函数空间理论和奇异积分算子理论组成了调和分析的重要部分.一般地,双倍条件在经典的调和分析结论中起着重要的作用.然而,多年来的许多研究结果表明,在非双倍条件下, $ {\Bbb R} ^{n} $上许多经典的函数空间理论以及奇异积分算子有界性的结论依然是成立的,参见文献[1-5].

2010年, Hytönen^[6]引入了满足几何双倍条件和上双倍条件的非齐度量测度空间$ ({\cal X}, d, \mu) $,这类空间同时包含了齐型空间和非双倍测度空间.杨大春和他的合作者们^[7]引入了非齐度量测度空间上的原子Hardy空间$ \widetilde{H}_{atb, \rho}^{p, q, \gamma}(\mu) $和分子Hardy空间$ \widetilde{H}_{mb, \rho}^{p, q, \gamma, \epsilon}(\mu) $,证明了Calderón-Zygmund算子和广义分数次积分算子的有界性.另外,我们引入了非齐度量测度空间上的Herz空间以及原子和分子Herz型Hardy空间,证明了相互关系

$ \widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu) = \widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)\subset \dot{K}_q^{\alpha, p}(\mu), $

并讨论了Calderón-Zygmund算子在这些空间上的有界性^[8].

Marcinkiewicz积分算子及其交换子的有界性问题一直以来受到许多作者的重视,参见文献[9-15].本文将主要讨论Marcinkiewicz积分算子及其与Campanato空间中的函数生成的交换子在非齐度量测度空间上的Herz空间以及Herz型Hardy空间的有界性,证明了Marcinkiewicz积分算子在$ \dot{K}_q^{\alpha, p}(\mu) $上的有界性,从$ \widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu) $到$ \widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, (\delta-v\alpha)/2}(\mu) $的有界性,以及交换子从$ \widetilde{H}\dot{K}_{atb, q_1, \rho}^{\alpha_1, p, \gamma}(\mu) $到$ \dot{K}_{q_2}^{\alpha_2, p}(\mu) $的有界性.

在这篇文章中, $ C $表示只依赖于主要参数的常数,在不同之处也许取值不同.对于度量空间$ ({\cal X}, d) $, $ \chi_E $表示$ {\cal X} $上子集$ E $的特征函数. $ {\cal X} $中的球$ B = B(x_0, r) = \left\{x \in {\cal X} : d(x_0, x) < r \right\} $,其中$ x_0 $表示球$ B $的中心, $ r $表示球$ B $的半径.

为了方便,下面介绍非齐度量测度空间上的一些基础知识和相关结论.

定义2.1^{[1, 16]}  若存在正整数$ N_0 \in {\Bbb N} $,使得对任意球$ B(x, r)\subset {\cal X} $, $ x \in {\cal X} $, $ r \in (0, \infty) $,都存在至多$ N_0 $个球$ \{B(x_i, r/2)\}_i $构成$ B(x, r) $的一个覆盖,则称度量空间$ ({\cal X}, d) $是几何双倍的.

定义2.2^[6]  如果$ \mu $,是$ {\cal X} $上的Borel测度,并存在一个控制函数$ \lambda: {\cal X}\times(0, \infty)\to(0, \infty) $,使得对每一个$ x\in{\cal X} $, $ \lambda(x, r) $关于$ r $都单调不减,且存在一个依赖于$ \lambda $的正常数$ C_{(\lambda)} $,使得对任意的$ x\in{\cal X} $和$ r\in(0, \infty) $,有

(2.1) $ \begin{eqnarray} \mu(B(x, r))\leqslant \lambda(x, r)\leqslant C_{(\lambda)}\lambda(x, \frac{r}{2}), \end{eqnarray} $

则称度量测度空间$ ({\cal X}, d, \mu) $是上双倍的.

