1 Introduction
In 2010, Hytönen in [
1] first introduced a new class of metric measure spaces which satisfy the so-called upper doubling and the geometrically doubling conditions (see also Definitions
1.1 and
1.2 below, respectively), for convenience, the new spaces are called
non-homogeneous metric measure spaces. As special cases, the new spaces not only contain the homogeneous type spaces (see [
2]), but also they include metric spaces endowed with measures satisfying the polynomial growth condition (see, for example, [
3‐
9]). Further, it is meaningful to pay much attention to a study of the properties of some classical operators, commutators, and function spaces on non-homogeneous metric measure spaces; see [
10‐
16]. In addition, we know that the harmonic analysis has important applications in many fields including geometrical analysis, functional analysis, partial differential equations, and fuzzy fractional differential equations, we refer the reader to [
17‐
20] and the references therein.
In the present paper, let
\((\mathcal{X}, d, \mu)\) be a non-homogeneous metric measure space in the sense of Hytönen [
1]. In 2007, Hu
et al. [
5] obtained the boundedness of the Marcinkiewicz with non-doubling measure. Besides, Lin and Yang [
13] established some equivalent boundedness of Marcinkiewicz integral on
\((\mathcal{X}, d, \mu)\). Inspired by this, we will mainly consider the boundedness of the fractional type Marcinkiewicz integrals introduced in [
21] on
\((\mathcal{X}, d, \mu)\).
To state the main consequences of this article, we first of all recall some necessary notions and notation. Hytönen [
1] originally introduced the following notions of the upper doubling condition and the geometrically doubling condition.
Hytönen
et al. [
16] have proved that there is another dominating function
λ̃ such that
\(\tilde{\lambda }\leq\lambda\),
\(C_{\tilde{\lambda}}\leq C_{\lambda}\), and
$$ \tilde{\lambda}(x,r)\leq C_{\tilde{\lambda}}\tilde{\lambda }(y,r), $$
(1.2)
where
\(x,y\in\mathcal{X}\) and
\(d(x,y)\leq r\). Based on this, we also assume the dominating function
λ that in (
1.1) satisfies (
1.2) in this paper.
Now we recall the definition of coefficient
\(K_{B,S}\) introduced by Hytönen in [
1], which is analogous to the quantity
\(K_{Q,R}\) introduced in [
4], that is, for any two balls
\(B\subset S\) in
\(\mathcal {X}\), define
$$ K_{B,S}:=1+ \int_{2S\setminus B}\frac{1}{\lambda (c_{B},d(x,c_{B}))}\,\mathrm{d}\mu(x), $$
(1.3)
where
\(c_{B}\) is the center of the ball
B.
Though the measure doubling condition is not assumed uniformly for all balls on
\((\mathcal{X},d, \mu)\), it was proved in [
1] that there still exist many balls satisfying the property of the
\((\alpha,\eta )\)-doubling, namely, we say that a ball
\(B\subset\mathcal{X}\) is
\((\alpha,\eta)\)-doubling if
\(\mu(\alpha B)\leq\eta\mu(B)\), for
\(\alpha,\eta>1\). In the rest of this paper, unless
α and
\(\eta_{\alpha}\) are specified, otherwise, by an
\((\alpha,\eta _{\alpha})\)-doubling ball we mean a
\((6,\beta_{6})\)-doubling ball with a fixed number
\(\eta_{6}>\max\{C^{3\log_{2}6}_{\lambda}, 6^{n}\}\), where
\(n:=\log _{2}N_{0}\) is viewed as a geometric dimension of the space. Moreover, the smallest
\((6,\eta_{6})\)-doubling ball of the from
\(6^{j}B\) with
\(j\in\mathbb{N}\) is denoted by
\(\tilde{B}^{6}\), and
\(\tilde{B}^{6}\) is simply denoted by
B̃.
Next, we recall the following definition of
\(\operatorname{RBMO}(\mu)\) from [
1].
From [
1], Hytönen showed that the space
\(\operatorname{RBMO}(\mu)\) is not dependent on the choice of
κ. Lin and Yang [
14] introduced the following definition of the space
\(\operatorname{RBLO}(\mu)\) and proved that
\(\operatorname{RBLO}(\mu)\subset\operatorname{RBMO}(\mu)\).
Now we give the notion of the fractional type Marcinkiewicz integral slightly changed from [
21].
The fractional type Marcinkiewicz integral
\(\mathcal{M}_{\beta,\rho ,q}(f)\) related to the above kernel
\(K(x,y)\) is formally defined by
$$ \mathcal{M}_{\beta,\rho,q}(f) (x):= \biggl( \int^{\infty}_{0} \biggl\vert \frac{1}{t^{\beta+\rho}} \int_{d(x,y)< t}\frac {K(x,y)}{[d(x,y)]^{1-\rho}}f(y) \,\mathrm{d}\mu(y)\biggr\vert ^{q}\frac{\mathrm{d}t}{t} \biggr)^{\frac {1}{q}}, $$
(1.6)
where
\(x\in\mathcal{X}\),
\(\rho>0\),
\(\beta\geq0\), and
\(q>1\).
Recently, many authors have studied the properties of the fractional type Marcinkiewicz integrals; see [
22‐
24]. To the fractional type Marcinkiewicz integral operator
\(\mathcal {M}_{\beta,\rho,q}\) as in (
1.6), one can return to the Marcinkiewicz integrals on different function spaces when the indices are replaced by some fixed numbers; see the following remark.
Further, we recall the notion of the atomic Hardy spaces given in [
16].
We say that a function \(f\in L^{1}(\mu)\) belongs to the atomic Hardy space \(H^{1,p}_{\mathrm{atb}}(\mu)\), if there exist \((p,1)_{\tau }\)-atomic blocks \(\{b_{i}\}^{\infty}_{i=1}\) such that \(f= {\sum } ^{\infty}_{i=1}b_{i}\) in \(L^{1}(\mu)\) and \({\sum } ^{\infty}_{i=1}|b_{i}|_{H^{1,p}_{\mathrm{atb}}(\mu)}<\infty \). The norm of f in \(H^{1,p}_{\mathrm{atb}}(\mu)\) is defined by \(\|f\|_{H^{1,p}_{\mathrm{atb}}(\mu)}:=\inf\{ {\sum} _{i}|b_{i}|_{H^{1,p}_{\mathrm{\mathrm{atb}}}(\mu)}\}\), where the infimum is taken over all the possible decompositions of f as above.
Also, in [
16], Hytönen
et al. proved that, for each
\(p\in (1,\infty]\), the atomic Hardy space
\(H^{1,p}_{\mathrm{atb}}(\mu)\) is independent of the choice of
ζ and that the spaces
\(H^{1,p}_{\mathrm{atb}}(\mu)\) and
\(H^{1,\infty}_{\mathrm{atb}}(\mu )\) have the same norms for all
\(p\in(1,\infty]\). Thus, we always denote
\(H^{1,p}_{\mathrm{atb}}(\mu)\) simply by
\(H^{1}(\mu)\).
Finally, we state the main results of this article.
By Theorem
1.9, Theorem
1.10, and Theorem 1.1 in [
15], it is easy to obtain the following corollary.
Throughout the paper, C represents for a positive constant which is independent of the main parameters involved, but it may be different from line to line. For a μ-measurable set E, \(\chi_{E}\) denotes its characteristic function. For any \(p\in[1,\infty]\), we denote by \(p'\) its conjugate index, that is, \(\frac{1}{p}+\frac{1}{p'}=1\).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.