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Georgian Mathematical Journal Volume 15 (2008), Number 2, 353–376

COMPACT COMMUTATORS ON MORREY SPACES WITH NON-DOUBLING MEASURES YOSHIHIRO SAWANO AND SATORU SHIRAI

Abstract. We study multi-commutators on the Morrey spaces generated by BMO functions and singular integral operators or by BMO functions and fractional integral operators. We place ourselves in the setting of Rd coming with a Radon measure µ which satisfies a certain growth condition. The Morrey-boundedness of commutators is established by M. Yan and D. Yang. However, the corresponding assertion of Morrey-compactness is still missing. The aim of this paper is to prove that the multi-commutators are compact if one of the BMO functions can be approximated with compactly supported smooth functions. 2000 Mathematics Subject Classification: Primary 42B35; Secondary 46B50. Key words and phrases: Morrey space, compact operator, commutator.

1. Introduction In this paper we give a criterion for the compactness of commutators on the Morrey spaces. As for the Lebesgue measure, the Morrey-boundedness of the commutators generated by functions and bounded linear operators is investigated intensively in [1, 16]. However, the compactness criterion of commutators is missing. In this present paper we shall obtain some compact commutators ∞ generated by nice functions close in some sense to Ccomp , the set of all compactly supported smooth functions and bounded linear operators. We assume that µ is a Radon measure satisfying µ(B(x, `)) ≤ c0 `n ,

0 < n ≤ d,

(1)

where B(x, `) is a ball with center x and of radius ` > 0. Harmonic analysis with the underlying measure µ satisfying (1) has been developing very rapidly in this decade. The theory expands very rapidly due to the pioneering work by Volberg, Nazarov and Treil and that by Tolsa (e.g., see [12, 13, 21]). We are mainly concerned with commutators given by Z [a, T ]f (x) := lim (a(x) − a(y))k(x, y)f (y) dµ(y) ε↓0

Rd \B(x,ε)

and multi-commutators. Here a is a function and k is a kernel of an integral operator T . Let us recall some definitions of integral operators associated with growth measures. Here and below, given a set of functions X, we denote by Xcomp the c Heldermann Verlag www.heldermann.de ISSN 1072-947X / $8.00 / °

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set of all compactly supported functions in X. We say that an L2 (µ)-bounded operator T is a singular integral operator if there exists a kernel function k : Rd × Rd → C satisfying the properties listed below. (i) There exists δ > 0 so that the quantity β(T ) defined below is finite: β(T ) := sup |k(x, y)| · |x − y|n

R

x,y∈ n x6=y

+

sup

R

x,y,z∈ n |x−y|>2|z−y|>0

(|k(x, y) − k(x, z)| + |k(y, x) − k(z, x)|)|x − y|n+δ . |z − y|δ

(ii) For all f ∈ L2comp (µ) we have Z T f (x) = k(x, y)f (y) dµ(y),

(2)

a.e. x ∈ / supp(µ).

Now we turn to the fractional integral operator defined by Z f (y) dµ(y). Iα f (x) = |x − y|n−α

Rd

We list [3, 7, 8, 18] as sources of the boundedness property of this operator. In [3], Garcia-Cuerva and Gatto considerd more generalized operators. Let n be a constant from (1). Let 0 < α < n and 0 < ε ≤ 1. A function kα : Rd × Rd → C is said to be a fractional kernel of order α with regularity ε if it satisfies γ(Kα ) = sup |kα (x, y)| · |x − y|n−α

R

x,y∈ d x6=y

+

sup

R

x,x0 ,y∈ d |x−y|≥2|x0 −x|>0

|kα (x, y) − kα (x0 , y)| · |x − y|n−α+ε < ∞. |x − x0 |ε

(3)

Let 1 < p < αn . We define the operator Kα for the kernel kα which satisfies the condition (3) Z Kα f (x) := kα (x, y)f (y) dµ(y).

Rd

Thus we can say that Iα is a fractional kernel of order 1. Now let us turn our attention to the function a in the commutator [a, T ]. Tolsa defined a nice substitute for BMO, which is called RBMO [21]. RBMO is an abbreviation of “regular bounded mean oscillation”. To describe RBMO let us fix some notation. Here and below we mean by a “cube” a compact cube with the sides parallel to the coordinate axis. For a cube Q we denote by `(Q) the sidelength of Q. Next, we define the set of doubling cubes: Q(µ) := {Q : Q is a cube with µ(Q) > 0}, Q(µ, 2) := {Q ∈ Q(µ) : 0 < µ(2Q) ≤ c1 µ(Q)},

(4)

COMPACT COMMUTATORS ON MORREY SPACES

355

where c1 is a constant larger than 2d+1 . Q(µ, 2) denotes the set of doubling cubes. Let Q ∈ Q(µ) be a cube which is not always doubling. By (1) a simple reduction to an absurdity argument shows us that the smallest doubling cube of the form 2j Q, j ∈ N, does exist and we denote it by Q∗ . Let 0 ≤ γ < n. Denote by cQ the center of Q ∈ Q(µ). Then we define `(R)µ Z (γ) KQ,R

:= 1 +

µ(B(cQ , `)) `n

¶1− nγ

d` `

`(Q) (0)

and KQ,R := KQ,R for Q, R ∈ Q(µ, 2) with Q ⊂ R. The coefficient KQ,R was (γ) introduced by Tolsa in [21] and the modified coefficient KQ,R was defined later (γ) by Chen and Sawyer [2]. The coefficient KQ,R is used for the definition of a sharp maximal operator M ],γ . For Q ∈ Q(µ, 2) we denote by mQ (a) the average of the L1loc (µ)-function a. Define Z 1 ],γ ¡ 3 ¢ |a(y) − mQ∗ (a)| dµ(y) M f (x) := sup x∈Q∈Q(µ) µ 2 Q Q

+

sup x∈Q⊂R Q,R∈Q(µ,2)

|mQ (a) − mR (a)| (γ)

KQ,R

and M ] f (x) := M ],0 f (x). With these definitions in mind we turn to the definition of elementary properties of RBMO due to Tolsa, such as the John–Nirenberg inequality. For details we refer to [21, Section 2]. We define © ª RBM O(µ) := a ∈ L1loc (µ) : kak∗ := kM ] a : L∞ (µ)k < ∞ . We can say that one of the advantages of RBMO is the following John–Nirenberg inequality. Many attempts to define a space of BMO type for the growth measures (1) have been made. However, many definitions did not recover the classical John–Nirenberg inequality. The John–Nirenberg inequality still holds in RBMO. Having defined the function space RBMO, let us formulate the JohnNirenberg inequality due to Tolsa. Proposition 1.1. Let f ∈RBMO and Q ∈ Q(µ). (1) There exist positive constants C and C 0 independent of f such that, for every λ > 0 and for every cube Q, µ ¶ µ ¶ C 0λ 3 Q exp − . µ {x ∈ Q : |f (x) − mQ∗ (f )| > λ} ≤ C µ 2 kf k∗ (2) Let q ∈ [1, ∞). Then there exists a constant C independent of f such that, for every cube Q ∈ Q(µ), Ã ! 1q Z 1 ¡ ¢ |f (x) − mQ∗ (f )|q dµ(x) ≤ C kf k∗ . µ 23 Q Q

