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Estimates for Marcinkiewicz commutators with Lipschitz functions under nondoubling measures

Abstract

Under the assumption that μ is a nondoubling measure on R d satisfying the growth condition, the author proves that the commutator M b generated by the Marcinkiewicz integral operator and the Lipschitz function is bounded from the Hardy space H fin 1 , , 0 (μ) into L q (μ) for 1/q=1β/n with the kernel satisfying a certain Hörmander-type condition. Moreover, the author shows that for p=n/β, M b is bounded from the Morrey space M q p (μ) into RBMO(μ), from L n / β (μ) into RBMO(μ) and from M q p (μ) into Lip ( β n p ) (μ), respectively.

MSC:42B25, 47B47, 42B20, 47A30.

1 Introduction

In recent years, harmonic analysis on spaces with nondoubling measures has become a very active research topic. There has been significant progress in the study of boundedness for singular integrals on these spaces; see [18]. Among a long list of research papers, some of them [911] are on the Marcinkiewicz integral operators. The motivation for developing the analysis with nondoubling measures and some important examples of nondoubling measures can be found in [12].

We recall that a nonnegative Radon measure μ on R d is said to be a nondoubling measure if there is a positive constant C 0 such that for all x R d and all r>0 it satisfies:

μ ( B ( x , r ) ) C 0 r n ,
(1.1)

where n is a positive constant and 0<nd, B(x,r) is the open ball centered at x and having radius r.

Let K(x,y) be a locally integrable function on R d × R d {(x,y):x=y}. Assume that there exists a constant C>0 such that for any x,y R d with xy,

| K ( x , y ) | C | x y | ( n 1 ) ,
(1.2)

and for any x,y, y R d ,

| x y | 2 | y y | [ | K ( x , y ) K ( x , y ) | + | K ( y , x ) K ( y , x ) | ] 1 | x y | dμ(x)C.
(1.3)

The Marcinkiewicz integral associated to the kernel K(x,y) and the measure μ as in (1.1) is defined by

M(f)(x)= ( 0 | | x y | t K ( x , y ) f ( y ) d μ ( y ) | 2 d t t 3 ) 1 / 2 ,x R d .
(1.4)

Let b L loc (μ), the Marcinkiewicz commutator M b is formally defined by

M b (f)(x)= ( 0 | | x y | t [ b ( x ) b ( y ) ] K ( x , y ) f ( y ) d μ ( y ) | 2 d t t 3 ) 1 / 2 ,x R d .
(1.5)

If μ is the d-dimensional Lebesgue measure in R d , and

K(x,y)= Ω ( x y ) | x y | d 1

with Ω homogeneous of degree zero and Ω Lip α ( S d 1 ) for some α(0,1], then it is easy to verify that K(x,y) satisfies (1.2) and (1.3), and in (1.4) is just the higher dimensional Marcinkiewicz integral M Ω defined by Stein in [13], which is important in classical harmonic analysis and is a focus of active research; see [1420]. Particularly, we should mention the work of Torchinsky and Wang [21], where they established the L p ( R d ) boundedness for the commutator generated by the Marcinkiewicz integral M Ω and BMO( R d ) function with p(1,). However, it is also worth to study the different behavior of another type commutator generated by the Marcinkiewicz integral M Ω and Lip β ( R d ) function, which was recently studied by Mo and Lu in [22] when Ω is homogeneous of degree zero and satisfies the cancellation condition. They obtained its boundedness from L p ( R d ) into L q ( R d ) for 1<p<n/β and 1/q=1/pβ/n.

When μ satisfies growth condition (1.1), as in (1.4) was first introduced by Hu et al. in [9], where the boundedness of such an operator in L p (μ) with 1<p< and the Hardy space H 1 (μ) were established under the assumption that is bounded on L 2 (μ) with the kernel K(x,y) satisfying (1.2) and (1.3). Moreover, they got the same estimates for the commutator M b defined as (1.5) with bRBMO(μ) when the kernel K(x,y) satisfies (1.2) and (1.6), which is slightly stronger than (1.3) and is defined as follows:

sup y , y R d , l > 0 , | y y | l k = 1 k 2 k l < | x y | 2 k + 1 l [ | K ( x , y ) K ( x , y ) | + | K ( y , x ) K ( y , x ) | ] 1 | x y | d μ ( x ) C .
(1.6)

However, in our problem, we discover that the kernels should satisfy some other kind of smoothness to replace condition (1.6).