引理2.1^[17]  若$ ({\cal X}, d, \mu) $是上双倍的, $ \lambda $是$ {\cal X}\times(0, \infty) $上的控制函数.则存在另一个控制函数$ \tilde{\lambda} $,使得$ \tilde{\lambda}\leqslant\lambda, C_{\tilde{(\lambda)}}\leqslant C_{(\lambda)} $,并且对于所有的$ x, y \in{\cal X} $,若$ d(x, y)\leqslant r $,就有

(2.2) $ \begin{eqnarray} \tilde{\lambda}(x, r) \leqslant C_{(\tilde{\lambda})} \tilde{\lambda}(y, r). \end{eqnarray} $

以下我们总假设$ ({\cal X}, d, \mu) $是一个非齐度量测度空间，其中控制函数$ \lambda $满足条件(2.2).

文献[6]中证明了如果$ ({\cal X}, d, \mu) $是一个满足上双倍条件的度量测度空间, $ \alpha, \beta>1 $并且$ \beta>C_{\lambda}^{\log_\alpha 2} $ = : $ \alpha^v $,则对于任意的球$ B\subset{\cal X} $,存在一个正整数$ j $使得$ \alpha^{j}B $是($ \alpha, \beta $) -倍的.

定义2.3^[6]  设$ \eta>0 $,若对所有的$ r\in(0, 2{\rm diam}({\cal X})) $和$ a\in(1, {2{\rm diam}({\cal X})}/{r}) $,存在一个只依赖于$ a $和$ {\cal X} $的常数$ C(a)>1 $,使得对于所有的$ x \in{\cal X} $, $ \lambda(x, ar)\geqslant C(a)\lambda(x, r) $,并且$ \sum\limits_{k = 1}^{\infty} {\frac{1}{[C(a^k)]^\eta}}<\infty. $则称控制函数$ \lambda $满足$ \eta $ -弱逆倍条件.

定义2.4^[6]  对于任意两个球$ B\subset S\subset {\cal X} $,定义

$ \widetilde{K}_{B, S}^{(\rho), p}: = \left\{ 1+\sum\limits_{k = -[\log_{\rho} 2]}^{N_{B, S}^{(\rho)}} \left[ \frac{\mu(\rho^{k}B)}{\lambda(c_B, \rho^{k}r_B)} \right]^p \right\} ^{1/p}, $

其中$ \rho>1, p\in(0, 1] $, $ N_{B, S}^{(\rho)} $是满足$ \rho^{N_{B, S}^{(\rho)}}r_B\geqslant r_S $的最小正整数, $ [\log_{\rho}2] $表示不大于$ \log_{\rho}2 $的最大整数.

引理2.2^[7]  假设$ ({\cal X}, d, \mu) $是一个非齐度量测度空间, $ p \in (0, 1] $.

$ \rm(i) $对任意$ \rho > $ 1,存在常数$ C_{(\rho)} >0 $,使得对所有球$ B \subset R \subset S $,有$ [\widetilde{K}_{B, R}^{(\rho), p}]^p\leqslant C_{(\rho)}[\widetilde{K}_{B, S}^{(\rho), p}]^p. $

$ \rm(ii) $对于任意的$ \alpha \geqslant 1, \rho > 1 $,存在一个常数$ C_{(\alpha, \rho)} > 0 $,使得对于所有的球$ B\subset S $,其中$ r_S\leqslant \alpha r_B $,有$ [\widetilde{K}_{B, S}^{(\rho), p}]^p\leqslant C_{(\alpha, \rho)} $.