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For later consideration it is also convenient to define modified maximal operators. Define Z 1 Mγ, 9 f (x) := sup |f (y)| dµ(y), 0 ≤ γ < n ¡ ¢1− nγ 8 x∈Q∈Q(µ) µ 9 Q 8 Q and M f (x) := M0, 9 f (x). 8 The aim of this paper is to establish the compactness of the commutator [a, T ] on the Morrey spaces. So let us recall the definition of the Morrey space Mpq (µ), where µ satisfies the growth condition (1). Let k > 1 and 1 ≤ q ≤ p < ∞. We define the Morrey space Mpq (k, µ) as © ª Mpq (k, µ) := f ∈ Lqloc (µ) : kf : Mpq (k, µ)k < ∞ , where the norm kf : Mpq (k, µ)k is given by kf : Mpq (k, µ)k := sup µ(k Q) Q∈Q(µ)

1 − 1q p

! 1q

à Z |f |q dµ

.

(5)

Q

By applying H¨older’s inequality to (5) it is easy to see that Lp (µ) = Mpp (k, µ) ⊂ Mpq1 (k, µ) ⊂ Mpq2 (k, µ)

(6)

for 1 ≤ q2 ≤ q1 ≤ p < ∞. The definition of the spaces does not depend on the constant k > 1. The norms for different choices of k > 1 are equivalent. More precisely, for k1 > k2 > 1 we have (see [18, Proposition 1.1]) ¶d µ k1 − 1 p p kf : Mpq (k1 , µ)k. (7) kf : Mq (k1 , µ)k ≤ kf : Mq (k2 , µ)k ≤ Cd k2 − 1 Nevertheless, for definiteness, we will assume k = 2 in the definition and denote Mpq (2, µ) by Mpq (µ), which reflects the fact that the precise value of k > 1 is not central for our arguments. Before we proceed further, let us recall the boundedness properties of operators. Proposition 1.2 ([18, 19]). The following statements are true: (i) Suppose that the parameters p, q, s, t and γ satisfy 1 γ q t 1 1 < q ≤ p < ∞, 1 < t ≤ s < ∞, 0 ≤ γ < n, = − and = . s p n p s p Then there exists a constant C > 0 such that, for all f ∈ Mq (µ), ° ° ° ° s 9 °Mγ, f : Mt (µ)° ≤ C kf : Mpq (µ)k.

(8)

8

(ii) Let 1 < q ≤ p < ∞ and T a singular integral operator. Then there exists a constant C > 0 such that, for every f ∈ Mpq (µ), kT f : Mpq (µ)k ≤ C kf : Mpq (µ)k, kf : Mpq (µ)k ≤ C kM ] f : Mpq (µ)k.

(9) (10)

COMPACT COMMUTATORS ON MORREY SPACES

357

(iii) Suppose that the parameters p, q, s, t and α satisfy 1 α q t 1 1 < q ≤ p < ∞, 1 < t ≤ s < ∞, 0 < α < n, = − and = . (11) s p n p s p Then there exists a constant C > 0 so that, for every f ∈ Mq (µ), kKα f : Mst (µ)k ≤ C kf : Mpq (µ)k. Let us present the definition of multi-commutators. Let T be an integral operator with the associated kernel k. Suppose that we are given a collection of RBM O(µ)-functions a1 , a2 , . . . , al . Then define [(a1 , a2 , . . . , al ), T ]f (x) := [a{1,2,...,l} , T ]f (x) Z l Y (aj (x) − aj (y)) k(x, y)f (y) dµ(y). := lim ε→0

Rd \B(x,ε)

j=1

It follows from the definition that if T is an integral operator, then [a{1,2,...,l} , T ] = [(aσ(1) , aσ(2) , . . . , aσ(l) ), T ],

(12)

[a{1,2,...,l} , T ] = [(a1 , a2 , . . . , al−1 ), [al , T ]] for all bijections σ : {1, 2, . . . , l} → {1, 2, . . . , l}. Let I = {i1 , i2 , . . . , im } ⊂ {1, 2, . . . , l}. It is convenient to write [aI , T ] = [(ai1 , ai2 , . . . , aim ), T ]. As for the boundedness of multi-commutators, Hu, Meng and Yang proved the following Proposition 1.3 ([5, 22]). Let a1 , a2 , . . . , al ∈ RBM O(µ) and T be a singular integral operator. Assume that 1 < q ≤ p < ∞. Then [a{1,2,...,l} , T ] is bounded on Mpq (µ) and the operator norm has the following estimate: ° ° ° [a{1,2,...,l} , T ] °

Mpq (µ)→Mpq (µ)

≤C

l Y

kaj k∗ ,

j=1

where C does not depend on a1 , . . . , al . Proposition 1.4 ([4, 22]). Let a1 , a2 , . . . , al ∈ RBM O(µ) and Kα be an integral operator with fractional order α ∈ (0, n) and regularity ε ∈ (0, 1]. Assume further (11). Then the multi-commutator [a{1,2,...,l} , Kα ] is bounded on Mpq (µ) and the operator norm has the following estimate: ° ° ° [a{1,2,...,l} , Kα ] °

Mpq (µ)→Mst (µ)

≤C

l Y

kaj k∗ ,

j=1

where C does not depend on a1 , . . . , al . To obtain compactness we turn our attention to the following function spaces. Definition 1.5. RV M O(µ) is a set of functions given by RV M O(µ) := {a ∈ RBM O(µ) : ∞ (Rd )}. a belongs to the closure with respect to k · k∗ of Ccomp

358

Y. SAWANO AND S. SHIRAI

That is, a ∈ RV M O(µ) if and only if there exists a sequence of compactly supported smooth functions {bj }∞ j=1 such that lim ka − bj k∗ = 0. j→∞

RVMO is the abbreviation of “regular vanishing mean oscillation”. With these definitions in mind, we present our main theorems. Theorem 1.6. In addition to the conditions in Proposition 1.3 assume that one of a1 , . . . , al belongs to RV M O(µ). Then [a{1,2,...,l} , T ] is compact on Mpq (µ). Theorem 1.7. In addition to the conditions in Proposition 1.4 assume that one of a1 , . . . , al belongs to RV M O(µ). Then [a{1,2,...,l} , Kα ] is a compact Mpq (µ)Mst (µ) operator. As a special case of Theorem 1.7, we obtain the following Corollary 1.8. In addition to the conditions in Proposition 1.4 assume that one of the a1 , . . . , al belongs to RV M O(µ). Then [a{1,2,...,l} , Iα ] is a compact Mpq (µ)-Mst (µ) operator. Finally, let us describe the organization of this paper. First, we give a simple criterion of the Morrey-compactness in Section 2. Next, in Section 3 we prove these theorems when l = 1 and we pass to the case for multi-commutators. Finally we consider the boundedness on the generalized Morrey space. Generalized Morrey spaces Lp,φ (µ) are defined in Section 4. Here we content ourselves with making a brief historical review of these spaces. In [11] Nakai defined the generalized Morrey space Lp,φ for the Lebesgue measure and [9, 10] Mizuhara proved that the commutator is bounded on this generalized Morrey space. In [17] the second author showed the converse: If, for example, the commutator generated by a function a and a singular integral operator T is bounded on Lp,φ , then the generating function a belongs to BMO. In this paper, based on the definition in [15], we investigate the boundedness and compactness of commutators on the generalized Morrey space. 2. A Compactness Criterion on the Morrey Spaces In this section we give a simple criterion of the Lp (µ)-Lq (µ) compactness of an integral operator T , where T is given by Z T f (x) := k(x, y)f (y) dµ(y). (13)