Definition 1.1 Let 1s<, 0<ε<1. We say that the kernel K satisfies a Hörmander-type condition if there exist c s >1 and C s >0 such that for any x R d and l> c s |x|,

sup l > 0 , y , y R d | y y | l k = 1 2 k ε ( 2 k l ) n ( 1 ( 2 k l ) n 2 k l < | x y | 2 k + 1 l [ ( | K ( x , y ) K ( x , y ) | + | K ( y , x ) K ( y , x ) | ) 1 | x y | ] s d μ ( x ) ) 1 / s C s .
(1.7)

Directly, one can see that condition (1.7) can be rewritten as

We note that this kind of smoothness was not new. Condition (1.7′) is similar to the Hörmander-type condition which allows that the integral operator can be controlled by a maximal operator in doubling measure spaces, and also useful in the research of Schrödinger operators; see [2325] for details. We denote by H s the class of kernels satisfying this condition. It is clear that these classes are nested,

H s 2 H s 1 H 1 ,1< s 1 < s 2 <.

We should point out that H 1 is not condition (1.6).

In [11], by supposing that the kernel K satisfies (1.2) and (1.3), the authors studied the commutator M b in the case of b Lip β (μ) and established that it is bounded from L p (μ) into L q (μ) for 1<p<n/β and 1/q=1/pβ/n. Furthermore, when condition (1.3) is replaced by (1.7), M b is bounded from L p (μ) into Lip β n / p (μ) for some 0<β<1/2 and n/β<p<, from L n / β (μ) into RBMO(μ) for some 0<β<1 and n/β<p<, respectively.

The purpose of this paper is to get some estimates for the commutator M b with the kernel K satisfying (1.2) and (1.7) on the Hardy-type space and RBMO(μ) spaces. To be precise, we establish the boundedness of M b in H fin 1 , (μ) for 1/q=1β/n in Section 2. In Section 3, we prove that M b is bounded from RBMO(μ) to the Morrey space M q p (μ), from RBMO(μ) to L n / β (μ) for p=n/β.

Before stating our result, we need to recall some necessary notation and definitions. For a cube Q R d , we mean a closed cube whose sides are parallel to the coordinate axes. We denote its center and its side length by x Q and (Q), respectively. Let α>1, αQ denote the cube with the same center as Q and (αQ)=α(Q). Given two cubes QR in R d , set

S Q , R =1+ k = 1 N Q , R μ ( 2 k Q ) [ ( 2 k Q ) ] n ,

where N Q , R is the smallest positive integer k such that ( 2 k Q)(R). The concept S Q , R was introduced in [1], where some useful properties of S Q , R can be found.

The following characterization of the Lipschitz space Lip β (μ) for 0<β1 in [26] plays a key role in the proof of theorems.

Lemma 1.1 For a function b L loc 1 (μ), conditions I, II and III below are equivalent.

  1. (I)

    There is a constant C 1 0 such that

    | b ( x ) b ( y ) | C 1 | x y | β

for μ-almost every x and y in the support of μ.

  1. (II)

    There exist some constant C 2 0 and a collection of numbers b Q such that these two properties hold: for any cube Q,

    1 μ ( 2 Q ) Q | b ( x ) b Q | dμ(x) C 2 ( Q ) β ,
    (1.8)

and for any cube R such that QR and (R)2(Q),

| m Q ( b ) m R ( b ) | C 2 ( Q ) β .
(1.9)
  1. (III)

    For any given p, 1p, there is a constant C(p)0 such that for every cube Q, we have

    [ 1 μ ( Q ) Q | b ( x ) m Q ( b ) | p d μ ( x ) ] 1 / p C(p) ( Q ) β ,
    (1.10)

where, and in the sequel,

m Q (b)= 1 μ ( Q ) Q b(y)dμ(y),

and also for any cube R such that QR and (R)2(Q),

| m Q ( b ) m R ( b ) | C(p) ( Q ) β .

In addition, the quantities inf{ C 1 }, inf{ C 2 } and inf{C(p)} with a fixed p are equivalent and denoted by b Lip β .

Remark 1.1 Lemma 1.1 is a slight variant of Theorem 2.3 in [26]. To be precise, if we replace all balls in Theorem 2.3 of [26] by cubes, we then obtain Lemma 1.1.

Remark 1.2 For 0<β1, (1.9) is equivalent to

| b Q b R | C 2 S Q , R ( R ) β
(1.11)

for any two cubes QR with (R)2(Q); see Remark 2.7 in [26]. Note that for β=0 (1.9) and (1.10) is just the space RBMO(μ) of Tolsa; see [27]. Therefore, the space Lip β (μ) for 0β1 can be seen as a member of a family containing RBMO(μ).

We also need the following lemma for the L p (μ)-boundedness of M b , which was proved in [11].

Lemma 1.2 Let b Lip β (μ), 0<β1. Suppose that K(x,y) satisfies (1.2) and (1.3) and that M b is as in (1.5). If is bounded on L 2 (μ), then there exists a positive constant C>0 such that for all bounded functions f with compact support,

M b ( f ) L q ( μ ) C b Lip β f L p ( μ ) ,
(1.12)

where 1<p<n/β and 1/q=1/pβ/n.