$ \rm(iii) $对于任意的$ \rho > 1 $,存在一个常数$ c_{(\rho, p, v)} > 0 $,使得对于所有的球$ B \subset R\subset S $,有$ [\widetilde{K}_{B, S}^{(\rho), p}]^p\leqslant [\widetilde{K}_{B, R}^{(\rho), p}]^p +c_{(\rho, p, v)}[\widetilde{K}_{R, S}^{(\rho), p}]^p. $

$ \rm(iv) $对任意$ \rho > 1 $,存在常数$ \tilde{c}_{(\rho, p, v)}>0 $,使对$ B \subset R\subset S $,有$ [\widetilde{K}_{R, S}^{(\rho), p}]^p\leqslant \tilde{c}_{(\rho, p, v)}[\tilde{K}_{B, S}^{(\rho), p}]^p. $

$ \rm(v) $对于任意的$ \rho_1, \rho_2 > 1 $,存在常数$ c_{(\rho_1, \rho_2, p, v)} >0 $和$ C_{(\rho_1, \rho_2, p, v)} >0 $,使得对于所有的球$ B \subset S $,有$ c_{({\rho}_1, \rho_2, p, v)}{\widetilde{K}_{B, S}^{(\rho_2), p}} \leqslant \widetilde{K}_{B, S}^{(\rho_1), p} \leqslant C_{(\rho_1, \rho_2, p, v)}{\widetilde{K}_{B, S}^{(\rho_2), p}}. $

下面是非齐度量测度空间上的Herz空间及Herz型Hardy空间的相关知识.

定义2.5^[8]  令$ -\infty < \alpha < \infty, 0 < p < \infty, 0 < q \leqslant \infty $.设$ ({\cal X}, d, \mu) $是一个非齐度量测度空间,则非齐度量测度空间上的齐次Herz空间定义为

(2.3) $ \begin{eqnarray} \dot{K}_q^{\alpha, p}(\mu) = \left\{f\in L_{\rm{loc}}^q ({\cal X}\ \backslash\left\{0\right\}): \left \|f \right \|_{\dot{K}_q^{\alpha, p}(\mu)} < \infty \right\}, \end{eqnarray} $

其中$ \left \|f \right \|_{\dot{K}_q^{\alpha, p}(\mu)} = \left\{\sum\limits_{k = -\infty}^{+\infty} [{\lambda(x_0, 2^k)}]^{\alpha p}\left \|f \chi_k \right \|_{L^q(\mu)}^p\right\}^{\frac{1}{p}} $.

定义2.6^[8]  令$ 0<\alpha < \infty, 1 \leqslant q < \infty $,则$ ({\cal X}, d, \mu) $上的函数$ b(x) $被称为中心($ \alpha, q $) -块,若$ b(x) $满足以下条件:

(i) supp $ b \subset B(x_0, r) $, $ r > 0 $; \ \ (ii) $ \left \|b\right \|_{L^q(\mu)} \leqslant [\lambda(x_0, r)]^{-\alpha}. $

引理2.3^[8]  令$ 0<\alpha < \infty, 0 < p < \infty, 1 \leqslant q < \infty $.设$ \lambda $满足$ \eta $ -弱逆倍条件, $ \eta \in (0, \min\{\frac{\alpha p}{2}, \frac{\alpha p}{2(p-1)}\}) $.则$ f \in {\dot{K}_q^{\alpha, p}(\mu)} $,当且仅当$ f(x) = \sum\limits_{k = -\infty}^{+\infty}{\lambda_k b_k(x)}, $其中$ b_k(x) $是中心$ (\alpha, q) $ -块,并且$ \sum\limits_{k = -\infty}^{+\infty}{{\left| \lambda_k \right|}^p} < \infty $.进一步, $ \left \|f \right \|_{\dot{K}_q^{\alpha, p}(\mu)} \sim \inf \left\{ {\sum\limits_{k = -\infty}^{+\infty}{{\left| \lambda_k \right|}^p}} \right\}^{\frac{1}{p}} $.