Rd Our machinery to show the compactness is the following. The next statement is well-known (see e.g. [6]), but for the sake of completeness we supply the proof. Proposition 2.1. Let 1 < p, q < ∞. Suppose that the kernel function k : R × Rd → C satisfies ) 1q ( Z Ã Z ! q0 p 0 0 dµ(x) < ∞. kk : Lq (Rnx ; Lp (Rny ))k := |k(x, y)|p dµ(y) d

Rd

Rd

COMPACT COMMUTATORS ON MORREY SPACES

359

Then the operator T defined by (13) is a compact Lp (µ)-Lq (µ) operator with the norm estimate 0 kT kLp (µ)→Lq (µ) ≤ kk : Lq (Rnx ; Lp (Rny ))k. Proof. We begin with the norm estimate. First by applying the H¨older inequality to Z |k(x, y)| · |f (y)| dµ(y)

|T f (x)| ≤

Rd we obtain a pointwise estimate à Z

! 10 Ã Z ! p1 p 0 |k(x, y)|p dµ(y) |f (y)|p dµ(y)

|T f (x)| ≤

Rd

Rd

for a.e. x ∈ Rd . For the norm estimate it remains to integrate both sides after taking q th -power. Once we obtain the norm estimate, it is not so hard to establish the compactness of T . Indeed, for any ε > 0 we can find a finite number of bounded Borel sets E1 , E2 , . . . , Ek , F1 , F2 , . . . , Fk and z1 , z2 , . . . , zk ∈ C such that 0

kk − k ε : Lq (Rnx ; Lp (Rny ))k < ε, P where we have put k ε (x, y) := kj=1 zj χEj (x)χFj (y). By the norm estimate obtained just now, we mean that T can be approximated by a finite rank operator T ε whose kernel is k ε . Therefore T is compact. The proof of Proposition 2.1 is now finished. ¤ Proposition 2.1 gives us a simple criterion of the Morrey compactness. Corollary 2.2. Suppose that the parameters p, q, s, t satisfy 1 < q ≤ p < ∞ and 1 < t ≤ s < ∞. Assume that k ∈ L∞ comp (µ ⊗ µ). Then T defined by (13) is a compact Mpq (µ)-Mst (µ) operator. Proof. Let Q0 ∈ Q(µ) be a large cube taken so that supp(k) ⊂ Q0 × Q0 . Define three linear operators T1 , T2 and T3 by T1 : f ∈ Mpq (µ) 7→ χQ0 · f ∈ Lq (µ), T2 : f ∈ Lq (µ) 7→ T f ∈ Ls (µ), T3 : f ∈ Ls (µ) 7→ χQ0 · f ∈ Mst (µ). Then T can be factorized into the composition T3 T2 T1 . Note that Proposition 2.1 shows that T2 is compact. Therefore so is T . ¤ 3. Proof of the Main Theorems With Proposition 2.1 in mind, we prove the main theorem. Instead of establishing the compactness of [a, T ] directly we prove it by means of approximations.

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3.1. Proof of Theorems 1.6 and 1.7 when l = 1. For the sake of simplicity we write a = a1 . Their proofs being similar, we will concentrate on Theorem 1.6 which deals with the commutator generated by an RV M O(µ) function and a singular integral operator. ∞ Claim 3.1. We may assume a ∈ Ccomp (Rd ).

Proof. Since we are assuming that a ∈ RV M O(µ), for all j ∈ N we can take ∞ bj ∈ Ccomp (Rd ) so that ka − bj k∗ ≤ j −1 holds. By Proposition 1.3 we have k [a, T ] − [bj , T ] kMpq (µ) ≤ C j −1 . Thus, once we establish the compactness of [bj , T ], that of [a, T ] immediately follows as a norm-limit of the sequence of compact operators. ¤ Lemma 3.2. Let x ∈ supp(µ) and ε > 0. Then there exists a constant c > 0 such that Z |f (y)| dµ(y) ≤ C εM f (x). (14) |x − y|n−1 B(x,ε)

Proof. Let us assume n > 1 for the time being. We shall rewrite the integral of the left-hand side of (14) with the identity 1 = (n − 1) |x − y|n−1

Z∞ χB(x,`) (y)

d` . `n

(15)

0

If we apply (15) to the left-hand side of (14) and then use Fubini theorem, we obtain   Z Z∞ Z |f (y)|   d` dµ(y) = C |f (y)| dµ(y) n  n−1 |x − y| ` 0

B(x,ε)

Z2ε ≤C 0

B(x,`)∩B(x,ε)

  

Z



 d` |f (y)| dµ(y) n , `

B(x,`)

where we have used n > 1 to obtain that Z∞ Z∞ ε−n+1 d` d` n−1 = 2 = < ∞. `n `n n−1 ε

2ε

Recall that we have been assuming (1), which allows us to use the modified maximal operator M . Using the modified maximal operator M , we obtain Z |f (y)| dµ(y) ≤ C εM f (x). (16) |x − y|n−1 B(x,ε)

COMPACT COMMUTATORS ON MORREY SPACES

361

Suppose instead that n ≤ 1, which gives us a rough estimate 1 ≤ ε−n+1 , |x − y|n−1 Using this estimate, we obtain Z Z |f (y)| −n+1 dµ(y) ≤ ε |x − y|n−1 B(x,ε)

y ∈ B(x, ε).

|f (y)| dµ(y) ≤ C εM f (x).

B(x,ε)

Therefore (16) is still valid for n ≤ 1.

¤

∞ In what follows we assume a ∈ Ccomp (Rd ). What counts in the proof is that we can truncate T near the singularities. More precisely, let ψ be a function such that χ(2,∞) ≤ ψ ≤ χ(1,∞) .

We define

µ

Z

[a, Tε ]f (x) :=

ψ

Rd

|x − y| ε

¶ (a(x) − a(y))k(x, y)f (y) dµ(y), ε ≤ 1.

