conversely, a space of homogeneous type automatically satisfies the second inequality of (1.4). (ii) It was proved in [34, Remark 1.1] that, if µ is doubling, then µ satisfies (1.4) if and only if there exist constants a0 , C0 ∈ (1, ∞) such that, for all x ∈ X and r ∈ (0, diam (X)/a0 ), µ(B(x, a0 r)) ≥ C0 µ(B(x, r)) (reverse doubling property) (if a0 = 2, this is the classical reverse doubling condition) and, equivalently, for all x ∈ X and r ∈ (0, diam (X)/a0 ), [B(x, a0 r) \ B(x, r)] 6= ∅, which is known in the topology as the uniform perfectness. See also [35] for more equivalent characterizations of RD-spaces. Throughout the entire article, we always assume that X is an RD-space with κ = n. Definition 1.3 A positive function ρ on X is said to be admissible if, for all τ ∈ (0, ∞), there exists a constant Cτ ∈ [1, ∞) such that, for all x, y ∈ X with d(x, y) ≤ τ ρ(x), (Cτ )−1 ρ(y) ≤ ρ(x) ≤ Cτ ρ(y).

(1.5)

Remark 1.4 Obviously, if ρ is a constant function, then ρ is admissible. It was proved in [16, Lemma 21] that ρL as in (1.1) satisfies (1.5), which implies that ρL is admissible. In Section 4, we show that, for all τ ∈ (0, 8), ρLa and ρLa satisfy (1.5). Moreover, this is enough in applying our results to some Laguerre operators; see Section 4 for details. In what follows, for an admissible function ρ as in (1.5), we always let Dρ := {B(x, r) : x ∈ X , r ∈ (0, ρ(x))}.

(1.6)

The following localized Morrey-Campanato space associated to the admissible function ρ as in (1.5) is a special case of the one introduced in [36]. Definition 1.5 Let ρ be an admissible function on X , Dρ be as in (1.6), p ∈ (0, ∞) and α ∈ R. Denote by B any ball of X . A function f ∈ Lploc (X ) is said to be in the localized Morrey-Campanato space Eρα, p (X ) if  1/p Z 1 p α, p kf kEρ (X ) := sup |f (y) − fB | dµ(y) [µ(B)]1+pα B B∈Dρ

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+ sup here and hereafter, fB :=

1 µ(B)

R

B ∈D / ρ

B



1 [µ(B)]1+pα

Z

B

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1/p |f (y)|p dµ(y) < ∞,

f (x) dµ(x).

Remark 1.6 When α = 0 and p ∈ [1, ∞), we denote Eρ0, p (X ) by BMOpρ (X ) and further denote BMO1ρ (X ) by BMOρ (X ). If X is the Euclidean space Rd and ρ ≡ 1, then BMOρ (X ) is just the localized BMO space of Goldberg [37]. Let α ∈ (0, ∞). A function f on X is said to be in the localized Lipschitz space Lipρ (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 ∈ Dρ , |f (x) − f (y)| ≤ C[µ(B)]α and, for all balls B ∈ / Dρ , kf kL∞(B) ≤ C[µ(B)]α . The minimal nonnegative constant C as above is called the norm of f in Lipρ (α; X ) and denoted by kf k Lipρ (α; X ) . If X is the Euclidean space Rd and ρ ≡ 1, then Lipρ (α; X ) with α ∈ (0, 1) is just the inhomogeneous Lipschitz space; see [37]. Let p ∈ [1, ∞) and α ∈ (−∞, 0). A function f ∈ Lploc (X ) is said to be in the Morrey space Mα, p (X ) if  1/p Z 1 p kf kMα, p (X ) := sup |f (x)| dµ(x) < ∞, [µ(B)]1+αp B B⊂X where the supremum is taken over all balls B of χ. Applying [36, Lemma 2.2] and some arguments similar to those used in the proofs of [38, Theorems 2.1 and 2.2], we obtain the following equivalent characterizations of the space Eρα, p (X ), the details being omitted. Proposition 1.7 Let p ∈ [1, ∞). Then the following hold: (i) when α ∈ (−∞, −1/p) and µ(X ) = ∞, Eρα, p (X ) = Mα, p (X ) = {0}; −1/p, p (ii) when α ∈ (−∞, −1/p) and µ(X ) < ∞, Eρα, p (X ) = Eρ (X ) and Mα, p (X ) = M−1/p, p (X ) = Lp (X ) with equivalent norms, respectively; (iii) when α ∈ [−1/p, 0), Eρα, p (X ) = Mα, p (X ) with equivalent norms; (iv) when α = 0, Eρα, p (X ) = BMOpρ (X ) = BMOρ (X ) with equivalent norms; (v) when α ∈ (0, ∞), Eρα, p (X ) = Lipρ (α; X ) with equivalent norms. Definition 1.8 Let p ∈ [1, ∞) and α ∈ R. A function g is called a pointwise multiplier on Eρα, p (X ) if, for all f ∈ Eρα, p (X ), the multiplication f g belongs to Eρα, p (X ) and kf gkEρα, p (X ) ≤ Ckf kEρα, p (X ) for some positive constant C independent of f . Our main result of this article is as follows, whose proof is given in Section 3 below. Theorem 1.9 Let κ = n be as in (1.4), p ∈ [1, ∞) and α ∈ (−1/p, 1/n]. Then a function α, p g is a pointwise multiplier on Eρα, p (X ) if and only if g ∈ L∞ (X ) ∩ EΨ (X ), where the space α α, p p EΨα (X ) is defined to be the collection of all g ∈ L loc (X ) such that  1/p Z 1 p α, p kgkEΨ (X ) := sup Ψα (B) |g(x) − gB | dµ(x) < ∞, α µ(B) B B∈Dρ

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where Ψα (B) is given by setting, for all B ∈ Dρ ,     1,     ρ(cB ) Ψα (B) := 1 + ln r , B   αn    ρ(c ) B   , rB

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α ∈ (−1/p, 0); α = 0;

(1.7)

α ∈ (0, 1/n].