定义2.7^[8]  假设$ ({\cal X}, d, \mu) $是一个非齐度量测度空间,令$ 0 < p \leqslant 1 \leqslant q \leqslant \infty, p \neq q $, $ \alpha \in (0, \infty) $, $ \rho\in(1, \infty), \gamma\in[1, \infty) $.若$ L^2(\mu) $上的函数$ b $满足以下条件:

$ \rm(i) $ supp $ b \subset B = B(x_0, r), r > 0 $;

$ \rm(ii) $$ \int_{\cal X} b(x)\, {\rm d}{\mu(x)} = 0 $;

$ \rm(iii) $对于$ j = 1, 2 $,存在支在球$ B_j \subset B $上的函数$ a_j $和常数$ \lambda_j \in {\Bbb C} $,使得$ b = \lambda_1 a_1 + \lambda_2 a_2 $,

$ \left \| a_j \right \|_{L^q(\mu)} \leqslant [\lambda(x_0, r_B)]^{-\alpha}[\widetilde{K}_{B_j, B}^{(\rho), p}]^{-\gamma}. $

则称$ b $是一个$ (\alpha, p, q, \gamma, \rho)_\lambda $ -原子块.并且,令$ \left| b \right|_{\widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu)} = \left| \lambda_1 \right| + \left| \lambda_2 \right| $.

若有($ \alpha, p, q, \gamma, \rho $)$ _{\lambda} $ -原子块列$ \left\{ b_i \right\}_{i = -\infty}^{+\infty} $,使得$ L^2(\mu) $中

$ f = \sum\limits_{i = -\infty}^{+\infty}b_i, \; \sum\limits_{i = -\infty}^{+\infty} \left| b_i \right|_{\widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu)}^p < \infty, $

则称$ f $是属于$ {\widetilde{{\Bbb H}} \dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu)} $的.并且

$ \left \| f \right \|_{\widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu)}: = \inf \left\{ \sum\limits_{i = -\infty}^{+\infty} \left| b_i \right|_{\widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu)}^p\right \}^{\frac{1}{p}} . $

原子Herz型Hardy空间$ \widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu) $定义为拟模$ \left \| \cdot \right\|_{\widetilde{H}\dot{K}_{atb, q, \rho}^ {\alpha, p, \gamma}(\mu)}^p $下$ {\widetilde{{\Bbb H}}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu)} $的完备化.

引理2.4^[8]  设$ 0 < p \leqslant 1 \leqslant q \leqslant \infty, p \neq q $.令$ 0<\alpha<\infty, 1<\rho<\infty $, $ 1\leqslant\gamma<\infty $.则$ f \in {\widetilde{{H}}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu)} $当且仅当存在$ (\alpha, p, q, \gamma, \rho $)$ _\lambda $ -原子块列$ \left\{ b_i \right\}_{i = -\infty}^{+\infty} $,使得$ f = \sum\limits_{i = -\infty}^{+\infty}b_i, $且$ \sum\limits_{i = -\infty}^{+\infty} \left| b_i \right|_{\widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu)}^p < \infty. $进一步有

$ \left \| f \right \|_{\widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu)}^p = \inf \left\{ \sum\limits_{i = -\infty}^{+\infty} \left| b_i \right|_{\widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu)}^p \right\}. $

定义2.8^[8]  假设$ ({\cal X}, d, \mu) $是一个非齐度量测度空间, $ 0 < p \leqslant 1 \leqslant q \leqslant \infty, p\neq q $,令$ \alpha \in (0, \infty) $, $ \rho\in(1, \infty), \gamma\in[1, \infty) $, $ \epsilon\in(0, \infty). $则若$ L^2(\mu) $上的函数$ b $满足以下条件:

$ \rm(i) $$ \int_{{\cal X}} b(x){\rm d}{\mu(x)} = 0 $;