Then the following proposition holds. Proposition 3.3. As ε → 0, [a, Tε ] tends to [a, T ] in the norm topology of Mpq (µ). Proof. We take f ∈ Mpq (µ) and write out [a, T ]f (x) − [a, Tε ]f (x) in full: [a, T ]f (x) − [a, Tε ]f (x) ¶ µ ¶¶ Z µ µ |x − y| |x − y| −ψ (a(x) − a(y))k(x, y)f (y) dµ(y). = lim ψ ρ→0 ε ρ

Rd

∞ Assuming a ∈ Ccomp (Rd ), we see that the kernel satisfies ¯ µ ¶ µ ¶¯ ¯ ¯ |x − y| |x − y| c ¯ψ ¯ · |a(x) − a(y)| · |k(x, y)| ≤ −ψ ¯ ¯ ε ρ |x − y|n−1

(17)

∞ with a constant c > 0 independent of ρ. Indeed, a ∈ Ccomp (Rd ) implies |a(x) − µ ¶ a(y)| ≤ sup |Da(z)| · |x − y|. In view of the condition of the kernel k and the

Rd

z∈

fact that ψ is a bounded function we obtain (17). Assume that M f (x) < ∞, which in turn implies Z |f (y)| dµ(y) < ∞. |x − y|n−1 B(x,²)

Note that this condition is satisfied for µ-almost all x ∈ Rd . Having eliminated the singularity near the diagonal, we are now in a position of using Lebesgue’s

362

Y. SAWANO AND S. SHIRAI

convergence theorem and Lemma 3.2 to obtain Z |f (y)| | [a, T ]f (x) − [a, Tε ]f (x)| ≤ C dµ(y) ≤ C ε M f (x) |x − y|n−1

(18)

B(x,ε)

for µ-almost all x ∈ Rd . As we have seen in the Introduction, the maximal operator M is Mpq (µ)-bounded (see Proposition 1.2). Thus, from (18) we have k ([a, T ] − [a, Tε ])f : Mpq (µ)k ≤ C ε kf : Mpq (µ)k for every f ∈ Mpq (µ), which is equivalent to k [a, T ] − [a, Tε ] kMpq (µ) ≤ C ε. Thus Proposition 3.3 is proved. ¤ Proposition 3.3 reduces to the argument of showing the compactness of [a, Tε ]. Once we eliminated the singularity near the diagonal, we are in a position of truncating the kernel at infinity. For ε < R, we now put µ ¶ Z |x − y| R [a, Tε ]f (x) := ψ (a(x) − a(y))k(x, y)f (y) dµ(y). ε B(x,R)

Proposition 3.4. Let 1 < q ≤ p < ∞. We have lim [a, TεR ] = [a, Tε ] in the R→∞

operator norm topology of Mpq (µ).

From this proposition the matters are again reduced to showing the compactness of [a, TεR ]. Proof. We take f ∈ Mpq (µ) and write out [a, Tε ]f (x) − [a, TεR ]f (x) in full again: Z R [a, Tε ]f (x) − [a, Tε ]f (x) = (a(x) − a(y))k(x, y)f (y) dµ(y). (19)

Rd \B(x,R) As before, we can find a cube Q0 engulfing supp(a). Note that Q0 is determined, ∞ once we are given a ∈ Ccomp . Now we bound the kernel by the simple estimate |a(x) − a(y)| C (χQ0 (x) + χQ0 (y)) ≤ . n |x − y| |x − y|n

(20)

By inserting (20) into the right-hand side of (19) we obtain Z (χQ0 (x) + χQ0 (y))|f (y)| R | [a, Tε ]f (x) − [a, Tε ]f (x)| ≤ C dµ(y). |x − y|n

Rd \B(x,R)

1 =n We return to the formula |x − y|n

Z∞ χB(x,`) (y)

d` `n+1

again to obtain

0

| [a, Tε ]f (x) − [a, TεR ]f (x)| ! Z∞ Ã Z d` ≤C (χQ0 (x) + χQ0 (y))|f (y)| dµ(y) n+1 . (21) ` R

B(x,`)

COMPACT COMMUTATORS ON MORREY SPACES

363

Thus, we are left with the task of estimating the right-hand side of (21). To do this, we need several estimates. By the growth condition (1) and the definition of the norm we have Z n |f (y)| dµ(y) ≤ C `n− p kf : Mp1 (µ)k. B(x,`)

Furthermore, the generalized Minkowski inequality and the growth condition (1) again give us ° Z∞ Ã Z ° ! ° ° d` ° ° χQ0 (y)|f (y)| dµ(y) n+1 : Mpq (µ)° ° ° ° ` R

B(·,`)

!

Z∞ Ã Z

kχB(·,`) (y) : Mpq (µ)k · |f (y)| dµ(y)

≤ R

d` `n+1

Q0

Z∞ µ

¶

d`

p

≤c

sup kχB(·,`) (y) : L (µ)k R

Rd

y∈

`n+1

· kf : Mp1 (µ)k

n

≤ cR−n+ p kf : Mp1 (µ)k. Using these estimates, then we have k[a, Tε ]f − [a, TεR ]f : Mpq (µ)k Z∞ n d` ≤ ckχQ0 : Mpq (µ)k · · kf : Mp1 (µ)k + cR−n+ p kf : Mp1 (µ)k. (22) n +1 `p R

Recall that Mp1 (µ) ⊃ Mpq (µ). Thus (22) reads as n

n

k [a, Tε ] − [a, TεR ] kMpq (µ) ≤ C (R− p + R−n+ p ). Thus our assertion is justified.

¤

With Propositions 2.1, 3.3 and 3.4 in mind, we finish the proof of Theorem 1.7. Propositions 3.3 and 3.4 reduce the proof of Theorem 1.7 to showing the compactness of [a, TεR ]. Note that [a, TεR ] can be expressed in terms of the kernel kεR , where kεR is given by µ ¶ |x − y| R kε (x, y) := χB(x,R) (y) · ψ · (a(x) − a(y)) · k(x, y). ε It is immediate to see from the definition of kεR that kεR ∈ L∞ comp (µ ⊗ µ). It remains to use Corollary 2.2. An application of this corollary readily shows that [a, TεR ] and hence [a, T ] is compact.

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Y. SAWANO AND S. SHIRAI

3.2. Proof of Theorems 1.6 and 1.7 for a general case. The passage to the multi-commutator [a{1,2,...,l} , Iα ] can be achieved in the same way as [a{1,2,...,l} , T ]. Thus we concentrate on the commutator [a{1,2,...,l} , T ]. To prove the compactness of the multi-commutator, we use the following Lemma 3.5 ([4], [5, Section 2]). Let η > 1 be a constant slightly larger than 1. (i) Under the same assumption as Proposition 1.3, we have M ] ([a{1,2,...,l} , T ]f )(x) l ´ ³ Y 1 η η ≤c kaj k∗ M (|T f | )(x) + β(T )M f (x) j=1