Moreover, the operator norm of g, denoted by kgkEρα, p (X )→Eρα, p (X ) , is comparable with kgkL∞ (X ) + kgkEΨα, p (X ) . α

α, p Observe that, in Theorem 1.9, if α ∈ (−1/p, 0), then L∞ (X ) ⊂ EΨ (X ). From this and α Theorem 1.9, we immediately deduce the following Corollary 1.10, the details being omitted.

Corollary 1.10 Let κ = n be as in (1.4), p ∈ [1, ∞) and α ∈ (−1/p, 0). Then a function g is a pointwise multiplier on Mα, p (X ) if and only if g ∈ L∞ (X ). Moreover, the operator norm of g, denoted by kgkMα, p (X )→Mα, p (X ) , is comparable with kgkL∞ (X ) . 1, q The following localized atomic Hardy space HD (X ) was introduced in [36].

Definition 1.11 Let D be a collection of balls in X and q ∈ (1, ∞]. R (i) A function a supported in a ball B ⊂ X is called a (1, q)-atom if X a(x) dµ(x) = 0 and kakLq (X ) ≤ [µ(B)]1/q−1 . (ii) A function b supported in a ball B ∈ D is called a (1, q)D -atom if kbkLq (X ) ≤ [µ(B)]1/q−1 . 1, q A function f ∈ L1 (X ) is said to be in HD (X ) if there exist {λj }j∈N , {νk }k∈N , (1, q)-atoms {aj }j∈N and (1, q)D -atoms {bk }k∈N such that X X f= λj aj + νk bk j∈N

in L1 (X ) and

P

j∈N

|λj | +

P

k∈N

k∈N

1, q |νk | < ∞. Moreover, the norm of f in HD (X ) is defined by

kf kH 1, q (X ) := inf D

(

X j∈N

|λj | +

X

k∈N

)

|νk | ,

where the infimum is taken over all the decompositions of f as above. 1, q Remark 1.12 It was proved in [36] that, for all q ∈ (1, ∞), the spaces HD (X ) and 1, ∞ 1, q HD (X ) coincide with equivalent norms. Thus, in what follows, we denote HD (X ) simply 1 1 by HD (X ) and, in the case that D := X \ Dρ with Dρ in (1.6), we denote HD (X ) by Hρ1 (X ).

From [36, Theorem 2.1], we deduce that the dual space of Hρ1 (X ) is BMOρ (X ). This, together with Theorem 1.9, gives us the following conclusion, the details being omitted.

Corollary 1.13 Let κ = n be as in (1.4). The following three statements are equivalent: (i) g is a pointwise multiplier on the space BMOρ (X ); (ii) g is a pointwise multiplier on the space Hρ1 (X ); (iii) g ∈ L∞ (X ) ∩ BMOln (X ), where B) Z 1 + ln ρ(c rB kgkBMOln (X ) := sup |g(x) − gB | dµ(x) < ∞. µ(B) B∈Dρ B Moreover, the operator norm of g is comparable with kgkL∞ (X ) + kgkBMOln (X ) .

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This article is organized as follows. In Section 2, we present some technical lemmas to be used in the proof of Theorem 1.9, which is given in Section 3. In Section 4, we give some examples of differential operators, including Schr¨ odinger operators and some Laguerre operators, which the results of this article can be applied to. We find an interesting phenomenon that some results of the atomic Hardy spaces associated with some Laguerre operators are included in the theory of the localized atomic Hardy spaces established in [36]; see Section 4 for details. We remark that some ideas used to prove Theorem 1.9 are inspired by Nakai and Yabuta [3, 4]. Applying Lemma 2.1, the proof of the sufficiency of Theorem 1.9 is not difficult. Moreover, B ) αn in Lemma 2.1, when α ∈ (0, 1/n] and B ∈ Dρ , the coefficient [ ρ(c appears in the upper rB ] bounded estimate by the doubling property of the measure (the second inequality of (1.4)). The proof of the necessary of Theorem 1.9 is more complicate. Indeed, to this end, we construct a special function (see Lemma 2.2 below) associated with a fixed ball B ∈ Dρ , while, if α ∈ B ) ακ (0, 1/n], the coefficient [ ρ(c appears in the lower bounded estimate by the reverse doubling rB ] property of the measure (the first inequality of (1.4)). To complete the proof of Theorem 1.9, we therefore need to assume that κ = n. We mention that RD-spaces with κ = n include Euclidean spaces, connected subsets of Rn equipped with a Lebesgue measure, Ahlfors regular metric measure spaces and Heisenberg groups. Finally, we make some conventions on notation. Denote by C a positive constant independent of the main parameters involved, which may vary at different occurrences. We use f . h or h & f to denote f ≤ Ch or h ≥ Cf , respectively. If f . h . f , we write f ∼ h. We also use B := B(cB , rB ) := {x ∈ X : d(x, cB ) < rB } to denote a ball of X , where cB and rB denote its center and radius, respectively.