Throughout this paper, we use the constant C with subscripts to indicate its dependence on the parameters. We denote simply by AB if there exists a constant C>0 such that ACB; and AB means that AB and BA. For a μ-measurable set E, χ E denotes its characteristic function. For any p[1,], we denote by p its conjugate index, namely, 1/p+1/ p =1.

2 Boundedness of M b in Hardy spaces

This section is devoted to the behavior of the commutator M b in Hardy spaces. In order to define the Hardy space H 1 (μ), Tolsa introduced the ‘grand’ maximal operator M Φ in [27].

Definition 2.1 Given f L loc 1 (μ), we define

M Φ f(x)= sup φ x | R d fφdμ|,

where the notation φx means that φ L 1 (μ) C 1 ( R d ) and satisfies

  1. (i)

    φ L 1 ( μ ) 1,

  2. (ii)

    0φ(y) 1 | y x | n for all y R d ,

  3. (iii)

    | φ (y)| 1 | y x | n + 1 for all y R d .

Based on Theorem 1.2 in [27], we can define the Hardy space H 1 (μ) as follows; see also [1].

Definition 2.2 The Hardy space H 1 (μ) is the set of all functions f L 1 (μ) satisfying that R d fdμ=0 and M Φ f L 1 (μ). Moreover, we define the norm of f H 1 (μ) by

f H 1 ( μ ) = f L 1 ( μ ) + M Φ f L 1 ( μ ) .

We recall the atomic Hardy space H atb 1 , , 0 (μ) as follows.

Definition 2.3 Let ρ>1. A function h L loc 1 (μ) is called an atomic block if

  1. (1)

    there exists some cube R such that supphR,

  2. (2)

    R d h(x)dμ(x)=0,

  3. (3)

    for i=1,2, there are functions a i supported on cubes Q i R and numbers λ i R such that h= λ 1 a 1 + λ 2 a 2 , and

    a i L ( μ ) [ μ ( ρ Q i ) S Q i , R ] 1 .

Then we define

| h | H atb 1 , , 0 ( μ ) =| λ 1 |+| λ 2 |.

Define H atb 1 , , 0 (μ) and H fin 1 , , 0 (μ) as follows:

f H atb 1 , , 0 ( μ ) =inf { j | h j | H atb 1 , , 0 ( μ ) : f = j = 1 h j , { h j } j N  are  ( 1 , , 0 ) -atoms }

and

f H fin 1 , , 0 ( μ ) =inf { j = 1 k | h j | H atb 1 , , 0 ( μ ) : f = j = 1 k h j , { h j } j = 1 k  are  ( 1 , , 0 ) -atoms } ,

where the infimum is taken over all possible decompositions of f in atomic blocks, H fin 1 , , 0 (μ) is the set of all finite linear combinations of (1,,0)-atoms.

Remark 2.1 It was proved in [1] that for each ρ>1, the atomic Hardy space H atb 1 , , 0 (μ) is independent of the choice of ρ.

To establish the boundedness of operators in Hardy-type spaces on R n , one usually appeals to the atomic decomposition characterization (see [28, 29]) of these spaces, which means that a function or distribution in Hardy-type spaces can be represented as a linear combination of atoms. Then the boundedness of linear operators in Hardy-type spaces can be deduced from their behavior on atoms in principle. However, Meyer [30] (see also [31]) gave an example of f H 1 ( R n ) whose norm cannot be achieved by its finite atomic decompositions via (1,,0)-atoms. Based on this fact, Bownik [31] (Theorem 2) constructed a surprising example of a linear functional defined on a dense subspace of H 1 ( R n ), which maps all (1,,0)-atoms into bounded scalars, but yet cannot extend to a bounded linear functional on the whole H 1 ( R n ).

Recently, in [32], a boundedness criterion was established via Lusin function characterizations of Hardy spaces on R n as follows: a sublinear operator T extends to a bounded sublinear operator from Hardy spaces H p ( R n ) with p(0,1] to some quasi-Banach space B if and only if T maps all (p,2,s)-atoms into uniformly bounded elements of B for some s[n(1/p1)]. Here and in what follows [t] means the integer part of real t. This result shows the structural difference between atomic characterization of H p ( R n ) via (p,2,s)-atoms and (p,,s)-atoms. On the other hand, Meda et al. [33] independently obtained some similar results by grand maximal function characterizations of Hardy spaces on R n . In fact, let p(0,1], p<q[1,] and integer s[n(1/p1)], and let H fin p , q , s ( R n ) be the set of all finite linear combinations of (p,q,s)-atoms. Denote by C( R n ) the set of all continuous functions. For any f H fin p , q , s ( R n ), when q< or f H fin p , q , s ( R n )C( R n ) when q=, Meda et al. in [33] proved that f H p ( R n ) can be achieved by a finite atomic decomposition via (p,q,s)-atom when q< or continuous (p,q,s)-atom when q=; from this, they further deduced that if T is a linear operator and maps all (1,q,0)-atoms with q(1,) or all continuous (1,q,0)-atoms with q= into uniformly bounded elements of some Banach space B, then T uniquely extends to a bounded linear operator from H 1 ( R n ) to B which coincides with T on these (1,q,0)-atoms.

