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Commutators of log-Dini-type parametric Marcinkiewicz operators on non-homogeneous metric measure spaces
Journal of Inequalities and Applications volume 2021, Article number: 115 (2021)
Abstract
Let \((\mathcal{X}, d, \mu )\) be a non-homogeneous metric measure space, which satisfies the geometrically doubling condition and the upper doubling condition. In this paper, the authors prove the boundedness in \(L^{p} (\mu )\) of mth-order commutators \(\mathcal{M}^{\rho }_{b,m}\) generated by the Log-Dini-type parametric Marcinkiewicz integral operators with RBMO functions on \((\mathcal{X}, d, \mu )\). In addition, the boundedness of the mth-order commutators \(\mathcal{M}^{\rho }_{b,m}\) on Morrey spaces \(M^{q}_{p}(\mu )\), \(1< p \leq q< \infty \), is also obtained for the parameter \(0<\rho <\infty \).
1 Introduction
Marcinkiewicz integral operators and their commutators play a very important role in harmonic analysis. Therefore, many authors have focused on studying the operators and their commutators. In 1960, Hörmander [1] introduced the parametric Marcinkiewicz integral, defined by,
where \(\rho \in (0,\infty )\). Let Ω be homogeneous of degree zero in \(R^{d}\) for \(d \geq 2\), integrable and have mean value zero on the unit sphere \(S^{d-1}\). Hörmander [1] proved that, if \(\Omega \in Lip_{\alpha }(S^{d-1})\) for some \(\alpha \in (0,1]\), then \(\mathcal{\mu }_{\Omega }^{\rho }\) is bounded on \(L^{p}(R^{d})\) for \(p\in (1,\infty )\). In 2001, Fan [2] obtained the boundedness of \(\mathcal{\mu }_{ \Omega }^{\rho }\) from \(L^{1}(R^{d})\) to \(L^{1,\infty }(R^{d})\) when \(\Omega \in L(\mathrm{logL}) (S^{d-1})\).
If \(\rho =1\) in (1.1), then it is the higher-dimensional Marcinkiewicz integral first introduced by Stein [3] in 1958, denoted \(\mathcal{\mu }_{\Omega }\). Stein [3] proved that \(\mathcal{\mu }_{\Omega }\) is bounded on \(L^{p}(R^{d})\) for any \(1< p\leq 2\), and is also bounded from \(L^{1}(R^{d})\) to \(L^{1,\infty }(R^{d})\). In 1990, Torchinsky and Wang [4] first introduced the commutator \(\mathcal{\mu }_{\Omega,b}\) generated by \(\mathcal{\mu }_{\Omega }\) and BMO function b, defined as follows:
and established its \(L^{p}(R^{d})\) boundedness for \(p\in (1,\infty )\). In 2002, Salman [5] reduced the condition of the kernel function to \(\Omega \in L(\mathrm{logL})^{\frac{1}{2}}(S^{{d}-1})\), and proved that \(\mathcal{\mu }_{\Omega }\) is bounded on \(L^{p}(R^{d})\) for \(p\in (1,\infty )\).
Recently, Gürbüz considered the boundedness of Marcinkiewicz integral operator with rough kernel associated with the Schrödinger operator and their commutators [6–8]. Gürbüz also proved some relevant conclusions about Marcinkiewicz operators, one may refer to [9–12]. In addition, Tao proved the boundedness of Marcinkiewicz integral operator with rough kernel [13–15]
In this paper, we will discuss the boundedness of commutators of the parametric Marcinkiewicz integral on the non-homogeneous metric space. Let \((\mathcal{X}, d)\) be a metric space, and let μ be a positive Borel measure on \(\mathcal{X}\) that satisfies the following growth condition: for all \(x\in \mathcal{X}, r>0\),
where \(C_{0}>0\) and \(B(x,r):=\{y \in \mathcal{X}:d(x,y)< r \}\).
It is well known that the analysis on \((\mathcal{X}, d, \mu )\) played key roles in many fields, for example, in solving Painlevé’s problem [16]. In 2010, Hytönen [17] introduced a non-homogeneous metric measure space, of which the measure satisfies the geometrically doubling condition and the upper doubling condition. From then on, many researchers considered singular integral operators on \((\mathcal{X}, d, \mu )\); see [18–20] for example. The purpose of this article is to consider the boundedness of the commutators generated by the Log-Dini-type parametric Marcinkiewicz integral with RBMO functions on \((\mathcal{X}, d, \mu )\). Before stating our results, we recall some notions of geometrically doubling and upper doubling measure [17].