2

Some Auxiliary Lemmas In this section, we establish some technical lemmas, which are used in the proof of Theorem

1.9. Lemma 2.1 Let p ∈ [1, ∞), α ∈ R, ρ be an admissible function on X and Dρ as in (1.6). Then there exists a positive constant C such that, for all f ∈ Eρα, p (X ), (i) for all balls B, B ′ ⊂ X with cB = cB ′ and rB < rB ′ ,   C[µ(B)]α kf kEρα, p (X ) ,         rB ′ 1 + ln C [µ(B)]α kf kEρα, p (X ) , |fB − fB ′ | ≤ r B    αn    r ′  C B [µ(B)]α kf kEρα, p (X ) , rB (ii) for all balls B ∈ Dρ ,   C[µ(B)]α kf kEρα, p (X ) ,         ρ(cB ) [µ(B)]α kf kEρα, p (X ) , |fB | ≤ C 1 + ln r B   αn     ρ(cB )  C [µ(B)]α kf kEρα, p (X ) , rB

α < 0, α = 0, α > 0;

α < 0, α = 0, α > 0.

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Proof To prove (i), let j0 be the smallest integer such that ej0 rB ≥ rB ′ . By (1.4) and Definition 1.5, we have Z 1 |fB(cB , ej0 rB ) − fB(cB , rB′ ) | . |f (x) − fB(cB , ej0 rB ) | dµ(x) µ(B(cB , ej0 rB )) B(cB , ej0 rB ) . [µ(B(cB , ej0 rB ))]α kf kEρα, p (X ) .

Similarly, we see that, for all j ∈ N ∪ {0}, |fB(cB , ej rB ) − fB(cB , ej+1 rB ) | . [µ(B(cB , ej+1 rB ))]α kf kEρα, p (X ) . Hence, |fB − fB ′ | .

jX 0 −1

|fB(cB , ej rB ) − fB(cB , ej+1 rB ) | + |fB(cB , ej0 rB ) − fB(cB , rB′ ) |

j=0

.

jX 0 −1 j=0

[µ(B(cB , ej+1 rB ))]α kf kEρα, p (X ) .

If α = 0, from the choice of j0 , we deduce that   rB ′ [µ(B)]α kf kEρα, p (X ) ; |fB − fB ′ | . 1 + ln rB if α 6= 0, by (1.4), we obtain  j −1 0   X ακ j+1  (e ) [µ(B)]α kf kEρα, p (X ) . [µ(B)]α kf kEρα, p (X ) ,    j=0 |fB − fB ′ | . j −1 αn  0  X  rB ′  αn j+1 α  [µ(B)]α kf kEρα, p (X ) , (e ) [µ(B)] kf kEρα, p (X ) .   rB

α < 0,

α > 0,

j=0

which implies that (i) holds true. Now we prove (ii). Notice that rB < ρ(cB ) for all balls B ∈ Dρ . By the H¨older inequality, we conclude that, for all f ∈ Eρα, p (X ) and B ∈ Dρ , Z 1 |fB | ≤ |fB − fB(cB , ρ(cB )) | + |f (x)| dµ(x) µ(B(cB , ρ(cB ))) B(cB , ρ(cB )) ≤ [µ(B(cB , ρ(cB )))]α kf kEρα, p (X ) + |fB − fB(cB , ρ(cB )) |,

which, together with (1.4), rB < ρ(cB ) and (i), implies that (ii) holds true. This finishes the proof of Lemma 2.1.  Lemma 2.2 Let κ = n be as in (1.4), p ∈ [1, ∞), α ∈ (−1/p, 1/n] and α 6= p. Given any ball B ∈ Dρ , define ( Z ) 3ρ(cB ) [µ(B(cB , t))]α f (x) := max 0, dt , x ∈ X . t d(x, cB ) Then there exists a positive constant C, independent of the ball B, such that (i) kf kEρα, p (X ) ≤ C; (ii) f (x) ≥ Ψα (B)[µ(B)]α for all x ∈ B. Proof We first prove (ii). Notice that rB < ρ(cB ) for any B ∈ Dρ and d(x, cB ) < rB for all x ∈ B. To prove (ii), we consider the following three cases for α.

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Case 1) α ∈ (−1/p, 0). In this case, by (1.4), for all x ∈ B, we have Z 3rB [µ(B(cB , t))]α f (x) ≥ dt & [µ(B(cB , 3rB ))]α & [µ(B)]α . t rB Case 2) α = 0. In this case, a trivial computation leads to that, for all x ∈ B, Z 3ρ(cB ) 3ρ(cB ) 1 f (x) ≥ dt = ln . t rB rB Case 3) α ∈ (0, 1/n]. In this case, by (1.4), we conclude that, for all x ∈ B,  αn Z 3ρ(cB ) ρ(cB ) [µ(B(cB , t))]α α dt & [µ(B(cB , ρ(cB )))] & [µ(B)]α . f (x) ≥ t rB ρ(cB ) Combining the above estimates, we obtain (ii). e∈ To prove (i), we first show that, for any ball B / Dρ , ( )1/p Z 1 p |f (x)| dµ(x) ≤ C, e 1+pα Be [µ(B)]