According to the theory of Meda et al. [33], we get the result as follows.

Theorem 2.1 Let 0<β1, b Lip β (μ) and 1/q=1β/n. Suppose that K satisfies (1.2) and H q condition. If f H fin 1 , , 0 (μ), then M b is bounded from the Hardy space into the Lebesgue space, namely, there exists a positive constant C such that

M b ( f ) L q ( μ ) C b Lip β f H fin 1 , , 0 ( μ ) .

Proof of Theorem 2.1 Via Remark 2.1, without loss of generality, we may assume that ρ=4 and f=h as a finite sum of atomic blocks defined in Definition 2.3. It is easy to see that we only need to prove the theorem for one atomic block h. Let R be a cube such that supphR, R d h(x)dμ(x)=0, and

h(x)= λ 1 a 1 (x)+ λ 2 a 2 (x),
(2.1)

where λ i for i=1,2, is a real number, | h | H atb 1 , ( μ ) =| λ 1 |+| λ 2 |, a i for i=1,2, is a bounded function supported on some cube Q i R and it satisfies

a i L ( μ ) [ μ ( 4 Q i ) S Q i , R ] 1 .
(2.2)

Write

M b ( h ) L q ( μ ) ( 2 R | M b ( h ) ( x ) | q d μ ( x ) ) 1 / q + ( R d ( 2 R ) | M b ( h ) ( x ) | q d μ ( x ) ) 1 / q ( 2 R | M b ( h ) ( x ) | q d μ ( x ) ) 1 / q + { R d ( 2 R ) ( 0 | x x R | + 2 ( R ) | | x y | t K ( x , y ) × [ b ( x ) b ( y ) ] h ( y ) d μ ( y ) | 2 d t t 3 ) q / 2 d μ ( x ) } 1 / q + { R d ( 2 R ) ( | x x R | + 2 ( R ) | | x y | t K ( x , y ) [ b ( x ) b ( y ) ] h ( y ) d μ ( y ) | 2 d t t 3 ) q / 2 d μ ( x ) } 1 / q = I + II + III .

By (2.1), we have

I | λ 1 | ( 2 R | M b ( a 1 ) ( x ) | q d μ ( x ) ) 1 / q + | λ 2 | ( 2 R | M b ( a 2 ) ( x ) | q d μ ( x ) ) 1 / q = I 1 + I 2 .

To estimate I 1 , we write

I 1 | λ 1 | ( 2 Q 1 | M b ( a 1 ) ( x ) | q d μ ( x ) ) 1 / q + | λ 1 | ( 2 R 2 Q 1 | M b ( a 1 ) ( x ) | q d μ ( x ) ) 1 / q = I 11 + I 12 .

Choose p 1 and q 1 such that 1< p 1 <n/β, 1<q< q 1 and 1/ q 1 =1/ p 1 β/n. By the Hölder inequality, the fact that S Q 1 , R 1 and the ( L p 1 (μ), L q 1 (μ))-boundedness of M b (Lemma 1.2), we have that

I 11 | λ 1 | [ 2 Q 1 | M b ( a 1 ) ( x ) | q 1 d μ ( x ) ] 1 / q 1 μ ( 2 Q 1 ) 1 / q 1 / q 1 b Lip β | λ 1 | a 1 L p 1 ( μ ) μ ( 2 Q 1 ) 1 / q 1 / q 1 b Lip β | λ 1 | a 1 L ( μ ) μ ( 2 Q 1 ) 1 / p 1 + 1 / q 1 / q 1 b Lip β | λ 1 | .

Denote N 2 Q 1 , 2 R simply by N 1 . Invoking the fact that a 1 L ( μ ) [ μ ( 4 Q 1 ) S Q 1 , R ] 1 , we thus get