Definition 1.1
([17])
Let \((\mathcal{X}, d)\) is a metric space; if there exists some \(N_{0}\in N\), and for any \(x\in \mathcal{X}, r>0\), such that any ball \(B(x,r)\subset \mathcal{X}\) can be covered by at most \(N_{0}\) balls \(B(x_{i},\frac{r}{2})\), we say \((\mathcal{X}, d)\) satisfies the geometrically doubling condition.
Definition 1.2
([17])
Let \((\mathcal{X}, d, \mu )\) is a metric measure space, if μ is a Borel measure on \(\mathcal{X}\) and there exist a dominating function \(\lambda (x,r):\mathcal{X} \times R_{+} \rightarrow R_{+}\) and a constant \(C_{\lambda }>0\) such that \(r \rightarrow \lambda (x,r)\) is increasing and
for all \(x\in \mathcal{X}, r>0\), then we say μ is an upper doubling measure.
We also need to recall other notions [17, 21].
Definition 1.3
For \(\alpha, \beta \in (1,\infty )\), a ball \(B\subset \mathcal{X}\) is called \((\alpha, \beta )\) doubling if
One can see from Lemma 3.2 of [17] that, if μ is upper doubling, for any \(\alpha, \beta \in (1,\infty )\) and \(\beta > C_{\lambda }^{\mathrm{log}_{2}{a}}=:\alpha ^{\nu }\), then for every ball \(B\subset \mathcal{X}\) there exists \(j\in n\), such that \(\alpha ^{j} B\) is \((\alpha, \beta )\) doubling ball. Moreover, we see from Lemma 3.3 of [17] that, if \((\mathcal{X}, d)\) is geometrically doubling, there exists \(n_{0}:=\mathrm{log}_{2}{N_{0}}\), such that \(\beta >\alpha ^{n_{0}}\), if μ is a Borel measure on \(\mathcal{X}\) which is finite on bounded sets, then, for μ-a.e. \(x\in \mathcal{X}\), there exist arbitrarily small \((\alpha, \beta )\) doubling balls centred at x. Moreover, for any preassigned \(r>0\), their radius can be chosen to be of the form \(\alpha ^{j} r\), \(j\in n\). Throughout this paper, fix \(\tau \geq 1\), B is a \((30\tau, \beta _{30\tau })\) doubling ball and
For any \(\tau \geq 1\), \(B\subset \mathcal{X}\), B̃ denotes the smallest \((30\tau, \beta _{30\tau })\) doubling ball of the form \((30\tau )^{j}B\).
As in [7], for any two balls \(B\subset S\), \(r_{B}\) and \(r_{S}\) denote the radius of the ball B and S, respectively. And \(x_{B}\) denotes the center of the ball B. We define \(K_{B,S}\) and \(\tilde{K}_{B,S}\) as follows:
Let \(N_{B,S}\) be the smallest integer satisfying \(6^{N_{B,S}}r_{B}\geq r_{S}\), we define
In the case that \(\lambda (x,ar)=a^{m}\lambda (x,r)\) for all \(x\in \mathcal{X}, a,r>0\), it is easy to show that \(K_{B,S}\simeq \tilde{K}_{B,S}\). Nevertheless, in general, we only have \(K_{B,S}\leq C \tilde{K}_{B,S}\).
Finally, we recall the definition of Morrey space [22] on \((\mathcal{X}, d, \mu )\).
Definition 1.4
Let \(\kappa >1\) and \(1\leq p\leq q<\infty \), the definition of Morrey space are as follows:
where
We remark that, for any \(\kappa _{1}, \kappa _{2}>1\), \(M^{q}_{p}(\kappa _{1},\mu )=M^{q}_{p}(\kappa _{2},\mu )\) (see [23]). Particularly, if μ is a doubling measure, then \(M^{q}_{p}(\kappa,\mu ) = M^{q}_{p}(1,\mu )\) for any \(\kappa >0\), and denote it by \(M^{q}_{p}(\mu )\) for brevity. Moreover, it is easily to see that the space \(M^{q}_{p}(\mu )\) becomes the classical Morrey space whenever \(d\mu =dx\).