(2.1)

e Notice that r e ≥ ρ(c e ) and f (x) = 0 where C is a positive constant independent of B and B. B B if x 6∈ B(cB , 3ρ(cB )). We consider the following two cases. Case a) d(cB , cBe ) ≤ 6ρ(cB ). In this case, by (1.5), we see that ρ(cB ) ∼ ρ(cBe ), which, together with (1.4), implies that µ(B(cB , 3ρ(cB ))) ∼ µ(B(cBe , ρ(cBe ))). Let J denote the lefthand side of (2.1). It then follows, from the Minkowski inequality and (1.4), that Z ( )1/p Z 3ρ(cB ) [µ(B(c , t))]α p 1 B J≤ dt dµ(x) e 1+pα B(cB , 3ρ(cB )) d(x, cB ) t [µ(B)] Z 3ρ(cB ) [µ(B(cB , t))]1/p+α 1 dt ≤ e 1/p+α 0 t [µ(B)]  1/p+α Z [µ(B(cB , 3ρ(cB )))]1/p+α 3ρ(cB ) t dt . . 1. 1/p+α e 3ρ(c ) t B [µ(B)] 0

e ∩ B(cB , 3ρ(cB )) = ∅, then it is easy to Case b) d(cB , cBe ) > 6ρ(cB ). In this case, if B e see that J = 0; if B ∩ B(cB , 3ρ(cB )) 6= ∅, then we conclude that rBe > 3ρ(cB ). In the later case, e ∩ B(cB , 3ρ(cB )), we find that letting x0 ∈ B B(cB , 3ρ(cB )) ⊂ B(x0 , 6ρ(cB )) ⊂ B(cBe , 3rBe ),

which, together with an argument similar to that used in the proof of Case a), also establishes (2.1) in this case. Thus, (2.1) always holds true. e ∈ Dρ , there exists some A e ∈ C By (2.1), to prove (i), it suffices to show that, for any B B such that ( )1/p Z 1 p |f (x) − ABe | dµ(x) ≤ C, (2.2) e 1+pα Be [µ(B)] e Observe that where C is a positive constant independent of B and B.

| max{0, a} − max{0, b}| ≤ |a − b| for all a, b ∈ C.

To prove (2.2), we consider the following two cases.

(2.3)

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e we have Case i) d(cB , cBe ) ≤ 2rBe . In this case, for all x ∈ B, d(x, cB ) ≤ d(x, cBe ) + d(cBe , cB ) < 3rBe .

e := B(c e , r e ) ⊂ B(cB , 3r e ) ⊂ B(c e , 5r e ), which, Since d(cB , cBe ) ≤ 2rBe , we see that B B B B B B e Choose together with (1.4), implies that µ(B(cB , 3rBe )) ∼ µ(B). ( Z ) 3ρ(cB ) [µ(B(cB , t))]α ABe := max 0, dt . t 3rB e From the above estimates, (2.3), the Minkowski inequality and (1.4), it follows that Z p Z Z 3rBe [µ(B(cB , t))]α p |f (x) − ABe | dµ(x) ≤ dt dµ(x) t e B(cB , 3rBe ) d(x, cB ) B Z 3r e  p B [µ(B(c , t))]α+1/p B ≤ dt t 0 e 1+pα , . [µ(B(cB , 3r e ))]1+pα ∼ [µ(B)] B

which implies that (2.2) holds true when d(cB , cBe ) ≤ 2rBe . Case ii) d(cB , cBe ) > 2rBe . In this case, choose ( Z ) 3ρ(cB ) [µ(B(cB , t))]α ABe := max 0, dt . t d(cB , cB e) e it is easy to see that For all x ∈ B,

3 1 d(cB , cBe ) ≤ d(x, cB ) ≤ d(cB , cBe ) and rBe > d(x, cBe ) ≥ d(x, cB ) − d(cB , cBe ) , 2 2 which, together with (2.3), the facts that B(cB , d(cB , cBe )) ⊂ B(cBe , 2d(cB , cBe )) and and (1.4), shows that

B(cBe , d(cB , cBe )) ⊂ B(cB , 2d(cB , cBe )),

Z d(x, cB ) [µ(B(c , t))]α rBe B |f (x) − ABe | ≤ dt . [µ(B(cB , d(cB , cBe )))]α d(cB , c e ) t d(c , cBe ) B B rBe ∼ [µ(B(cBe , d(cB , cBe )))]α . d(cB , cBe )

(2.4)

Notice that d(cB , cBe ) > 2rBe . Let K be the left-hand side of (2.2). If α ∈ (−1/p, 0], then K.

[µ(B(cBe , d(cB , cBe )))]α . 1. e α [µ(B)]

If α ∈ (0, 1/n], then 1 − αn ≥ 0, which, together with (1.4), implies that  1−αn rBe [µ(B(cBe , d(cB , cBe )))]α rBe . . 1. K. e α d(cB , cBe ) d(cB , cBe ) [µ(B)]

Combining the above estimates, we see that (2.2) holds true also when d(cB , cBe ) > 2rBe , which completes the proof of (i) and hence Lemma 2.2.  Remark 2.3 For α ∈ (0, 1/n], using the first inequality of (1.4), we know that f (x) & B ) αn for all x ∈ B, while Ψα (B) = [ ρ(c . If we do not assume that κ = n, then rB ] α the inequality f (x) ≥ Ψα (B)[µ(B)] for all x ∈ B may not be true.

B ) ακ α [ ρ(c rB ] [µ(B)]

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The following lemma is a variant of [4, Lemma 3.4]. Lemma 2.4 Suppose that p ∈ [1, ∞) and α ∈ [−1/p, ∞). If f ∈ Eρα, p (X ) and g ∈ L (X ), then f g ∈ Eρα, p (X ) if and only if ∞

F (f, g) := sup |fB | B∈Dρ



1 [µ(B)]1+pα

Z

B

1/p |g(x) − gB | dµ(x) < ∞. p

In this case, |kf gkEρα, p (X ) − F (f, g)| ≤ 3kf kEρα, p (X ) kgkL∞ (X ) . Proof 

For all balls B ∈ / Dρ , since f ∈ Eρα, p (X ) and g ∈ L∞ (X ), it is easy to see that

1 [µ(B)]1+pα

Z

B

1/p  |(f g)(x)|p dµ(x) ≤

1 [µ(B)]1+pα

Z

B

1/p |f (x)|p dµ(x) kgkL∞ (X )

≤ kf kEρα, p (X ) kgkL∞ (X ) .