I 12 | λ 1 | { k = 1 N 1 + 1 2 k + 1 Q 1 2 k Q 1 [ 0 | | x y | t [ b ( x ) b ( y ) ] | x y | n 1 a 1 ( y ) d μ ( y ) | 2 d t t 3 ] q / 2 d μ ( x ) } 1 / q | λ 1 | { k = 1 N 1 + 1 2 k + 1 Q 1 2 k Q 1 [ Q 1 | b ( x ) b ( y ) | | x y | n 1 | a 1 ( y ) | ( | x y | d t t 3 ) 1 / 2 d μ ( y ) ] q d μ ( x ) } 1 / q | λ 1 | { k = 1 N 1 + 1 2 k + 1 Q 1 2 k Q 1 [ Q 1 | b ( x ) b ( y ) | | x y | n | a 1 ( y ) | d μ ( y ) ] q d μ ( x ) } 1 / q b Lip β | λ 1 | { k = 1 N 1 + 1 ( 2 k Q 1 ) q ( β n ) 2 k + 1 Q 1 2 k Q 1 [ Q 1 | a 1 ( y ) | d μ ( y ) ] q d μ ( x ) } 1 / q b Lip β | λ 1 | { k = 1 N 1 + 1 ( 2 k Q 1 ) q ( β n ) μ ( 2 ( k + 1 ) Q 1 ) a 1 L ( μ ) q μ ( Q 1 ) q } 1 / q b Lip β | λ 1 | { k = 1 N 1 + 1 ( 2 k Q 1 ) q ( β n ) μ ( 4 Q 1 ) q S Q 1 , R q μ ( 2 ( k + 1 ) Q 1 ) μ ( Q 1 ) q } 1 / q b Lip β | λ 1 | { S Q 1 , R q k = 2 N 1 + 1 μ ( 2 k Q 1 ) ( 2 k Q 1 ) n } 1 / q b Lip β | λ 1 | ,

here we have used the fact that

k = 2 N 1 + 1 μ ( 2 k Q ) l ( 2 k Q ) n C S Q 1 , R ;

see [1, 27] for details.

The estimates for I 11 and I 12 give the desired estimate for I 1 . A similar argument tells us that

I 2 b Lip β | λ 2 |.

Combining the estimates for I 1 and I 2 yields the desired estimate for I.

For i=1,2, y Q i R, x R d (2R), we have |xy||x x R ||x x R |+2(R). By the Minkowski inequality, we get

II { R d ( 2 R ) [ R d ( | x y | | x x R | + 2 ( R ) d t t 3 ) 1 / 2 | h ( y ) | | x y | n 1 | b ( x ) b ( y ) | d μ ( y ) ] q d μ ( x ) } 1 / q R { R d ( 2 R ) [ ( 1 ( | x x R | + 2 ( R ) ) 2 1 | x y | 2 ) 1 / 2 × | h ( y ) | | x y | n 1 | b ( x ) b ( y ) | ] q d μ ( x ) } 1 / q d μ ( y ) R { R d ( 2 R ) ( ( R ) 1 / 2 | x y | 3 / 2 | h ( y ) | | x y | n 1 | b ( x ) b ( y ) | ) q d μ ( x ) } 1 / q d μ ( y ) R { k = 1 2 ( k + 1 ) R 2 k R ( ( R ) 1 / 2 | x y | n β + 1 / 2 b Lip β ) q d μ ( x ) } 1 / q | h ( y ) | d μ ( y ) b Lip β ( j = 1 2 | λ j | a j L 1 ( μ ) ) { k = 1 ( R ) 1 / 2 ( 2 k R ) n + β 1 / 2 μ ( 2 k + 1 R ) 1 / q } b Lip β ( j = 1 2 | λ j | ) .

For any yR, we have t|x x R |+2(R)|x x R |+|y x R ||xy|. It follows that

III { R d ( 2 R ) | R K ( x , y ) [ b ( x ) b ( y ) ] h ( y ) d μ ( y ) ( | x x R | + 2 ( R ) d t t 3 ) 1 / 2 | q d μ ( x ) } 1 / q { R d ( 2 R ) | R K ( x , y ) [ b ( x ) b ( y ) ] h ( y ) d μ ( y ) 1 | x x R | + 2 ( R ) | q d μ ( x ) } 1 / q { R d ( 2 R ) | R K ( x , y ) h ( y ) | x x R | + 2 ( R ) [ b ( x ) m R ( b ) ] d μ ( y ) | q d μ ( x ) } 1 / q + { R d ( 2 R ) | R K ( x , y ) h ( y ) | x x R | + 2 ( R ) [ m R ( b ) b ( y ) ] d μ ( y ) | q d μ ( x ) } 1 / q = III 1 + III 2 .

For III 1 , by the Minkowski inequality, we have

III 1 = { R d ( 2 R ) | [ b ( x ) m R ( b ) ] R K ( x , y ) K ( x , x R ) | x x R | + 2 ( R ) h ( y ) d μ ( y ) | q d μ ( x ) } 1 / q R k = 1 m ( 2 k + 1 R 2 k R [ b Lip β | x x R | β | K ( x , y ) K ( x , x R ) | | x y | ] q d μ ( x ) ) 1 / q | h ( y ) | d μ ( y ) b Lip β R | h ( y ) | k = 1 m ( 2 k + 1 R 2 k R [ ( 2 k R ) β | K ( x , y ) K ( x , x R ) | | x y | ] q d μ ( x ) ) 1 / q d μ ( y ) b Lip β R | h ( y ) | k = 1 m 2 k ( β ε n + n / q ) ( R ) β n + n / q 2 k ε [ 2 k ( R ) ] n × ( 1 [ 2 k ( R ) ] n 2 k ( R ) < | x y | 2 k + 1 ( R ) [ | K ( x , y ) K ( x , x R ) | | x y | ] q d μ ( x ) ) 1 / q d μ ( y ) b Lip β j = 1 2 | λ j | a j L 1 ( μ ) b Lip β j = 1 2 | λ j | ,

here we used the fact that 1/q=1β/n and 0<ε1.

