Estimates for fractional type Marcinkiewicz integrals with non-doubling measures

Let μ be a nonnegative Radon measure on R d which satisfies the following growth condition: for all x ∈ R d and all r > 0,

where C 0 and n are positive constants and n ∈ ( 0 , d ], B ( x , r ) is the open ball centered at x and having radius r. So μ is claimed to be non-doubling measure. If there exists a positive constant C such that for any x ∈ supp ( μ ) and r > 0, μ ( B ( x , 2 r ) ) ≤ C μ ( B ( x , r ) ), the μ is said to be doubling measure. It is well known that the doubling condition on underlying measures is a key assumption in the classical theory of harmonic analysis. Especially, in recent years, many classical results concerning the theory of Calderón-Zygmund operators and function spaces have been proved still valid if the underlying measure is a nonnegative Radon measure on R d which only satisfies (1.1) (see [1‐8]). The motivation for developing the analysis with non-doubling measures and some examples of non-doubling measures can be found in [9]. We only point out that the analysis with non-doubling measures played a striking role in solving the long-standing open Painlevé’s problem by Tolsa in [10].

Let K ( x , y ) be a μ-locally integrable function on R d × R d ∖ { ( x , y ) : x = y }. Assume that there exists a positive constant C such that for any x , y ∈ R d with x ≠ y,

and for any x , y , y ′ ∈ R d,

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

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

with Ω homogeneous of degree zero and Ω ∈ Lip γ ( S d − 1 ) for some γ ∈ ( 0 , 1 ], then K satisfies (1.2) and (1.3). Under these conditions, ℳ in (1.4) is introduced by Si et al. in [11]. As a special case, by letting α = 0, we recapture the classical Marcinkiewicz integral operators that Stein introduced in 1958 (see [12]). Since then, many works have appeared about Marcinkiewicz type integral operators. A nice survey has been given by Lu in [13].

In 2007, the Hörmander-type condition was introduced by Hu et al. in [14], which was slightly stronger than (1.3) and was defined as follows:

However, in this paper, we discover that the kernel should satisfy some other kind of smoothness condition to replace (1.6).

Definition 1.1 Let 1 ≤ s < ∞, 0 < ε < 1. The kernel K is said to satisfy 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 |,

We denote by H s the class of kernels satisfying this condition. It is clear that these classes are nested,

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

The purpose of this paper is to get some estimates for the fractional type Marcinkiewicz integral ℳ with kernel K satisfying (1.2) and (1.7) on the Hardy-type space and the RBMO ( μ ) space. To be precise, we establish the boundedness of ℳ in H fin 1 , ∞ , 0 ( μ ) for 1 q = 1 − α n in Section 2. In Section 3, we prove that ℳ is bounded from the space RBMO ( μ ) to the Morrey space M q p ( μ ), from the space RBMO ( μ ) to the Lebesgue space L n α ( μ ) for p = n α.

Before stating our results, 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 ℓ ( Q ), respectively. Let η > 1, ηQ denote the cube with the same center as Q and ℓ ( η Q ) = η ℓ ( Q ). Given two cubes Q ⊂ R in R d, set

where N Q , R is the smallest positive integer k such that ℓ ( 2 k Q ) ≥ ℓ ( R ). The concept S Q , R was introduced in [15], where some useful properties of S Q , R can be found.

Lemma 1.2 For a function b ∈ L loc 1 ( μ ), 0 < β ≤ 1, conditions (i) and (ii) below are equivalent.

and for any cube R such that Q ⊂ R and ℓ ( R ) ≤ 2 ℓ ( Q ),

where

and also for any cube R such that Q ⊂ R and ℓ ( R ) ≤ 2 ℓ ( Q ),

Remark 1.3 Lemma 1.2 is a slight variant of Theorem 2.3 in [16]. To be precise, if we replace all balls in Theorem 2.3 of [16] by cubes, we then obtain Lemma 1.2.

Remark 1.4 For 0 < β ≤ 1, (1.9) is equivalent to

for any two cubes Q ⊂ R with ℓ ( R ) ≤ 2 ℓ ( Q ) (see Remark 2.7 in [16]).

Lemma 1.5 Let 0 < α < n, 1 < p < n α, 1 r = 1 p − α n and q ≥ n n − α. Then the fractional integral operator I α defined by

is bounded from L p ( μ ) to L r ( μ ) (see [17]).

Lemma 1.6 Let 0 < α < n, 1 < p < n α, 1 q = 1 p − α n. Suppose that K ( x , y ) satisfies (1.2) and (1.3) and ℳ is as in (1.4). Then there exists a positive constant C > 0 such that for all bounded functions f with compact support,

Proof of Lemma 1.6 By Minkowski’s inequality, we have

By Lemma 1.5 then

 □

Throughout this paper, we use the constant C with subscripts to indicate its dependence on the parameters. 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.