For all balls B ∈ Dρ , we have Z 1/p Z 1/p − |fB | |(f g)(x) − (f g)B |p dµ(x) |g(x) − gB |p dµ(x) B B Z 1/p ≤ |(f g)(x) − (f g)B − fB g(x) + fB gB |p dµ(x) B

≤

Z

B

≤2

Z

1/p |[f (x) − fB ]g(x)| dµ(x) + |(f g)B − fB gB |[µ(B)]1/p p

B

1/p |[f (x) − fB ]g(x)|p dµ(x)

≤ 2[µ(B)]1/p+α kf kEρα, p (X ) kgkL∞ (X ) ,

which further implies that  1/p Z 1 p |(f g)(x) − (f g)B | dµ(x) [µ(B)]1+pα B  1/p Z 1 p −|fB | |g(x) − gB | dµ(x) ≤ 2kf kEρα, p (X ) kgkL∞(X ) . [µ(B)]1+pα B From the above estimates and Definition 1.5, we conclude the proof of Lemma 2.4.

3



Proof of Theorem 1.9

The main purpose of this section is to prove Theorem 1.9 by using the lemmas established in the previous section. Proof of Theorem 1.9 To show the sufficiency, we need to prove that, for any functions α, p f ∈ Eρα, p (X ) and g ∈ L∞ (X ) ∩ EΨ (X ), α h i (3.1) kf gkEρα, p (X ) . kf kEρα, p (X ) kgkL∞ (X ) + kgkEΨα, p (X ) . α

Indeed, for all balls B ∈ Dρ , by Lemma 2.1 and the definition of Ψα (B), we see that |fB | . Ψα (B)[µ(B)]α kf kEρα, p (X ) ,

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which further implies that  1/p Z 1 p |fB | |g(x) − g | dµ(x) B [µ(B)]1+pα B  1/p Z 1 p |g(x) − gB | dµ(x) . kf kEρα, p (X ) kgkEΨα, p (X ) . . kf kEρα, p (X ) Ψα (B) α µ(B) B From this and Lemma 2.4, we deduce that f g ∈ Eρα, p (X ) and (3.1) holds true, which proves the sufficiency part. Now we show the necessary part of Theorem 1.9. For any ball B ∈ Dρ , let f be the function (associated to B) defined as in Lemma 2.2. With such an f , by Lemmas 2.2 and 2.1, the fact that Ψα (B) & 1, we conclude that  1/p Z 1 p |g(x)| dµ(x) µ(B) B  1/p Z 1 1 p . |f (x)g(x)| dµ(x) Ψα (B) [µ(B)]1+pα B 1/p  Z |(f g)B | 1 p |f (x)g(x) − (f g)B | dµ(x) + . 1+pα [µ(B)] Ψα (B)[µ(B)]α B . kf gkEρα, p (X ) . kgkEρα, p (X )→Eρα, p (X ) .

This, via the Lebesgue differentiation theorem, yields that g ∈ L∞ (X ) and kgkL∞ (X ) . kgkEρα, p (X )→Eρα, p (X ) .

(3.2)

On the other hand, for any ball B ∈ Dρ , with f as in Lemma 2.2, since f, f g ∈ Eρα, p (X ) and g ∈ L∞ (X ), applying Lemmas 2.2 and 2.4, we have  1/p  1/p Z Z 1 1 p p Ψα (B) |g(x) − gB | dµ(x) . |fB | |g(x) − gB | dµ(x) µ(B) B [µ(B)]1+pα B . kf gkEρα, p (X ) + kf kEρα, p (X ) kgkL∞ (X ) . kgkEρα, p (X )→Eρα, p (X ) . Taking supremum over all balls B ∈ Dρ , we conclude that kgkEΨα, p (X ) . kgkEρα, p (X )→Eρα, p (X ) , α

(3.3)

α, p which, together with (3.2), implies that g ∈ L∞ (X ) ∩ EΨ (X ). This proves the necessary part α of Theorem 1.9 and hence finishes the proof of Theorem 1.9. 

4

Applications

This section is divided into two subsections. We apply Theorem 1.9, respectively, to Schr¨odinger operators and some Laguerre operators. 4.1

Schr¨ odinger Operators

Let L := −∆ + V be the Schr¨ odinger operator on Rn , n ≥ 3, where the potential V is a nonnegative locally integrable function. A nonnegative potential V is said to be in Bq (Rn )

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(q ∈ (1, ∞]) if there exists a positive constant C such that the reverse H¨older inequality  1/q Z Z 1 C q [V (y)] dy ≤ V (y)dy |B| B |B| B

holds for all balls B of Rn . For all V ∈ Bq (Rn ) with q ∈ (1, ∞] and x ∈ Rn , set ( ) Z 1 ρ(x) := sup r ∈ (0, ∞) : V (y)dy ≤ 1 ; rn−2 B(x, r)

(4.1)

see [9]. It was proved in [9] that ρ in (4.1) satisfies (1.1) if n ≥ 3 and V ∈ Bn/2 (Rn ). Moreover, Yang and Zhou [16] showed that ρ in (4.1) satisfies (1.1) if n ≥ 1, q ∈ [max{1, n/2}, ∞) and V ∈ Bq (Rn ). The following atomic Hardy space HL1, ∞ (Rn ) was originally introduced by Dziuba´ nski and Zienkiewicz [15]; see also [14]. Definition 4.1 Let ρ be as in (4.1). A measurable function a on Rn is called an H 1, ∞ (Rn )L -atom associated to the ball B(x, r) of Rn , if (i) supp a ⊂ B(x, r) for some x ∈ Rn and r ∈ (0, ∞); (ii) kakL∞ (Rn ) ≤ |B(x, r)|−1 ; R (iii) if r ∈ (0, ρ(x)), then Rn a(y)dy = 0. A function f ∈ L1 (Rn ) is said to be in the atomic Hardy space HL1, ∞ (Rn ) if there exist P P HL1, ∞ (Rn )-atoms {aj }j∈N and {λj }j∈N ⊂ C such that f = λj aj in L1 (X ) and |λj | < ∞. j∈N