Next, we introduce the conditions of kernel discussed in this article.
Definition 1.5
Let \(\omega:[0,\infty )\rightarrow [0,\infty )\) be non-decreasing function that satisfies
Let \(K(x, y) \in L^{1}_{\mathrm{loc}} ( (\mathcal{X})^{2} \setminus \{(x,y):x=y\} )\), we say \(K(x, y)\) is the parametric Marcinkiewicz kernel of Log-Dini type, if there exists \(C>0\) such that the following size estimate and smoothness estimates hold:
-
(i)
For \(x, y \in \mathcal{X}\) with \(x\neq y\),
$$\begin{aligned} \bigl\vert K(x, y) \bigr\vert \leq C \frac{d(x,y)}{\lambda (x,d(x,y))}. \end{aligned}$$(1.9) -
(ii)
For \(x, x',y \in \mathcal{X}\) and if \(2d(x,x')\leq d(x,y)\),
$$\begin{aligned} \bigl\vert K(x, y)-K\bigl(x', y\bigr) \bigr\vert \leq C \frac{d(x,y)}{\lambda (x,d(x,y))} \omega \biggl( \frac{d(x,x')}{d(x,y)} \biggr). \end{aligned}$$(1.10) -
(iii)
For \(x, y',y \in \mathcal{X}\) and if \(2d(y,y')\leq d(x,y)\),
$$\begin{aligned} \bigl\vert K(x, y)-K\bigl(x, y'\bigr) \bigr\vert \leq C \frac{d(x,y)}{\lambda (x,d(x,y))} \omega \biggl( \frac{d(y,y')}{d(x,y)} \biggr). \end{aligned}$$(1.11)
The parametric Marcinkiewicz integral \(\mathcal{M}^{\rho }\) with Log-Dini-type kernel \(K(x, y)\) satisfying (1.9), (1.10) and (1.11) is then defined, initially for \(f \in L^{\infty }\) with compact support, by
In case \(\rho =1\), \(\mathcal{M}^{\rho }\), denoted by \(\mathcal{M}\), is just the Marcinkiewicz integral operator on \((\mathcal{X}, d, \mu )\) with Log-Dini-type kernel.
In 2014, Lin and Yang [24] proved that \(\mathcal{M}\) is bounded on \(L^{p}(\mu )\) if and only if \(\mathcal{M}\) is bounded from \(L^{1}(\mu )\) to \(L^{1,\infty }(\mu )\), if the kernel \(K(x, y)\) satisfies (1.9) and for all \(x,y,y'\in \mathcal{X}\)
In 2016, Fu and Lin [25] proved that when the kernel \(K(x, y)\) satisfies (1.9) and (1.13), if \(\mathcal{M}^{\rho }\) is bounded on \(L^{p_{0}}(\mu )\) with some \(1< p_{0}<\infty \) then \(\mathcal{M}^{\rho }\) is bounded from \(L^{1}(\mu )\) to \(L^{1,\infty }(\mu )\).
Given \(b\in \operatorname{RBMO}(\mu )\), the commutators \(\mathcal{M}^{\rho }_{b}\) generated by \(\mathcal{M}^{\rho }\) with RBMO function b is defined by
In general, for all \(m \in \mathrm{N}\), the mth-order commutators \(\mathcal{M}^{\rho }_{b,m}\) is defined by
In 2015, Zhou [26] showed that the commutator \(\mathcal{M}_{b}\) is bounded on \(L^{p}(\mu )\), if \(\mathcal{M}\) is bounded on \(L^{2}(\mu )\), and the kernel \(K(x, y)\) satisfies (1.9) and the following Hörmander type condition:
In 2019, Tao [27] proved that, if the kernel satisfies (1.9) and (1.16), then \(\mathcal{M}_{b}\) is bounded on \(M^{q}_{p}(\mu )\). In fact, we can see that (1.16) is stronger than (1.13).
In case \(\mathcal{X}=R^{d}\), the non-homogeneous Euclidean space, then for the kernel \(K(x, y)\) in the Marcinkiewicz integral it can be assumed that \(K(x, y) \in L^{1}_{\mathrm{loc}} ( R^{d} \times R^{d} \setminus \{(x,y):x=y\} )\) satisfies the following conditions with a constant \(C>0\):
and
for all \(x,y,y'\in R^{d}\) with \(x\neq y\). And the Marcinkiewicz integral \(\mathcal{M}\) is defined by
In 2007, Hu [28] obtained \(\mathcal{M}\) is bounded on \(L^{p}(\mu ), 1< p<\infty \), and is bounded from \(L^{1}(\mu )\) to \(L^{1,\infty }(\mu )\). Later, Zhang [29] proved \(\mathcal{M}\) is bounded on \(M^{q}_{p}(\mu )\).