We now turn to estimate III 2 . Note that for any yR, x R d 2R, we have |xy||x x R |+2(R), so by the Minkowski inequality,

III 2 R k = 1 ( 2 k + 1 R 2 k R [ | K ( x , y ) | | x y | | m R ( b ) b ( y ) | ] q d μ ( x ) ) 1 / q | h ( y ) | d μ ( y ) R k = 1 ( 2 k + 1 R 2 k R [ | m R ( b ) b ( y ) | | x y | n ] q d μ ( x ) ) 1 / q | h ( y ) | d μ ( y ) R k = 1 b Lip β ( R ) β ( 2 k R ) n μ ( 2 k + 1 R ) 1 / q j = 1 2 | λ j | | a j ( y ) | d μ ( y ) b Lip β j = 1 2 | λ j | a j L 1 ( μ ) b Lip β j = 1 2 | λ j | .

Then

III b Lip β j = 1 2 | λ j |.

Combining the estimates for I, II and III yields that

M b ( h ) L q ( μ ) C | h | H atb 1 , , 0 ( μ ) ,

and this is the result of Theorem 2.1. □

3 Boundedness of M b in RBMO(μ) space

In this section, we investigate the boundedness for the commutator M b as in (1.5) in the space RBMO(μ) for f M q p (μ) and f L n / β (μ), respectively.

Firstly, we recall the definition of the Morrey space with nondoubling measure denoted by M q p (μ), which was introduced by Sawano and Tanaka in [3436].

Definition 3.1 Let k>1 and 1qp<. We define the Morrey space M q p (μ) as

M q p (μ):= { f L loc q ( μ ) | f M q p ( μ ) < } ,

where the norm f M q p ( μ ) is given by

f M q p ( μ ) := sup Q μ ( k Q ) 1 p 1 q ( Q | f | q d μ ) 1 q .

We should note that the parameter k>1 appearing in the definition does not affect the definition of the space M q p (μ), and the space M q p (μ) is a Banach space with its norm; see [34]. By using the Hölder inequality to (1.5), it is easy to see that for all 1 q 2 q 1 p, we have

L p (μ)= M p p (μ) M q 1 p (μ) M q 2 p (μ).

Theorem 3.1 Let b Lip β (μ), 0<β1, 1q<p= n β . Suppose that K satisfies (1.2) and H p condition, is bounded on L 2 (μ) and M b is defined as in (1.5). Then there exists a positive constant C such that for all f M q p (μ),

M b ( f ) RBMO ( μ ) C b Lip β f M q p ( μ ) .

Theorem 3.2 Let b Lip β (μ), 0<β1 and p=n/β. Suppose that K satisfies (1.2) and H n / ( n β ) condition. If is bounded on L 2 (μ) and M b is defined as in (1.5), then there is a constant C>0 such that for all bounded functions f with compact support,

M b ( f ) RBMO ( μ ) C b Lip β f L n / β ( μ ) .

Theorem 3.3 Let b Lip β (μ), 0<β1, 1q<p and p>n/β. Suppose that K satisfies (1.2) and H p condition, is bounded on L 2 (μ) and M b is defined as in (1.5). Then there exists a positive constant C such that for all f M q p (μ),

M b ( f ) Lip ( β n p ) C b Lip β f M q p ( μ ) .

Remark 3.1 By the Minkowski inequality and the kernel condition, we get that

M b ( f ) ( x ) = ( 0 | | x y | t [ b ( x ) b ( y ) ] K ( x , y ) f ( y ) d μ ( y ) | 2 d t t 3 ) 1 / 2 = ( 0 | 1 t | x y | t [ b ( x ) b ( y ) ] K ( x , y ) f ( y ) d μ ( y ) | 2 d t t ) 1 / 2 = R d | K ( x , y ) [ b ( x ) b ( y ) ] f ( y ) | ( | x y | t d t t 3 ) 1 / 2 d μ ( y ) R d | [ b ( x ) b ( y ) ] f ( y ) | | x y | n 1 | x y | 1 d μ ( y ) b Lip β R d | f ( y ) | | x y | n β d μ ( y ) b Lip β I β ( | f | ) ( x ) ,

where I β is the fractional integral operator. Then M b (f) L loc 1 (μ).

Remark 3.2 Theorem 3.2 can be deduced as a conclusion of Theorem 3.1 in the case of p=q= n β .

Remark 3.3 Applying Lemma 1.1, a slight change in the proof of Theorem 3.1 actually shows Theorem 3.3 and we leave the details to the reader.