Moreover, the norm of f in HL1, ∞ (Rn ) is defined by ( kf kH 1, ∞ (Rn ) := inf L

X j∈N

j∈N

)

|λj | ,

where the infimum is taken over all the decompositions of f as above. Remark 4.2 Let ρ be as in (4.1) and D := X \ Dρ with Dρ in (1.6). It is easy to see that 1 the atomic Hardy space HD (Rn ) in Definition 1.11 and the atomic Hardy space HL1, ∞ (Rn ) in Definition 4.1 coincide with equivalent norms. Moreover, the space BMOρ (Rn ) in Definition 1.5 is just the space BMOL (Rn ) in [14], and it was proved in [14] that BMOL (Rn ) is the dual space of the space HL1, ∞ (Rn ). From Remark 4.2 and Corollary 1.13, we immediately deduce the following conclusion, the details being omitted.

Proposition 4.3 Let L := −∆ + V be the Schr¨odinger operator on Rn , n ≥ 3, with V ∈ Bn/2 (Rn ). Let ρ be as in (4.1) and Dρ as in (1.6). Then the following three statements are equivalent: (i) g is a pointwise multiplier on the space BMOL (Rn ); (ii) g is a pointwise multiplier on the space HL1, ∞ (Rn ); (iii) g ∈ L∞ (Rn ) ∩ BMOln (Rn ), where B) Z 1 + ln ρ(c rB kgkBMOln (Rn ) := sup |g(x) − gB |dx < ∞. |B| B∈Dρ B Moreover, the operator norm of g is comparable with kgkL∞(Rn ) + kgkBMOln (Rn ) . Especially, functions in Cc∞ (Rn ) are, respectively, pointwise multipliers on BMOL (Rn ) and 1, ∞ HL (Rn ).

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4.2

Laguerre Operators

The setting we consider in this subsection is the space (0, ∞) equipped with the Euclidean distance | · | and the Lebesgue measure dx. For every n ∈ {0, 1, · · · }, the n-th Laguerre polynomial of type a is defined by 1 x −a dn −x a+n e x (e x ), x ∈ (0, ∞) n! dxn (see, for example, [39, Chapter 4]). It is well known that, for a fixed a ∈ (−1, ∞), the systems a ∞ {Lan }∞ n=1 and {ϕn }n=0 with Lan (x) :=

1

Lan (x) := (n!/Γ(n + a + 1)) 2 Lan (x)e−x/2 xa/2

1

and ϕan (x) := Lan (x2 )(2x) 2 ,

x ∈ (0, ∞)

are orthogonal on L2 ((0, ∞)). They are, respectively, the eigenfunctions of the differential operators   d2 d x a2 a L := x 2 + − + dx dx 4 4x

and

La :=

d2 1 − x2 − 2 dx2 x

  1 a2 − . 4

The functions ρLa and ρLa in (1.2) are associated with the operators La and La , respectively. In the remainder of this subsection, we study the properties of the Laguerre operator La with a ∈ (− 12 , ∞). We point out that all the results obtained above are also true for the Laguerre operator La with a ∈ (0, ∞). Since the arguments of the proofs are similar and easier, we omit the details for brevity. Lemma 4.4 Let τ ∈ (0, ∞). Then there exists a positive constant Cτ such that, if x, y ∈ (0, ∞) and |x − y| ≤ τ ρLa (x), then Cτ−1 ρLa (y) ≤ ρLa (x) ≤ Cτ ρLa (y)

(4.2)

if and only if τ ∈ (0, 8). Proof

To show the sufficiency, we let τ ∈ (0, 8). We consider the following two cases for

x. Case 1) x ∈ (0, 1]. In this case, ρLa (x) = x8 and (1 − τ8 )x ≤ y ≤ (1 + τ8 )x. If y ∈ (0, 1], then ρLa (y) = y8 , therefore we only need to choose Cτ > max{(1 − τ8 )−1 , 1 + τ8 }. If y ∈ (1, ∞),
1 then ρLa (y) = 8y and, moreover, 1 + τ8 x > 1. Notice that (1+1τ )x ≤ y1 ≤ (1−1τ )x . If we choose Cτ >

(1+ τ8 )2 1− τ8 ,

8

8

then

(Cτ )−1 x = and Cτ x =

(Cτ )−1 x(1 + τ8 )x (Cτ )−1 (1 + τ8 ) 1 ≤ < τ τ (1 + 8 )x (1 + 8 )x (1 + τ8 )x

Cτ x(1 − τ8 )x Cτ (1 − τ8 ) [(1 + τ8 )x]2 1 = > , τ τ 2 τ (1 − 8 )x (1 + 8 ) (1 − 8 )x (1 − τ8 )x

which imply that (4.2) holds true in this case. 1 τ τ Case 2) x ∈ (1, ∞). In this case, ρLa (x) = 8x and x − 8x ≤ y ≤ x + 8x and, moreover, 1 1 1 1 τ −1 ≤ ≤ . Choose C > max{(1 − ) , 1 + τ8 }, then τ τ . If y ∈ (1, ∞), then ρLa (y) = τ x+ y x− 8y 8 8x