For \(m\in N\) and \(b\in\operatorname{RBMO}\), the mth-order commutator for Marcinkiewicz integral is denoted by
In 2007, Hu [28] proved that \(\mathcal{M}_{b,m}\) is bounded on \(L^{p}(\mu )\) if the kernel \(K(x, y)\) satisfies (1.17) and the following condition:
It is easy to see that (1.21) is stronger than (1.18). In 2010, Zhang [29] proved that \(\mathcal{M}_{b}\) is bounded on \(M^{q}_{p}(\mu )\) under the same assumptions.
Now we turn to stating the main results of this paper.
Theorem 1.1
Let K satisfy (1.9), (1.10), and (1.11). \(\mathcal{M}^{\rho }\), \(\mathcal{M}^{\rho }_{b}\) be as in (1.12) and (1.14), respectively. Suppose that \(\mathcal{M}^{\rho }\) is bounded on \(L^{2}(\mu )\), \(b\in \operatorname{RBMO}(\mu )\), \(0<\rho <\infty \). If ω satisfies
then, for all \(f \in {L^{p} (\mu )}, 1< p<\infty \), there exists a constant \(C>0\) such that
In fact we will prove the \(L^{p} (\mu )\) boundedness for a more general mth-order commutator for the parametric Marcinkiewicz integral.
Theorem 1.2
Under the same conditions of Theorem 1.1and \(\mathcal{M}^{\rho }_{b,m}\) be as in (1.15). If ω satisfies the following condition:
then for all \(f \in {L^{p} (\mu )}, 1< p<\infty \), there exists a constant \(C>0\) such that
Theorem 1.1 is the special case of Theorem 1.2 in which one can take \(m=1\). We will prove Theorem 1.2 in Sect. 2.
Moreover, we will establish the boundedness of \(\mathcal{M}^{\rho }_{b,m}\) on the Morrey space.
Theorem 1.3
Assume the same conditions of Theorem 1.1and \(\mathcal{M}^{\rho }_{b,m}\) as in (1.15). If ω satisfies (1.24), then there exists a constant \(C>0\), for all \(f \in {M^{q}_{p} (\mu )}, 1< p\leq q<\infty \), such that
By checking the proofs of Theorem 1.2 and Theorem 1.3, we can obtain the following two corollaries, which extend the results in [26] and [27].
Corollary 1.4
Let the kernel \(K(x, y)\) satisfy (1.9) and (1.16), \(\mathcal{M}^{\rho }\) and \(\mathcal{M}^{\rho }_{b,m}\) be as in (1.12) and (1.15), respectively. Suppose that \(\mathcal{M}^{\rho }\) is bounded on \(L^{2}(\mu )\), \(b\in \operatorname{RBMO}(\mu ), 0<\rho <\infty \). If ω satisfies (1.24), then there exists a constant \(C>0\), for all \(f \in {L^{p} (\mu )}, 1< p<\infty \), such that
Corollary 1.5
Under the same conditions of Corollary 1.4, there exists a constant \(C>0\), for all \(f \in {M^{q}_{p} (\mu )}, 1< p\leq q<\infty \), such that
Remark 1.6
If \(\rho =1,m=1\) on Corollary 1.4, which is Theorem 1.10 of [26]; if \(\rho =1,m=1\) on Corollary 1.5, which is Theorem 1.8 of [27], so our results contain their conclusions.
Throughout this paper, d is the dimension of space; C denotes a positive constant that is independent of the parameters, furthermore, it value may differ from line to line; \(x_{B}\) denotes the center of the ball B, \(r_{B}\) denotes the radius of the ball B; for any \(p\in (1,\infty )\), we denote by \(p'=\frac{p}{p-1}\) its conjugate index; \(m_{B}(b)\) is the mean value of B on B, namely \(m_{B}(b)=\frac{1}{\mu (B)} \int _{B} b(x)\,d\mu (x)\).
