Estimates for fractional type Marcinkiewicz integrals with non-doubling measures

Under the assumption that μ is a non-doubling measure on Rd satisfying the growth condition, the authors prove that the fractional type Marcinkiewicz integral ℳ is bounded from the Hardy space Hfin1,∞,0(μ) to the Lebesgue space Lq(μ) for 1q=1−αn with kernel satisfying a certain Hörmander-type condition. In addition, the authors show that for p=nα, ℳ is bounded from the Morrey space Mqp(μ) to the space RBMO(μ) and from the Lebesgue space Lnα(μ) to the space RBMO(μ). MSC:46A20, 42B25, 42B35.


Introduction
Let μ be a nonnegative Radon measure on R d which satisfies the following growth condition: for all x ∈ R d and all r > , μ B(x, r) ≤ C  r n , (.) where C  and n are positive constants and n ∈ (, 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 > , μ(B(x, r)) ≤ Cμ(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 (.) (see [ and for any x, y, y ∈ R d , |x-y|≥|y-y | K(x, y) -K x, y + K(y, x) -K y , The fractional type Marcinkiewicz integral M associated to the above kernel K(x, y) and the measure μ as in (.) is defined by If μ is the d-dimensional Lebesgue measure in R d , and However, in this paper, we discover that the kernel should satisfy some other kind of smoothness condition to replace (.).
Definition . Let  ≤ s < ∞,  < ε < . The kernel K is said to satisfy a Hörmander-type condition if there exist c s >  and C s >  such that for any x ∈ R d and > 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  is not condition (.). http://www.journalofinequalitiesandapplications.com/content/2014/1/285 The purpose of this paper is to get some estimates for the fractional type Marcinkiewicz integral M with kernel K satisfying (.) and (.) on the Hardy-type space and the RBMO(μ) space. To be precise, we establish the boundedness of M in H ,∞, fin (μ) for  q =  -α n in Section . In Section , we prove that M is bounded from the space RBMO(μ) to the Morrey space M p q (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 (Q), respectively. Let η > , ηQ denote the cube with the same center as Q and (ηQ) = η (Q). Given two cubes Q ⊂ R in R d , set where N Q,R is the smallest positive integer k such that ( k Q) ≥ (R). The concept S Q,R was introduced in [], where some useful properties of S Q,R can be found.
(i) There exist some constant C  and a collection of numbers b Q such that these two properties hold: for any cube Q, and for any cube R such that Q ⊂ R and (R) ≤  (Q), and also for any cube R such that Q ⊂ R and (R) ≤  (Q), Then the fractional integral operator I α defined by 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 ∈ [, ∞], we denote by p its conjugate index, namely  p +  p = .

Boundedness of M in Hardy spaces
This section is devoted to the behavior of M in Hardy spaces. In order to define the Hardy space H  (μ), Tolsa introduced the grand maximal operator M φ in [].
Based on Theorem . in [], we can define the Hardy space H  (μ) as follows (see []).
We recall the atomic Hardy space H ,∞, atb (μ) as follows.
Then define Define H ,∞, atb (μ) and H ,∞, fin (μ) as follows: Proof of Theorem . Without loss of generality, we may assume that ρ =  and f = h as a finite of atomic blocks defined in Definition .. 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) dμ(x) = , and where λ i for i = ,  is a real number, |h i | H ,∞, atb (μ) = λ  + λ  , a i for i = ,  is a bounded function supported on some cubes Q i ⊂ R and it satisfies
By (.), we have To estimate I  , we write Choose p  and q  such that  < p  < n α ,  < q < q  and  q  =  p  -n α . By the Hölder inequality, the fact that S Q  ,R ≥  and the (L p  (μ), L q  (μ))-boundedness of M (see Lemma .), we http://www.journalofinequalitiesandapplications.com/content/2014/1/285 have that Denote N Q  ,R simply by N  . Invoking the fact that a  L ∞ (μ) ≤ [μ(Q i )S Q i ,R ] - , we thus get Here we have used the fact that |λ j | . http://www.journalofinequalitiesandapplications.com/content/2014/1/285 Here we have used the fact that  q =  -α n . Combining the estimates for I, II and III yields that and this is the result of Theorem ..

Boundedness of M in RBMO(μ) spaces
In this section, we discuss the boundedness for M as in (.) in the space RBMO(μ) for f ∈ M q p (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 p q (μ), which was introduced by Sawano and Tanaka in [].
where the norm f M p q (μ) is given by We should note that the parameter ν >  appearing in the definition does not affect the definition of the space M p q (μ), and M p q (μ) is a Banach space with its norms (see []). By using the Hölder inequality to (.), it is easy to see that for all  ≤ q  ≤ q  ≤ p, then Theorem . Let  < α < n,  ≤ q < p = n α . Suppose that K(x, y) satisfies (.) and the H p condition, M is defined as in (.). Then there exists a positive constant C such that for all f ∈ M p q (μ), Theorem . Let  < α < n and p = n α . Suppose that K(x, y) satisfies (.) and the H Remark . As a special condition, we take p = q = n α , Theorem . can be deduced with a similar method of Theorem ..
Proof of Theorem . For any cubes Q and R in R d such that Q ⊂ R satisfies (R) ≤  (Q), let and It is easy to see that a Q and a R are real numbers. By Lemma ., we need to show that for some fixed r > q, there exists a constant C >  such that Let us first prove estimate (.). For a fixed cube Q and x ∈ Q, For  r =  q -α n and p = α n , it follows that Now let us estimate the term I  , By a similar argument, it follows that Finally, by the condition H P , which the kernel K(x, y) conditions, applying Minkowski's inequality, and the fact that α = n p , we have http://www.journalofinequalitiesandapplications.com/content/2014/1/285 Combining these estimates, we conclude that and so estimate (.) is proved. We proceed to show (.). For any cubes Q ⊂ R with x ∈ Q, denote N Q,R+ simply by N . Write As in the estimate for the term I  , then We conclude from y ∈ R, z ∈  N Q \   Q that Taking mean over y ∈ R, we obtain Analysis similar to that in the estimates for E  shows that Finally, we get (.) and this is precisely the assertion of Theorem ..