8x

Cτ (1 − 8xτ 2 ) Cτ (1 − τ8 ) Cτ 1 = > > τ τ τ x x − 8x x − 8x x − 8x

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C −1 (1 + τ8 ) C −1 (1 + 8xτ 2 ) 1 Cτ−1 < τ < = τ τ τ τ , x x + 8x x + 8x x + 8x

τ which imply (4.2). If y ∈ (0, 1], then ρLa (y) = y8 and, moreover, x − 8x ≤ 1. It then follows √ √ 1+ 5 τ −1 3+ 5 τ that x ∈ (1, 2 ). If we choose Cτ > max{(1 − 8 ) , 2 + 8 }, then  Cτ Cτ  τ  Cτ τ  τ √ = 2 τ x+ > x+ >x+ τ 3+ 5 x x +8 8x 8x 8x +8 2

and

C −1  τ  C −1  τ  τ Cτ−1 = 2τ τ x− < τ τ x−
which also imply that (4.2) holds true in this case. This finishes the proof of the sufficiency. Notice that τ ∈ (0, ∞). By the sufficiency of Lemma 4.4, to show its necessary, we only 1 need to prove that (4.2) fails if τ ∈ [8, ∞). Indeed, let τ ∈ [8, ∞) and x = 12 . Then ρLa ( 12 ) = 16 1 1 1 1 and τ ρLa ( 2 ) ≥ 2 , which implies that, for all y ∈ (0, 1), | 2 − y| ≤ τ ρLa ( 2 ). On the other hand, if y ∈ (0, 1), then ρLa (y) = 81 y. It then follows that ρLa ( 12 ) → ∞ as y → 0. ρLa (y) Thus, we can not find a positive constant C such that (4.2) holds true for all y ∈ (0, 1), which implies that (4.2) fails if τ ∈ [8, ∞). This proves the necessary of Lemma 4.4, and hence finishes the proof of Lemma 4.4.  The following lemmas are, respectively, [20, Lemma 1] and [20, Corollary 1]. Lemma 4.5 Let x0 :=1,

xk := xk−1 + ρLa (xk−1 ) for k ∈ N,

and xk := xk+1 − ρLa (xk+1 ) for k ∈ {−1, −2, · · · }.

(4.3)

Then the family of critical balls, B := {Bk }k∈Z with Bk := B(xk , ρLa (xk )) for all k ∈ Z, satisfies that S (i) Bk = (0, ∞); k∈Z

(ii) for any k ∈ Z, Bk ∩ Bj = ∅ provided j ∈ {k − 1, k, k + 1}; (iii) for any y0 ∈ (0, ∞), at most three balls in B have nonempty intersection with B(y0 , ρLa (y0 )). Lemma 4.6 There exists a positive constant C such that, for all balls B(x, r) ⊂ (0, ∞) with r ∈ (ρLa , ∞), X |B(x, r)| ≤ |Bk | ≤ C|B(x, r)|. {Bk ∈B: Bk ∩B(x, r)6=∅}

Definition 4.7 (i) Let q ∈ (1, ∞]. A measurable function b is called a (1, q)ρLa -atom associated to the ball B(x, r) of (0, ∞), if (A1) supp b ⊂ B(x, r) for some x ∈ (0, ∞) and r ∈ (0, ∞); (A2) kbkLq ((0, ∞)) ≤ |B(x, r)|1/q−1 ; R∞ (A3) if r ∈ (0, ρLa (x)/2], then 0 b(y)dy = 0. ^ (ii) A measurable function b is called a (1, ∞)ρLa -atom associated to the ball B(x, r) of (0, ∞), if b satisfies (A1) with r ≤ ρLa (x), (A2) with q = ∞, and (A3).

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Definition 4.8 (i) Let q ∈ (1, ∞]. A function f ∈ L1 ((0, ∞)) is said to be in the Hardy space HL1,aq ((0, ∞)) if there exist (1, q)ρLa -atoms {bj }j∈N and {λj }j∈N ⊂ C such that P P f = λj bj in L1 ((0, ∞)) and |λj | < ∞. Moreover, the norm of f in HL1,aq ((0, ∞)) is j∈N

j∈N

defined by

kf kH 1,aq ((0, ∞)) := inf L

(

X j∈N

)

|λj | ,

where the infimum is taken over all the decompositions of f as above. e 1,a∞ ((0, ∞)) is defined as in (i) with (1, q)ρ a -atoms replaced (ii) The atomic Hardy space H L L ^ by (1, ∞) -atoms. ρLa

e 1,a∞ ((0, ∞)) was introduced by Dziuba´ The atomic Hardy space H nski [18]. Indeed, we L have the following conclusion. e 1,a∞ ((0, ∞)) coincide Proposition 4.9 For all q ∈ (1, ∞], the spaces H 1,aq ((0, ∞)) and H L

L

with equivalent norms.