2 Proof of Theorem 1.2
We first recall the definition of a sharp maximal operator \(M^{\sharp }f(x)\) [21] over \((\mathcal{X}, d, \mu )\). For any \(f\in L^{1}_{\mathrm{loc}}(\mu )\),
here \(\Delta _{x}=\{(B,S):x\in B\subset S, B,S \text{are doubling balls}\}\). As usual, we let \(M^{\sharp }_{\delta }(f)(x) = [M^{\sharp }(|f(x)|^{\delta })]^{ \frac{1}{\delta }} \).
We will use the following lemma about sharp maximal function on \((\mathcal{X}, d, \mu )\) proved by Fu [18].
Lemma 2.1
\((\mathrm{i})\) Let \(p>1\), \(s\in [1,p), \zeta \in [5,\infty )\). For all \(f \in L^{1}_{\mathrm{loc}}(\mu )\) and \(x\in \mathcal{X}\),
is bounded on \(L^{p}(\mu )\) and also bounded from \(L^{1}(\mu )\) to \(L^{1,\infty }(\mu )\). If \(s=1\), then \(M_{s,\zeta } f =M_{(\zeta )} f\).
\((\mathrm{ii})\) For any \(\delta \in (0,1)\) and for \(f\in L^{1}_{\mathrm{loc}}(\mu )\), define
then, for μ-almost every \(x\in \mathcal{X}\),
According to Theorem 4.2 in [21], we can easily get the following lemma.
Lemma 2.2
Let \(f\in L^{1}_{\mathrm{loc}}(\mu )\) satisfy \(\int _{\mathcal{X}} fd\mu =0\) when \(\| \mu \|:=\mu (\mathcal{X})<\infty \). Assume that \(\inf \{1,N_{\delta }f\}\in L^{p}(\mu )\), for any \(p\in (1,\infty )\), \(\delta \in (0,1)\), then there exists a constant \(C>0\),
The next two lemmas can be found in [30].
Lemma 2.3
Let \(\varrho >1\), for \(b\in L^{1}_{\mathrm{loc}}(\mu )\). The following statements are equivalent:
\((\mathrm{i})\) \(b\in \operatorname{RBMO}(\mu )\).
\((\mathrm{ii})\) There exists a constant \(C>0\), such that, for all balls B,
and for all \((30\tau,\beta _{30\tau })\) doubling balls \(B\subset S\),
where \(m_{B}(b)=\frac{1}{\mu (B)} \int _{B} b(x)\,d\mu (x)\). Furthermore, the infimum of all positive constants C satisfying (2.5) and (2.6) is an equivalent RBMO norm of b, denoted by \(\|b\|_{\operatorname{RBMO}(\mu )}\).
Lemma 2.4
Let \(\varrho >1, p\in [1,\infty )\), if \(b\in \operatorname{RBMO}\), for any ball B, then there exists a constant \(C>0\), we have
We need the following lemma about the boundedness of parametric Marcinkiewicz integral operators.
Lemma 2.5
Let kernel \(K(x, y) \in L^{1}_{\mathrm{loc}} ( (\mathcal{X})^{2} \setminus \{(x,y):x=y\} )\) satisfy (1.9), (1.10) and (1.11), \(\mathcal{M}^{\rho }\) be as in (1.12), \(0<\rho <\infty \). If \(\mathcal{M}^{\rho }\) is bounded \(L^{p_{0}}(\mu ), 1< p_{0}<\infty \), then \(\mathcal{M}^{\rho }\) is bounded from \(L^{1}(\mu )\) to \(L^{1,\infty }(\mu )\).
Proof
In Theorem 2.1 of [25], the kernel function satisfies (1.9) and (1.13). It is easily to see that (1.13) is weaker than (1.10) and (1.11). So by similar argument as that in Theorem 2.1 of [25], we can prove the lemma. Hence, we omit the details. □
To prove Theorem 1.2, we should first establish the following lemma.
Lemma 2.6
Let \(K(x,y)\) satisfy (1.9), (1.10) and (1.11). Suppose \(\mathcal{M}^{\rho }\) be as in (1.12) is bounded on \(L^{2}(\mu )\), \(b\in \operatorname{RBMO}(\mu )\). If \(0<\rho <\infty \), \(\delta \in (0,1)\) and ω satisfies (1.24), then there exists a constant \(C>0\), for all \(f \in {L^{p} (\mu )}\), such that
Here \(\mathcal{M}^{\rho }_{b,1}=\mathcal{M}^{\rho }_{b}\) and \(\mathcal{M}^{\rho }_{b,0}=\mathcal{M}^{\rho }\).