Proof of Theorem 3.1 For any cubes Q and R in R d such that QR satisfies (R)2(Q), let

a Q = m Q [ M b ( f χ R d 3 2 Q ) ]

and

a R = m R [ M b ( f χ R d 3 2 R ) ] .

It is easy to see that a Q and a R are real numbers. By Lemma 1.1, we need to show that for some fixed r>q there exists a constant C>0 such that

( 1 μ ( 2 Q ) Q | M b ( f ) ( x ) a Q | r d μ ( x ) ) 1 / r b Lip β f M q p ( μ )
(3.1)

and

| a Q a R | b Lip β f M q p ( μ ) .
(3.2)

Let us first prove estimate (3.1). For a fixed cube Q and xQ, decompose f= f 1 + f 2 , where f 1 =f χ 3 2 Q and f 2 =f f 1 . Write that

1 μ ( 2 Q ) Q | M b ( f ) ( x ) a Q | r d μ ( x ) 1 μ ( 2 Q ) Q | M b ( f 1 ) ( x ) | r d μ ( x ) + 1 μ ( 2 Q ) Q | M b ( f 2 ) ( x ) a Q | r d μ ( x ) = I 1 + I 2 .

For 1/r=1/qβ/n and p=n/β, it follows that

I 1 1 μ ( 2 Q ) [ Q | M b ( f 1 ) ( x ) | r d μ ( x ) ] 1 / r 1 μ ( 2 Q ) b Lip β r ( 3 2 Q | f ( x ) q | d μ ( x ) ) r / q 1 μ ( 2 Q ) b Lip β r { ( μ ( 2 Q ) 1 / p 1 / q 3 2 Q | f ( x ) | q d μ ( x ) ) 1 / q } r μ ( 2 Q ) ( 1 / p 1 / q ) r b Lip β r f M q p ( μ ) r μ ( 2 Q ) ( 1 / q 1 / p ) r 1 b Lip β r f M q p ( μ ) r μ ( 2 Q ) ( 1 / r + β / n 1 / p ) r 1 b Lip β r f M q p ( μ ) r .

In order to estimate the term I 2 , set

D 1 ( x , y ) = ( 0 [ | x z | t < | y z | | K ( x , z ) | | b ( z ) m Q ( b ) | | f 2 ( z ) | d μ ( z ) ] 2 d t t 3 ) 1 / 2 , D 2 ( x , y ) = ( 0 [ | y z | t < | x z | | K ( y , z ) | | b ( z ) m Q ( b ) | | f 2 ( z ) | d μ ( z ) ] 2 d t t 3 ) 1 / 2

and

D 3 (x,y)= ( 0 [ | y z | t | x z | t | K ( x , z ) K ( y , z ) | | b ( z ) m Q ( b ) | | f 2 ( z ) | d μ ( z ) ] 2 d t t 3 ) 1 / 2 .

It is easy to get that for any x,yQ,

| M b ( f 2 ) ( x ) M b ( f 2 ) ( y ) | = | ( 0 | | x z | t [ b ( x ) b ( z ) ] K ( x , z ) f 2 ( z ) d μ ( z ) | 2 d t t 3 ) 1 / 2 ( 0 | | y z | t [ b ( y ) b ( z ) ] K ( y , z ) f 2 ( z ) d μ ( z ) | 2 d t t 3 ) 1 / 2 | j = 1 3 D j ( x , y ) .

For D 1 (x,y), since x,yQ, z 3 2 Q, we thus get

D 1 ( 0 [ | x z | t < | y z | | b ( z ) m Q ( b ) | | x z | n 1 | f 2 ( z ) | d μ ( z ) ] 2 d t t 3 ) 1 / 2 | x z | < | y z | | b ( z ) m Q ( b ) | | x z | n 1 | f 2 ( z ) | [ | y z | | x z | d t t 3 ] 1 / 2 d μ ( z ) | x z | < | y z | | b ( z ) m Q ( b ) | | x z | n 1 | f 2 ( z ) | ( Q ) 1 / 2 | x z | 3 / 2 d μ ( z ) ( Q ) 1 / 2 R d 3 2 Q | b ( z ) m Q ( b ) | | x z | n + 1 / 2 | f ( z ) | d μ ( z ) ( Q ) 1 / 2 k = 1 3 2 2 k Q 3 2 2 k 1 Q | b ( z ) m Q ( b ) | | x z | n + 1 / 2 | f ( z ) | d μ ( z ) ( Q ) 1 / 2 k = 1 1 ( 3 2 2 k Q ) n + 1 / 2 3 2 2 k Q | b ( z ) m Q ( b ) | | f ( z ) | d μ ( z ) k = 1 2 k / 2 1 ( 2 k Q ) n ( 3 2 2 k Q | b ( z ) m Q ( b ) | q d μ ( z ) ) 1 / q ( 3 2 2 k Q | f ( z ) | q d μ ( z ) ) 1 / q b Lip β f M q p ( μ ) k = 1 k 2 k / 2 ( 2 k Q ) β n μ ( 3 2 2 k Q ) 1 1 / q μ ( 2 k + 2 Q ) 1 / q 1 / p b Lip β f M q p ( μ ) k = 1 k 2 k / 2 b Lip β f M q p ( μ ) ,

here we used the Minkowski inequality, β=n/p, (1.10) of Lemma 1.1 and the fact that

| b ( z ) m Q ( b ) | ( 2 k Q ) β b Lip β for z R d 3 2 Q.