+c

 X

 Y



(

I {1,2,...,l}

kaj k∗  · M ([aI , T ]f )(x),

j∈{1,2,...,l}\I

where β(T ) is a number given by (2). (ii) Under the same assumptions as Proposition 1.4, we have M ],α ([a{1,2,...,l} , Kα ]f )(x) l ³ ´ Y 1 ≤c kaj k∗ M (|Kα f |η )(x) η + γ(Kα )Mα, 9 f (x) 8

j=1

+c

 X

(



I {1,2,...,l}

 Y

kaj k∗  · M ([aI , Kα ]f )(x),

j∈{1,2,...,l}\I

where γ(Kα ) is a constant appearing in (3). First, we take some reduction steps before we deal with this commutator. By Propositions 1.3 and 1.4 as before we can strengthen the assumption and ∞ assume that aj ∈ Ccomp for some j = 1, 2, . . . , l. (12) allows us to assume that j = l. Furthermore, since Theorem 1.6 has already been proved in the case where l = 1, it can be assumed that l ≥ 2. In view of (12), again, after relabeling the aj , we have only to prove that [a{1,2,...,l−1} , [al , T ]] ∞ and a1 , a2 , . . . , al−1 ∈ RBM O(µ). is compact, whenever al ∈ Ccomp Having clarified what to prove, we now turn to the truncation near the diagonal. We shall prove

° ° °[a{1,2,...,l−1} , [al , T ]] − [a{1,2,...,l−1} , [al , Tε ]]°

Mpq (µ)

≤ Cε

l−1 Y

kaj k∗

(23)

j=1

inductively with the same notation as before. As we have seen, this is the case where l = 1. It is understood that (23) reads k[a, T ] − [a, Tε ]kMpq (µ) ≤ Cε

COMPACT COMMUTATORS ON MORREY SPACES

365

when l = 1. Let l0 ≥ 1 and assume that (23) is true for l whenever l ≤ l0 . Let l = l0 + 1. Then let us invoke Lemma 3.5 and the identity [(a1 , a2 , . . . , al ), T ] − [(a1 , a2 , . . . , al ), Tε ] = [a{1,2,...,l0 } , [al , T − Tε ]]. Recall that M is bounded on Mpq (µ). Take η so that 1 < η < q. Then M is p/η bounded on Mq/η (µ) as well. Therefore we obtain kM ] ([a{1,2,...,l0 } , [al , T − Tε ]])f : Mpq (µ)k l0 ´ ³ Y 1 ≤C kaj k∗ kM (|[al , T − Tε ]f |η ) η : Mpq (µ)k + β([al , T − Tε ])kf, : Mpq (µ)k j=1

Ã

X

+C

(

I:I {1,2,...,l0 }

≤C

l0 Y

Y

! · kM ([aI , [al , T − Tε ]]f ) : Mpq (µ)k

kaj k∗

j∈{1,2,...,l0 }\I

¢ ¡ kaj k∗ k[al , T − Tε ]f : Mpq (µ)k + β([al , T − Tε ]) · kf : Mpq (µ)k

j=1

Ã

X

+C

(

I:I {1,2,...,l0 }

Y

! kaj k∗

· k[aI , [al , T − Tε ]]f : Mpq (µ)k.

j∈{1,2,...,l0 }\I

The crux of the proof is to view [al , T −Tε ] as a single singular integral operators. We observe that β([al , T − Tε ]) ≤ C ε, ε ≤ 1. (24) Indeed, to prove (24) we need to establish sup |x − y|n |al (x) − al (y)| |k(x, y) − kε (x, y)| ≤ C ε

R

(25)

x,y∈ d x6=y

and sup

R

x,y,z∈ d |x−y|>2|z−y|

sup

R

x,y,z∈ d |x−y|>2|z−y|

|al (x) − al (y)| |k(x, y) − k(x, z) − kε (x, y) + kε (x, z)| ≤ C ε, |x − y|−n−δ |z − y|δ

(26)

|al (x) − al (y)| |k(y, x) − k(z, x) − kε (y, x) + kε (z, x)| ≤ C ε. |x − y|−n−δ |z − y|δ

(27)

Here kε is the kernel corresponding to T² . However, the proof of (27) is analogous to (26) because of symmetry. Therefore we have only to prove (25) and (26). Writing it out in full, we obtain sup |x − y|n |al (x) − al (y)| |k(x, y) − kε (x, y)|

R

x,y∈ d x6=y

¯ ¶¯ µ ¯ ¯ |x − y| ¯ = sup |x − y| |k(x, y)| |al (x) − al (y)| ¯¯1 − ψ ¯ ε d n

R

x,y∈ x6=y

366

Y. SAWANO AND S. SHIRAI

¯ µ ¶¯ ¯ ¯ |x − y| ¯ ≤ C sup |al (x) − al (y)| ¯¯1 − ψ ¯ ε x,y∈Rd ¯ µ ¶¯ ¯ |x − y| ¯¯ ¯ ≤ C sup |x − y| ¯1 − ψ ¯ ε x,y∈Rd ¯ ¶¯ µ |x − y| ¯¯ |x − y| ¯¯ ≤ C ε sup ¯1 − ψ ¯ ≤ C ε. ε ε x,y∈Rd As for (26), we first decompose k(x, y) − k(x, z) − kε (x, y) + kε (x, z) ¶¶ µ µ |x − y| = (k(x, y) − k(x, z)) 1 − ψ ε µ µ ¶ µ ¶¶ |x − z| |x − y| + k(x, z) ψ −ψ ε ε and prove

¯ ¶¯ µ ¯ ¯ |x − z| ¯ |al (x) − al (y)| |k(x, y) − k(x, z)| ¯¯1 − ψ ¯ ε

sup

R

x,y,z∈ d |x−y|>2|z−y|

sup

R

x,y,z∈ d |x−y|>2|z−y|

|x − y|−n−δ |z − y|δ

≤ C ε,

¯ µ ¶ µ ¶¯ ¯ |x − y| |x − z| ¯¯ ¯ |al (x) − al (y)| |k(x, z)| ¯ψ −ψ ¯ ε ε ≤ C ε. |x − y|−n−δ |z − y|δ

(28)

(29)

Let us consider (28). By the kernel condition and the mean value theorem we have ¯ ¶¯ µ ¯ ¯ |x − z| ¯ Left hand side of (28) ≤ C sup |al (x) − al (y)| ¯¯1 − ψ ¯ ε d

R

x,y,z∈ |x−y|>2|z−y|

≤C

sup

R

x,y,z∈ d |x−y|>2|z−y|

¯ µ ¶¯ ¯ ¯ |x − z| ¯ |x − y| ¯¯1 − ψ ¯ ε

¯ µ ¶¯ |x − z| ¯¯ |x − z| ¯¯ 1−ψ ≤ C ε sup ¯ = C ε. ε ¯ ε x,z∈Rd What remains to be done for the proof of (24) is to confirm (29). In view of the kernel estimate and the fact that al is bounded, (29) is reduced to showing that ¯ µ ¶ µ ¶¯ ¯ |x − y| |x − z| ¯¯ ¯ |al (x) − al (y)| ¯ψ −ψ ¯ ε ε ≤ C ε. (30) sup |x − y|−δ |z − y|δ x,y,z∈Rd |x−y|>2|z−y|