Proof Let D := {B(x, r) : r ∈ [ρLa (x)/2, ∞)}. Then it is easy to see that, for all 1 q ∈ (1, ∞], the spaces HL1,aq ((0, ∞)) and HD ((0, ∞)) coincide with equivalent norms. This, together with Remark 1.12, shows that, for all q ∈ (1, ∞), the spaces HL1,aq ((0, ∞)) and HL1,a∞ ((0, ∞)) coincide with equivalent norms. Thus, to prove Proposition 4.9, it suffices to e 1,a∞ ((0, ∞)) coincide with equivalent norms. Obviprove that the spaces HL1,a∞ ((0, ∞)) and H L e 1,a∞ ((0, ∞)) ⊂ H 1,a∞ ((0, ∞)). To prove the converse inclusion, we only need to prove ously, H L L that any (1, q)ρLa -atom a, with a ball B(x0 , r) with r ∈ (ρLa (x0 ), ∞), can be written into P ^ a = k λk bk , where, for any k, bk is a (1, ∞)ρLa -atom supported in some ball B(xk , rk ) with P rk ∈ (0, ρLa (xk )] and λk ∈ C, and k |λk | ≤ C with a positive constant C independent of a. We use some arguments used in the proof of [20, Lemma 1] to prove this. Indeed, for any k ∈ Z, let xk be as in (4.3),     aχ[xk , xk +ρLa (xk )) , if k > 0 |B(x0 , r)| ak := aχ(xk −ρLa (xk ), xk ] , if k < 0 and bk := ak ,  ρLa (xk )   aχBk , if k = 0. where, for any set E ⊂ (0, ∞), χE denotes its characteristic function. It then follows that bk for ^ any k is a (1, ∞)ρLa -atom. Recall that, by Lemma 4.6, there exist two integers i0 and j0 , and j0 j0 S P a positive constant C such that B(x0 , r) ⊂ Bk and |B(x0 , r)| ≤ |Bk | ≤ C|B(x0 , r)|. k=i0

k=i0

From this, we deduce that

a=

j0 X

λk bk

with λk :=

k=i0

ρLa (xk ) |B(x0 , r)|

and, moreover, j0 X

k=i0

|λk | ≤

j0 X

k=i0

|Bk | ≤ C. |B(x0 , r)|

Combining the above estimates, we then obtain the converse inclusion, which completes the proof of Proposition 4.9. 

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Remark 4.10 Based on Proposition 4.9, from now on, we denote HL1,aq ((0, ∞)) simply p eρ a := {B(x, r) : x ∈ (0, ∞), r ∈ ( 1 ρLa (x), ∞)} and BMO ^ La ((0, ∞)) by Hρ1La ((0, ∞)). Let D L 2 p eρ a . When p = 1, we denote BMO ^ La ((0, ∞)) be as in Definition 1.5 with Dρ replaced by D L ^ La ((0, ∞)). It then follows, from [36, Lemma 2.2], that, for all p ∈ (1, ∞), the simply by BMO p ^La ((0, ∞)) and BMO ^La ((0, ∞)) coincide with equivalent norms. Moreover, by [36, spaces BMO ^ La ((0, ∞)). Theorem 2.1], we see that the dual space of Hρ1 a ((0, ∞)) is BMO L

Proposition 4.11 Let p ∈ [1, ∞), DρLa be as in (1.6) and BMOpLa ((0, ∞)) as in Definition 1.5 with Dρ replaced by DρLa . Then, for all p ∈ [1, ∞), the spaces BMOpLa ((0, ∞)) and ^La ((0, ∞)) coincide with equivalent norms. BMO Proof When p = 1, we denote BMOpLa ((0, ∞)) simply by BMOLa ((0, ∞)). From [36, Lemma 2.2], it then follows that, for all p ∈ (1, ∞), the spaces BMOpLa ((0, ∞)) and BMOLa ((0, ∞)) coincide with equivalent norms. Thus, to prove Proposition 4.11, we only ^La ((0, ∞)) coincide with equivalent need to prove that the spaces BMOLa ((0, ∞)) and BMO R 1 ^La ((0, ∞)) ⊂ norms. Notice that |B| B |f (x) − fB |dx ≤ 2|f |B . It is easy to see that BMO BMOLa ((0, ∞)). Now we consider the converse inclusion. Let f ∈ BMOLa ((0, ∞)). We claim that, for all B := B(x0 , r) with r ∈ [ 12 ρLa (x0 ), ρLa (x0 )), |f |B . kf kBMOLa ((0, ∞)) . Indeed, by Lemma 4.5, we see that at most three balls in B have nonempty intersection with B(x0 , r). Without loss of generality, we may assume that the balls are Bk−1 , Bk and Bk+1 . It then S follows that B ⊂ Bk+i and |B| ∼ |Bk−1 | ∼ |Bk | ∼ |Bk+1 | by Lemma 4.4, which, i∈{−1, 0, 1}

together with the definitions of Bk and BMOLa ((0, ∞)), further implies that " # Z Z 1 X 1 1 |f |B = |f (y)|dy . |f (y) − fBk+i |dy + |f |Bk+i |B| B |B| Bk+i i=−1 .

1 X

i=−1

|f |Bk+i . kf kBMOLa ((0, ∞)) .

Thus, the above claim holds true. From this claim, we easily deduce that ^ La ((0, ∞)), BMOLa ((0, ∞)) ⊂ BMO which completes the proof of Proposition 4.12.



By Remark 4.10 and Proposition 4.12, we see that the dual space of Hρ1La ((0, ∞)) is BMOLa ((0, ∞)). We remark that this result was independently proved by Cha and Liu [20]. Notice that, in the proof of Theorem 1.9, we only use the fact that (1.5) holds for τ ∈ (0, 6] (see the proof of (2.1)). This, together with Lemma 4.4 and Corollary 1.13, tells us the following conclusion, the details being omitted. Proposition 4.12 Let ρLa be the admissible function associated with the Laguerre operator La with a ∈ (− 21 , ∞), and DρLa as in (1.6). Then the following three statements are equivalent: (i) g is a pointwise multiplier on the space BMOLa ((0, ∞)); (ii) g is a pointwise multiplier on the space Hρ1La ((0, ∞)); (iii) g ∈ L∞ ((0, ∞)) ∩ BMOln ((0, ∞)), where a (cB ) Z 1 + ln ρL rB kgkBMOln ((0, ∞)) := sup |g(x) − gB |dx < ∞. |B| B∈DρLa B

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