By a similar argument, it follows that

D 2 b Lip β f M q p ( μ ) .

Finally, by the condition H p , which the kernel K satisfies, and the fact that β=n/p, applying the Minkowski inequality, we have

D 3 ( x , y ) = ( 0 [ | y z | t | x z | t | K ( x , z ) K ( y , z ) | | b ( z ) m Q ( b ) | | f 2 ( z ) | d μ ( z ) ] 2 d t t 3 ) 1 / 2 R d | K ( x , z ) K ( y , z ) | | b ( z ) m Q ( b ) | | f 2 ( z ) | [ | y z | t | x z | t d t t 3 ] 1 / 2 d μ ( z ) k = 1 3 2 2 k Q 3 2 2 k 1 Q | K ( x , z ) K ( y , z ) | | b ( z ) m Q ( b ) | | f ( z ) | | y z | d μ ( z ) b Lip β k = 1 ( 2 k Q ) β 3 2 2 k Q 3 2 2 k 1 Q | K ( x , z ) K ( y , z ) | | f ( z ) | | y z | d μ ( z ) b Lip β f M q p ( μ ) k = 1 ( 2 k Q ) β n / p ( 2 k Q ) n / q × ( 3 2 2 k Q 3 2 2 k 1 Q [ | K ( x , z ) K ( y , z ) | 1 | y z | ] q d μ ( z ) ) 1 / q b Lip β f M q p ( μ ) k = 1 ( 2 k Q ) n × ( 1 ( 2 k Q ) n 3 2 2 k Q 3 2 2 k 1 Q [ | K ( x , z ) K ( y , z ) | 1 | y z | ] q d μ ( z ) ) 1 / q b Lip β f M q p ( μ ) .

Combining these estimates, we conclude that

I 2 b Lip β r f M q p ( μ ) r ,

and so estimate (3.1) is proved. □

We proceed to show (3.2). For any cubes QR with xQ, where Q is arbitrary and R is a doubling cube with (R)(Q), denote N Q , R +1 simply by N. Write

| a Q a R | | m R [ M b ( f χ R d 2 N Q ) ] m Q [ M b ( f χ R d 2 N Q ) ] | + | m Q [ M b ( f χ 2 N Q 3 2 Q ) ] | + | m R [ M b ( f χ 2 N Q 3 2 R ) ] | = E 1 + E 2 + E 3 .

As in the estimate for the term I 2 , we have

E 1 b Lip β f M q p ( μ ) .

We conclude from yR, z 2 N Q 3 2 Q that

M b ( f χ 2 N Q 3 2 R ) ( y ) 2 N Q 3 2 R | K ( y , z ) ( b ( y ) b ( z ) ) f ( z ) | ( | y z | d t t 3 ) 1 / 2 d μ ( z ) 2 N Q 3 2 R | b ( y ) b ( z ) | | y z | n | f ( z ) | d μ ( z ) b Lip β 2 N Q 3 2 R | f ( z ) | | y z | n β d μ ( z ) b Lip β ( R ) β n ( 2 N Q | f ( z ) | q d μ ( z ) ) 1 / q μ ( 2 N Q ) 1 / q b Lip β f M q p ( μ ) ( 2 N Q ) β n + n n / q + n / q n / p b Lip β f M q p ( μ ) .

Taking mean over yR, we obtain

E 3 b Lip β f M q p ( μ ) .

Analysis similar to that in the estimate for E 3 shows that

E 2 b Lip β f M q p ( μ ) .

Finally, we get (3.2) and this is precisely the assertion of Theorem 3.1.

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Acknowledgements

The author would like to thank the referee for his very careful reading and valuable remarks which made this article more readable. This work was supported by NSFC of China (No. 11261055), NSFC of Xinjiang (No. 2012211B28), Graduate Innovative Research Program Foundation of Jiangsu Province (2012) and Science Research Foundation of Yili Normal University (No. 2011YNZD009).

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Li, L. Estimates for Marcinkiewicz commutators with Lipschitz functions under nondoubling measures. J Inequal Appl 2013, 388 (2013). https://doi.org/10.1186/1029-242X-2013-388

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Keywords

  • nondoubling measure
  • Marcinkiewicz integral
  • commutator
  • Lip β (μ)