COMPACT COMMUTATORS ON MORREY SPACES

367

To prove (30) let x, y, z be fixed so that |x − y| > 2|z − y|. We have to show ¯ µ ¶ µ ¶¯ ¯ |x − y| |x − z| ¯¯ ¯ |al (x) − al (y)| ¯ψ −ψ ¯ ε ε ≤Cε |x − y|−δ |z − y|δ with the constant C independent of x, y, z. To do this, we may assume that ¶ ¶ µ µ |x − y| |x − z| ψ −ψ 6= 0. ε ε A geometrical observation shows that 1 |x − y| ≤ |x − z| ≤ 2|x − y| 2 from the condition |x − y| > 2|z − y|. Therefore we obtain ¯ µ ¶ µ ¶¯ ¯ ¯ ¯ψ |x − y| − ψ |x − z| ¯ ≤ |y − z| sup |ψ 0 (t)|. ¯ ¯ ε ε ε 2|x−y| |x−y| 2ε

≤t≤

ε

0

Note that supp(ψ ) ⊂ [1, 2]. From this we conclude that it is necessary that · ¸ |x − y| 2|x − y| [1, 2] ∩ , 6= ∅, 2ε ε ¯ µ ¶ µ ¶¯ ¯ ¯ |x − z| |x − y| ¯ 6= 0. In this i.e., ε ≤ |x − y| ≤ 4ε in order that ¯¯ψ −ψ ¯ ε ε case we have the two-sided estimate 12 ε ≤ |x − z| ≤ 4ε. The above inequalities yield (30). This observation and the induction assumption thus give ° ° °[(a1 , a2 , . . . , al ), T ] − [(a1 , a2 , . . . , al ), Tε ]°

Mpq (µ)

l0 Y

≤ Cε

kaj k∗ .

j=1

Therefore it follows that, in the operator topology, [a{1,2,...,l0 } , [al , Tε ]] → [a{1,2,...,l0 } , [al , T ]] as ε → 0. In the same way as before, we shall prove [a{1,2,...,l0 } , [al , TεR ]] → [a{1,2,...,l0 } , [al , Tε ]] in the operator topology as R → ∞. Indeed, take a doubling cube Q0 that is centered at the origin 0 and engulfs supp(a) as before. Let R > 2`(Q0 ). Then what we have to estimate can be summarized as follows: X (1) (2) (31) kBI f + BI f : Mpq (µ)k ≤ cR · kf : Mpq (µ)k I : I⊂{1,2,...,l0 }

with Z∞à Z (1)

BI f (x) :=

! χQ0 (x) · AI (x, y) · |f (y)| dµ(y)

R

B(x,`)

d` `n+1

,

368

Y. SAWANO AND S. SHIRAI

Z∞à Z (2) BI f (x)

!

:=

χQ0 (y) · AI (x, y) · |f (y)| dµ(y) R

d` `n+1

,

B(x,`)

where cR is a quantity tending to 0 as R → ∞ and  Ã ! Y Y AI (x, y) = |aj (x) − mQ0 (aj )|  |aj (y) − mQ0 (aj )| . j∈I

j ∈I /

Let I ⊂ {1, 2, . . . , l} be fixed. Then (31) can be decomposed into two parts, Our present task is to find positive constants cR,1 and cR,2 satisfying, for every f ∈ Mpq (µ), (1)

kBI f : Mpq (µ)k ≤ cR,1 kf : Mpq (µ)k,

(32)

(2) kBI f

(33)

: Mpq (µ)k ≤ cR,2 kf : Mpq (µ)k,

for any subset I of {1, 2, . . . , l} and lim (cR,1 + cR,2 ) = 0. We estimate the R→∞

right-hand side of (32). Let us assume that I is a proper subset of {1, 2, . . . , l}. With a minor change of the calculation below, the resulting estimate is also available for I = {1, 2, . . . , l}. Take an auxiliary constant u ∈ (1, ∞) such that l−]I + 1q = 1. u (1)

First, we observe that BI f is factorized as follows : Ã ! Y (1) BI f (x) = χQ0 (x) |aj (x) − mQ0 (aj )| j∈I

Z∞ Ã Z

Ã

× R

B(x,`)

Y

! |aj (y) − mQ0 (aj )|

! · |f (y)| dµ(y)

j ∈I /

d` `n+1

.

We use the generalized H¨older inequality kϕ1 ϕ2 . . . ϕk kp ≤

k Y j=1

kϕj kpj ,

1 1 1 1 = + + ··· + , p p1 p2 pk

to separate the RBMO functions {aj }j∈I and the Morrey function f . ! Z Ã Y |aj (y) − mQ0 (aj )| · |f (y)| dµ(y) j ∈I /

B(x,`)

≤

Y j ∈I /

à Z B(x,`)

! u1 Ã Z |aj (y) − mQ0 (aj )|u dµ(y) B(x,`)

! 1q |f (y)|q dµ(y)

.

COMPACT COMMUTATORS ON MORREY SPACES

369

We remark that this inequality is valid even for I = {1, 2, . . . , l}. By the John– Nirenberg inequality (Proposition 1.1) we obtain à Z ! u1 Y l−]I l−]I ≤ C log u (2 + `)·µ(B(x, `)) u . (34) |aj (y)−mQ0 (aj )|u dµ(y) j ∈I /

B(x,`)

Here we have set loga x := (log x)a for a > 0 and x > 1. From the definition of the Morrey norm we have à Z ! 1q |f (y)|q dµ(y)

n

n

≤ C ` q − p kf : Mpq (µ)k.

(35)

B(x,`)

Also observe that ° °Ã ! ° ° Y ° ° |aj − mQ0 (aj )| : Mpq (µ)° ° χQ0 ° ° j∈I °Ã ° ! ° ° Y ° ° p ≤ ° χQ0 |aj − mQ0 (aj )| : L (µ)° ≤ c < ∞, ° °

(36)

j∈I

where c depends only on aj and the cube Q0 which has been taken beforehand. From (34)–(36) we obtain a pointwise estimate à Z∞ ! l−]I u (2 + `) d` log (1) BI f (x) ≤ C χQ0 (x) kf : Mpq (µ)k (37) n +1 p ` R

with the help of the Minkowski inequality. Let us set l−]I Z∞ log u (2 + `) d` c cR,1 = , n kχQ0 : Mpq (µ)k ` p +1 R

where c is a constant in (37). Since p < ∞, we see that lim cR,1 = 0. Indeed, if R > 3, then we have l−]I Z∞ log u (2 + `) d` R

`

n +1 p

R→∞

Z∞ ≤ R

log `2 d` `

n +1 p

Z∞

n

2t e− p t dt → 0

= log R

as R → ∞. This proves that cR,1 = 0. Therefore (32) has been proved. To prove (33) we proceed in almost the same way as in the case of (32). We indicate the necessary change. A geometric observation shows that x ∈ B(0, 2`) whenever ` > R > 2`(Q0 ), |x − y| < ` and y ∈ Q0 . Thus, instead of estimating (34), we can replace χB(x,`) (y) by χB(0,2`) (x). Therefore the matter is again reduced to proving the compactness of [(a1 , a2 , . . . , al ), TεR ].