Proof
Without loss of generality, we may assume that \(\|b\|_{\operatorname{RBMO}(\mu )}=1\). In order to prove (2.8), it suffices to prove that, for all \(x\in \mathcal{X}\) and balls \(B \ni x\),
and for all balls \(B\subset S\), S is a \((30\tau,\beta _{30\tau })\) doubling ball,
where
Now we decompose the function f into two parts, i.e., \(f=f{\chi _{6B}}+f{\chi _{\mathcal{X} \setminus {6B}}}:=f_{1}+f_{2}\). We can write
Thus, we obtain
Since \(0<\delta < 1\),
To estimate \(E_{1}\), let \(\gamma,\eta >1\), be such that
By Hölder’s inequality and Lemma 2.4, we have
For \(E_{2}\), by the Kolmogorov inequality, Lemma 2.5, Hölder’s inequality and Lemma 2.4, we have
As to the estimate \(E_{3}\), we observe that
Hence
In fact, for \(y, w\in B\), we observe that
In order to estimate \(E_{3}\), it suffices to estimate \(F_{1}, F_{2}, \text{and} F_{3}\). To estimate \(F_{1}\), for all \(y, w\in B\), we have \(d(y,z)\sim d(w,z)\sim d(c_{B},z)\). By the Minkowski inequality, (1.9), Hölder’s inequality, and Lemma 2.4, we get
We have used the following fact in the last inequality:
Similarly, we get
To estimate \(F_{3}\), for all \(y,w\in B\), we have \(d(y,z)\sim d(w,z)\sim d(c_{B},z)\), using the Minkowski inequality, we get
Next we estimate \(F_{31}\) and \(F_{32}\), respectively. For \(F_{31}\), by (1.10), Hölder’s inequality, Lemma 2.4 and (1.24), we have
We have used the following fact in the last inequality:
To estimate \(B_{32}\), for all \(y,w\in B\), if \(\rho \in (0,\infty )\), by (1.9), Hölder’s inequality and Lemma 2.4, we get
Then we have
Moreover, combining the estimates of \(E_{1}, E_{2}, F_{1}, F_{2}\), and \(F_{3}\), we obtain the desired inequality (2.9).
Next we give the proof of (2.10). Write
To estimate \(G_{1}\), it being similar to \(E_{3}\), we get
For \(G_{2}\), we use
So we have
Therefore,
With the same argument as for \(E_{1}\), we get
The estimates of \(H_{2}\) and \(H_{3}\) is very similar to \(G_{4}\) and \(E_{2}\), respectively, then we have
Therefore, combining \(H_{1},H_{2},H_{3}\), we have
For \(G_{3}\), by (1.9), the Minkowski inequality, Hölder’s inequality, we obtain
Therefore,
Similarly,
Since (2.10) has been proved, Lemma 2.6 follows directly.
By Lemma 2.5 and the Marcinkiewicz interpolation theorem, we have
Then using (2.3), (2.4), Lemma 2.6 and Lemma 2.1, we get
We set
Finally, using mathematical induction, (2.3), (2.4), Lemma 2.6 and Lemma 2.1, we obtain
Then Theorem 1.2 is proved. □
3 Proof of Theorem 1.3
In this section, we will prove Theorem 1.3. We recall the boundedness in Morrey space \(M^{q}_{p}(\mu )\) of the sharp maximal function on \((\mathcal{X},d,\mu )\) [22, 31].
Lemma 3.1
Let \(f\in L^{1}_{\mathrm{loc}}(\mu )\) satisfy \(\int _{\mathcal{X}} f(x) \,d\mu (x)=0\) when \(\| \mu \|:=\mu (\mathcal{X})<\infty \). Let \(1< p\leq q<\infty \), \(\delta \in (0,1)\). If \(\inf \{1,N_{\delta }f\} \in M^{q}_{p}(\mu )\), then there exists a constant \(C>0\), such that
Lemma 3.2
Let \(\zeta >1, 1< s< p\leq q<\infty \), \(M_{s,\zeta }f\) be as in (2.2) is bounded on Morrey space \(M^{q}_{p}(\mu )\), that is,
Next we give the proof of Theorem 1.3.