This section is devoted to the behavior of ℳ in Hardy spaces. In order to define the Hardy space H 1 ( μ ), Tolsa introduced the grand maximal operator M ϕ in [18].

Definition 2.1 Given f ∈ L loc 1 ( μ ), M ϕ f is defined as

where the notation φ ∼ x means that φ ∈ L 1 ( μ ) ∩ C 1 ( R d ) and satisfies

Based on Theorem 1.2 in [18], we can define the Hardy space H 1 ( μ ) as follows (see [15]).

Definition 2.2 The Hardy space H 1 ( μ ) is the set of all functions f ∈ L 1 ( μ ) satisfying that ∫ R d f d μ = 0 and M ϕ f ∈ L 1 ( μ ). Moreover, the norm of f ∈ H 1 ( μ ) is defined by

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

Then define

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

and

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 2.4 It was proved in [15] that for each ρ > 1, the atomic Hardy space H atb 1 , ∞ , 0 ( μ ) is independent of the choice of ρ.

Applying the theory of Meda et al. in [19], we easily get the result as follows.

Theorem 2.5 Let 0 < α < n, 1 q = 1 − α n. Suppose that K satisfies (1.2) and the H q condition and f ∈ H fin 1 , ∞ , 0 ( μ ). Then ℳ is bounded from the Hardy space into the Lebesgue space, namely there exists a positive constant C such that

Proof of Theorem 2.5 Without loss of generality, we may assume that ρ = 4 and f = ∑ h as a finite 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 supp h ⊂ R, ∫ R d h ( x ) d μ ( x ) = 0, and

where λ i for i = 1 , 2 is a real number, | h i | H atb 1 , ∞ , 0 ( μ ) = λ 1 + λ 2, a i for i = 1 , 2 is a bounded function supported on some cubes Q i ⊂ R and it satisfies

Write

By (2.1), we have

To estimate I 1, we write

Choose p 1 and q 1 such that 1 < p 1 < n α, 1 < q < q 1 and 1 q 1 = 1 p 1 − n α. 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 (see Lemma 1.6), we have that

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

Here we have used the fact that

see [16] for details.

The estimates for I 11 and I 12 give the desired estimate for I 1. With a similar argument, we have

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

For i = 1 , 2, y ∈ Q i ⊂ R, x ∈ R d ∖ ( 2 R ), we have | x − y | ∼ | x − x R | ∼ | x − x R | + 2 ℓ ( R ), by Minkowski’s inequality, we get

For any y ∈ R, we have | x − y | ≤ | x − x R | + | y − x R | ≤ | x − x R | + 2 ℓ ( R ) ≤ t. It follows that

Here we have used the fact that 1 q = 1 − α n.

Combining the estimates for I, II and III yields that

and this is the result of Theorem 2.5. □

In this section, we discuss the boundedness for ℳ as in (1.4) in the space RBMO ( μ ) for f ∈ M p q ( μ ) and f ∈ L n α ( μ ), respectively.

Firstly, we need to recall the definition of Morrey space with non-doubling measure denoted by M q p ( μ ), which was introduced by Sawano and Tanaka in [20].

Definition 3.1 Let ν > 1 and 1 ≤ q ≤ p < ∞. The Morrey space M q p ( μ ) is defined by

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

We should note that the parameter ν > 1 appearing in the definition does not affect the definition of the space M q p ( μ ), and M q p ( μ ) is a Banach space with its norms (see [20]). By using the Hölder inequality to (1.4), it is easy to see that for all 1 ≤ q 2 ≤ q 1 ≤ p, then

Theorem 3.2 Let 0 < α < n, 1 ≤ q < p = n α. Suppose that K ( x , y ) satisfies (1.2) and the H p ′ condition, ℳ is defined as in (1.4). Then there exists a positive constant C such that for all f ∈ M q p ( μ ),

Theorem 3.3 Let 0 < α < n and p = n α. Suppose that K ( x , y ) satisfies (1.2) and the H n n − α condition, ℳ is defined as in (1.4). Then there exists a positive constant C such that for all bounded functions f with compact support,

Remark 3.4 As a special condition, we take p = q = n α, Theorem 3.3 can be deduced with a similar method of Theorem 3.2.

Proof of Theorem 3.2 For any cubes Q and R in R d such that Q ⊂ R satisfies ℓ ( R ) ≤ 2 ℓ ( Q ), let

and

It is easy to see that a Q and a R are real numbers. By Lemma 1.2, we need to show that for some fixed r > q, there exists a constant C > 0 such that

and

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

For 1 r = 1 q − α n and p = α n, it follows that

Now let us estimate the term I 2,

In order to estimate | M ( f 2 ) ( x ) − M ( f 2 ) ( y ) |, we write

and

It is easy to get that for any x , y ∈ Q,