370

Y. SAWANO AND S. SHIRAI

However, once we truncate both the singularity near the diagonal and that at [ infinity, the integral kernel belongs to Lpcomp (µ ⊗ µ). Thus we are still in 1≤p k2 > 1 and 1 ≤ p < ∞. Suppose that φ : (0, ∞) → (0, ∞) is an increasing function. Then there exists a constant Cd depending only on k1 , k2 , d such that ¶d µ k2 − 1 p,φ p,φ kf : L (k1 , µ)k ≤ kf : L (k2 , µ)k ≤ Cd kf : Lp,φ (k1 , µ)k. (38) k1 − 1 In particular, Lp,φ (k1 , µ) and Lp,φ (k2 , µ) coincide as sets and their norms are mutually equivalent. Keeping Proposition 4.2 in mind, we denote Lp,φ (µ) = Lp,φ (2, µ), which again reflects that fact that the precise value of k > 1 in Lp,φ (µ) is not any longer important. 4.1. Some elementary facts on generalized Morrey spaces. Here and below, for the sake of simplicity, we shall postulate some of the following conditions on φ according to the operators we deal with: is almost decreasing: There exists a constant (i) The mapping t 7→ φ(t) t c > 0 such that φ(s) φ(t) ≤C for s ≤ t. (39) t s

COMPACT COMMUTATORS ON MORREY SPACES

(ii) There exists a constant c > 0 such that, for every r > 0, Z∞ φ(t) dt φ(r) ≤C . t t r

371

(40)

r

(iii) There exists a constant c > 0 such that, for every r > 0, Z∞ φ(t) dt φ(r) ≤C p . p tq t rq

(41)

r

If φ satisfies (39), then there exists a constant c > 0 such that for every r, s with 0 < s ≤ r ≤ 2s, φ(r) ≤ C φ(s). (42) Below we shall summarize the boundedness properties of the operators. Proposition 4.3 ([15, Theorem 2.3]). Let 1 ≤ p < ∞. Assume that φ satisfies (39). Then there exists a constant c > 0 such that, for every f ∈ Lp,φ (µ), kM f : Lp,φ (µ)k ≤ C kf : Lp,φ (µ)k. Proposition 4.4 ([15, Theorem 4.1]). Suppose that 1 < p < ∞ and φ satisfies (40). Let T be a singular integral operator. Then we have kT f : Lp,φ (µ)k ≤ C kf : Lp,φ (µ)k. Proposition 4.5 ([15, Corollary 3.5]). Suppose further that the parameters p, q, α satisfy 1 1 α 0 < α < n, 1 < p < q < ∞, = − . q p n Assume that φ satisfies (41). Then the fractional integral operator Iα enjoys the following boundedness: q p

kIα f : Lq,φ (µ)k ≤ C kf : Lp,φ (µ)k. 4.2. Boundedness and compactness of commutators. Having set down the preliminary facts in the preceding sections, we now turn to the commutators. To proceed further, we need some lemmas. We define the doubling noncentered maximal operator N by N f (x) =

sup

mQ (|f |).

x∈Q∈Q(µ,2)

Lemma 4.6 ([20]). Suppose that w belongs to Lp (µ) for some p ∈ (1, ∞]. For every f ∈ L1loc (µ), Q0 ∈ Q(µ), q ∈ [1, ∞) and α ∈ (0, 1), there exists a constant C independent of f such that ! 1q à Z ! 1q à Z |f (x) − m(Q0 )∗ (f )|q w(x)α dµ(x) ≤C M ] f (x)q W (x)α dµ(x) . Q0

3 Q 2 0

372

Y. SAWANO AND S. SHIRAI

Here, denoting N j as the j-th composition of the operator N , we put ∞ X W (x) := (2β)1−j N j w(x), β ≥ kN kp0 .

(43)

j=1

In particular, given Q0 ∈ Q(µ), we have, for every 1 ≤ q < ∞, ! 1q à Z à Z ! 1q |f (x) − m(Q0 )∗ (f )|q dµ(x) ≤C M ] f (x)q dµ(x) . Q0

3 Q 2 0

An immediate corollary of this fact is the following, which is a generalized Morrey space version of the sharp maximal inequality in [19, Theorem 1.3]. Theorem 4.7. Assume that φ is an increasing function with (39). Then we have the norm equivalence kf : Lp,φ (µ)k ' kM ] f : Lp,φ (µ)k + kf : Lp,φ (µ)k∗ , where kf : Lp,φ (µ)k∗ is an auxiliary norm given by p,φ

kf : L

∗

(µ)k =

sup Q∈Q(µ,2)

Z

1 1

(44)

|f (y)| dµ(y).

1

φ(µ(Q)) p µ(Q)1− p

Q

Proof. By virtue of the fact that M ] is dominated by M and the H¨older inequality, kM ] f : Lp,φ (µ)k + kf : Lp,φ (µ)k∗ ≤ C kf : Lp,φ (µ)k is clear. Let us prove the inverse inequality. To do this, we take Q ∈ Q(µ) is arbitrarily. Then we have, by virtue of Lemma 4.6 and the fact that t 7→ φ(t) t almost increasing, Ã ! p1 Z 1 |f (y)|p dµ(y) φ(µ(2Q)) Ã ≤

Q

1 φ(µ(2Q)) Ã

≤C Ã ≤C

! p1

Z p

|f (y) − mQ∗ (f )| dµ(y) Q

1 φ(µ(2Q))

! p1

Z ]

p

M f (y) dµ(y) Q

µ

µ +

µ(Q)mQ∗ (f ) φ(µ(2Q))

µ(2Q)mQ∗ (f ) + φ(µ(2Q)) !

¶

¶

kM ] f : Lp,φ (µ)k + kf : Lp,φ (µ)k∗ .

Thus this is the desired result.

¤

Remark 4.8. A minor change of the argument in [20] shows that (44) is still valid for M ],γ with 0 ≤ γ < n.

COMPACT COMMUTATORS ON MORREY SPACES

373

Furthermore, to deal with multi-commutators, the following lemma is indispensable. Lemma 4.9 ([11, Lemma 2]). Suppose that φ : (0, ∞) → (0, ∞) satisfies Z∞ du φ(u) ≤ C φ(r) for all r ∈ (0, ∞) u r

for some c > 0. Then there exists ε > 0 and c > 0 such that Z∞ du φ(u) uε ≤ C φ(r) rε for all r ∈ (0, ∞). u r

Now we are going to discuss the boundedness of commutators. A similar argument will work for the proof of the multi-commutators generated by RBMO functions and the integral operator of fractional kernel. Thus we concentrate on the generated by RBMO functions and the singular integral operator. We shall content ourselves with stating the main results of the former. To show the boundedness of a commutator, the following lemma is one of the keys of the proof. Lemma 4.10. Let 1 < p < ∞ and assume φ satisfies (39) and (40). Suppose further that a1 , a2 , . . . , al ∈ RBM O(µ). Let Q ∈ Q(µ, 2) be a fixed doubling cube and assume that f ∈ Lp,φ (µ) vanishes for µ-a.e. x ∈ 23 Q. Then µ ¶ ` | [aI , T ]f (x)| ≤ C log 2 + `(Q) ! Ã Y 1 1 × µ(B(x, 2`))1− p φ(µ(B(x, 2`))) p kaj k∗ kf : Lp,φ k∗ . j ∈I /