In combination with Lemma 2.6, the differences between the proof of Theorem 1.2 and Theorem 1.3 are as follows:
By (1.7) and (2.12), it is easy to see
Then using (2.3), (3.1), Lemma 2.6 and (2.2), we have
We set
Finally, by mathematical induction, (2.3), (3.1), Lemma 2.6 and (2.2), we get
So, the proof of Theorem 1.3 is finished.
4 Proof of Corollary 1.4
If \(\rho =1,m=1\) on Corollary 1.4, which is Theorem 1.10 of [26]. The different between Corollary 1.4 and Theorem 1.10 of [26] is to estimate \(F_{31}\). So, in order to complete the proof of Corollary 1.4, it suffices to show that
Using Lemma 2.3, Lemma 2.4 in [26], (1.9) and (1.16), we have
and
This completes the proof of Corollary 1.4.
Using the similar to the argument in the proof of Corollary 1.4, we can get Corollary 1.5.
5 Some applications
Now we give the applications of Theorem 1.2 and Theorem 1.3 for the classical parametric Marcinkiewicz integral.
Let Ω be homogeneous of degree zero in \(R^{d}\) for \(d \geq 2\), integrable and have mean value zero on the unit sphere \(S^{d-1}\). In addition, Ω satisfies the following condition: with a constant \(C > 0\), for \(x, x^{\prime },y \in R^{d}\) and \(\vert x-x^{\prime } \vert \leq \frac{|x-y|}{2}\),
where ω satisfies (1.24).
\({\mu }_{\Omega }^{\rho }\) be as in (1.1), where Ω satisfies the above condition (5.1). Moreover, \({\mu }^{\rho }_{\Omega,b,m}\) is generated by \({\mu }_{\Omega }^{\rho }\) with RBMO functions b, defined by
where \(0<\rho <d\).
Theorem 5.1
Let \(0<\rho <d\) and Ω satisfies (5.1). \({\mu }_{\Omega,b,m}^{\rho }(f)\) be as in (5.2), and ω satisfies (1.24), then there exists a constant \(C>0\), for all \(f \in {L^{p} (R^{d})}, 1< p<\infty \) such that
For all \(f \in {M^{q}_{p} (R^{d})},1< p\leq q<\infty \), such that
Next, we give the applications of Theorem 1.2 and Theorem 1.3 for the parametric Marcinkiewicz integral operator in Euclidean space where μ satisfies the growth condition (1.2).
Let ω satisfy (1.24), K satisfy (1.17) and the following conditions hold with a constant \(C > 0\):
where \(x, x^{\prime },y \in R^{d}\) and \(\vert x-x^{\prime } \vert \leq \frac{|x-y|}{2}\).
where \(x, y^{\prime },y \in R^{d}\) and \(\vert y-y^{\prime } \vert \leq \frac{|x-y|}{2}\).
Define the parametric Marcinkiewicz integral operator \(M^{\rho }\) with respect to the kernel above as follows:
\(M_{b,m}^{\rho }\) is generated by \(M^{\rho }\) with RBMO functions b, defined by
Theorem 5.2
Let \(0<\rho <\infty \), and K satisfythe above conditions (1.17), \((\mathrm{b}')\) and \((\mathrm{c}')\). Let \(M^{\rho }, M_{b,m}^{\rho }\) be as in (5.3) and (5.4). Suppose that \(M^{\rho }\) is bounded on \(L^{2}(\mu )\), \(b\in \operatorname{RBMO}(\mu )\), ω satisfies (1.24), then there exists a constant \(C>0\), for all \(f \in {L^{p} (\mu )}, 1< p<\infty \) such that
For all \(f \in {M^{q}_{p} (R^{d})},1< p\leq q<\infty \), we have
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The authors thank the anonymous referee for reading the paper carefully and giving some useful advices.
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This work was supported partially by the NNSF of China (No. 11771399, 11961056) and Zhejiang University of Science and Technology graduate research innovation fund.
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Xiangxing, T., Qiange, Z. Commutators of log-Dini-type parametric Marcinkiewicz operators on non-homogeneous metric measure spaces. J Inequal Appl 2021, 115 (2021). https://doi.org/10.1186/s13660-021-02651-6
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DOI: https://doi.org/10.1186/s13660-021-02651-6