Proof. First we have, taking into account the kernel condition and the size of support, Ã ! Y X |aj (x) − mQ (aj )| |[aI , T ]f (x)| ≤ C I : I⊂{1,2,...,l}

Z ×

Rd \ 32 Q Z∞ As before, we insert

j∈I

1 · |y − cQ |n

Ã

1 = (n−1) |x − y|n

Y

|aj (y) − mQ (aj )| |f (y)| dµ(y).

j ∈I /

Z∞ χB(x,`) (y) 0

0

!

d` `n+1

and change the order

of integration. Thus, the matters are reduced to the estimate of the following term with a subset I fixed. ! ! Z∞ Ã Z Ã Y Y d` |aj (y) − mQ (aj )| |f (y)| dµ(y) n+1 . |aj (x) − mQ (a)| ` j∈I 3 `(Q) 2

B(x,`)

j ∈I /

374

Y. SAWANO AND S. SHIRAI

Now we use à Z à Y

!

!

|aj (y) − mQ (aj )| |f (y)| dµ(y)

j ∈I /

B(x,`)

µ ≤ C log 2 +

` `(Q)

µ

` ≤ C log 2 + `(Q)

Ã

¶ 1

µ(B(x, 2`))1− p

Y

!Ã Z

|f (y)|p dµ(y)

kaj k∗

j ∈I /

B(x,`)

Ã

¶ 1− p1

µ(B(x, 2`))

! p1

φ(µ(B(x, 2`)))

1 p

Y

! kaj k∗ kf : Lp,φ k∗ .

j ∈I /

Insert this formula, and we obtain the desired result by Lemma 4.9.

¤

Now we are able to prove the Lp,φ -boundedness of the multi-commutators generated by RBMO functions and singular integral operators. Theorem 4.11. Let 1 < p < ∞ and assume that φ satisfies (39) and (40). Suppose further that we are given a singular integral operator T and a1 , a2 , . . . , al ∈ RBM O(µ). (i) Let T be a singular integral operator. Then we have à ! Y k [a{1,2,...,l} , T ]f : Lp,φ (µ)k ≤ C kaj k∗ · kf : Lp,φ (µ)k. j∈I

(ii) In addition to the assumption above, suppose that one of aj belongs to RV M O(µ). Then [a{1,2,...,l} , T ] is a compact Lp,φ (µ)-operator. Proof. Compactness can be proved similarly to the above, we concentrate on (ii). It suffices to prove kM ] ([aI , T ]f ) : Lp,φ (µ)k ≤ C kf : Lp,φ (µ)k

(45)

k[aI , T ]f : Lp,φ (µ)k∗ ≤ C kf : Lp,φ (µ)k.

(46)

The proof of (45) is simple. We now turn to the proof of (46). Let Q ∈ Q(µ, 2) be given. Then we are to estimate Z 1 | [aI , T ]f (y)| dµ(y). 1 1 φ (µ (Q)) p µ(Q)1− p Q

We decompose f with respect to 32 Q. Let f1 = χ 3 Q and f2 = f − f1 . The esti2 mate for f2 is valid from Lemma 4.10. Since [a, T ] is Lp -bounded, the estimate for f1 is also simple : Ã ! p1 Z Z 1 1 | [aI , T ]f1 (y)| dµ(y) ≤ | [aI , T ]f1 (y)|p dµ(y) 1 1− p1 φ(µ(Q)) p φ(µ(Q)) µ(Q) Q

Q

COMPACT COMMUTATORS ON MORREY SPACES

à ≤C

1 φ(µ(Q))

375

! p1

Z

|f (y)|p dµ(y)

≤ C kf : Lp,φ (µ)k.

3 Q 2

This is the desired result.

¤

Finally we shall present the corresponding assertion for the multi-commutators generated by RBMO(µ)-functions and Kα . Theorem 4.12. Suppose that the parameters p, q satisfy 1 < p < q < ∞, = p1 − αn and φ satisfies (39) and (41). Suppose further that we are given a1 , a2 , . . . , al ∈ RBM O(µ)and an integral operator Kα with a fractional kernel of order α and regularity ε. Then (i) There exists a constant c > 0 such that for every f ∈ Lp,φ (µ), Ã ! ° ° q Y ° ° p ° [a{1,2,...,l} , Kα ]f : Lq,φ (µ)° ≤ C kaj k∗ · kf : Lp,φ (µ)k. ° ° 1 q

j∈I

(ii) In addition to the assumption above, suppose that one of aj belongs to q p

RV M O(µ). Then [a{1,2,...,l} , Kα ] is a compact Lp,φ (µ)-Lq,φ (µ) operator. Acknowledgment The authors are very grateful to Professor T. Mizuhara for his useful discussion. The first author is supported financially by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Finally the authors are thankful to the anonymous referee for having pointed out a vital mistake in the calculation. References 1. F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. Appl. (7) 7(1987), No. 3-4, 273–279 (1988). 2. W. Chen and E. Sawyer, A note on commutators of fractional integrals with RBMO(µ) functions. Illinois J. Math. 46(2002), No. 4, 1287–1298 3. J. Garc´ıa-Cuerva and E. Gatto, Boundedness properties of fractional integral operators associated to non-doubling measures. Studia Math. 162(2004), No. 3, 245–261. 4. G. Hu, Y. Meng, and D. Yang, Multilinear commutators for fractional integrals in non-homogeneous spaces. Publ. Mat. 48(2004), No. 2, 335–367. 5. G. Hu, Y. Meng, and D. Yang, Multilinear commutators of singular integrals with non doubling measures. Integral Equations Operator Theory 51(2005), No. 2, 235–255. 6. L. P. Kantorovich and G. P. Akilov, Functional analysis. (Translated from the Russian) Second edition. Pergamon Press, Oxford–Elmsford, N.Y., 1982. 7. V. Kokilashvili, Weighted estimates for classical integral operators. Nonlinear analysis, function spaces and applications, Vol. 4 (Roudnice nad Labem, 1990), 86–103, TeubnerTexte Math., 119, Teubner, Leipzig, 1990. 8. V. Kokilashvili and A. Meskhi, Fractional integrals on measure spaces. Fract. Calc. Appl. Anal. 4(2001), No. 1, 1–24.

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(Received 20.02.2007) Authors’ addresses: Y. Sawano Gakushuin University, Department of Mathematics 1-5-1, Mejiro, Toshimaku, Tokyo, 171-8588, Japan E-mail: [email protected] S. Shirai Yoneyama High School 215 Yoneyamacho Nakatuyamaaza Doubadome Mote City, Miyagi, 987-0331, Japan E-mail: bmo [email protected]