$ \rm(ii) $存在一些球$ B : = B(x_0, r_B $),其中$ r_B > 0 $和常数$ \widetilde{M}, M \in {\Bbb N} $使得对于所有的$ k \in {\Bbb Z}_{+} $, $ j \in \left\{ 1, \ldots, M_k\right\} $,当$ k = 0 $时, $ M_0 : = \widetilde{M} $,当$ k >0 $时, $ M_k : = M $;存在支在球$ B_{k, j}\subset U_k(B) $上的函数$ m_{k, j} $,其中当$ k = 0 $时, $ U_0(B) : = \rho^2 B $,当$ k>0 $时, $ U_k(B): = \rho^{k+2}B \setminus \rho^{k-2}B $以及常数$ \lambda_{k, j}\in {\Bbb C} $使得在$ L^2(\mu) $中$ b = \sum\limits_{k = 0}^{\infty} \sum\limits_{j = 1}^{M_k} \lambda_{k, j}m_{k, j} $,且有

$ \left \| m_{k, j} \right \|_{L^q(\mu)} \leqslant \rho^{-k\epsilon} [\lambda(x_0, \rho^{k+2}r_B)]^{-\alpha}[\widetilde{K}_{B_{k, j}, \rho^{k+2}B}^{(\rho), p}]^{-\gamma}, $

$ \left| b \right|_{\widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)}^p = \sum\limits_{k = 0}^{\infty} \sum\limits_{j = 1}^{M_k}\left| \lambda_{k, j} \right|^p < \infty, $

则称$ b $是一个$ (\alpha, p, q, \gamma, \epsilon, \rho)_\lambda $ -分子块.

若存在$ (\alpha, p, q, \gamma, \epsilon, \rho)_{\lambda} $ -分子块列$ \left\{ b_i\right\}_{i = -\infty}^{+\infty} $,使得在$ L^2(\mu) $中$ f = \sum\limits_{i = -\infty}^{+\infty}b_i $,且有

$ \sum\limits_{i = -\infty}^{+\infty} \left| b_i \right|_{\widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)}^p < \infty, $

则函数$ f $称为属于$ {\widetilde{{\Bbb H}}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)} $.并且定义

$ \left \| f \right \|_{\widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)}: = \inf \left\{ \sum\limits_{i = -\infty}^{+\infty} \left| b_i \right|_{\widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)}^p \right\}^{\frac{1}{p}} . $

分子Herz型Hardy空间$ \widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu) $定义为拟模$ \left \| \cdot \right\|_{\widetilde{H}\dot{K}_{mb, q, \rho}^ {\alpha, p, \gamma, \epsilon}(\mu)}^p $下$ {\widetilde{{\Bbb H}}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)} $的完备化.

引理2.5^[8]  设$ 0 < p \leqslant 1 \leqslant q \leqslant \infty, p \neq q $.令$ 0<\alpha<\infty, 1<\rho<\infty, 1\leqslant\gamma<\infty, $$ \epsilon>0 $,则$ f \in {\widetilde{{H}}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)} $当且仅当存在$ (\alpha, p, q, \gamma, \epsilon, \rho $)$ _\lambda $ -分子块列$ \left\{ b_i \right\}_{i = -\infty}^{+\infty} $,使在$ {\widetilde{{H}}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)} $中, $ f = \sum\limits_{i = -\infty}^{+\infty}b_i, $且$ \sum\limits_{i = -\infty}^{+\infty} \left| b_i \right|_{\widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)}^p < \infty. $进一步

$ \left \| f \right \|_{\widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)}^p = \inf\left \{ \sum\limits_{i = -\infty}^{+\infty} \left| b_i \right|_{\widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)}^p \right\}. $

注2.1  假设$ ({\cal X}, d, \mu) $是一个非齐度量测度空间,令$ 0 < p \leqslant 1 \leqslant q \leqslant \infty, p \neq q $, $ \alpha \in (0, \infty) $, $ \rho\in(1, \infty), $且$ \gamma\in[1, \infty) $. $ \widetilde{H}_{atb, \rho}^{p, q, \gamma}(\mu) $和$ \widetilde{H}_{mb, \rho}^{p, q, \gamma, \epsilon}(\mu) $均与$ \rho $和$ \gamma $的取值无关.若$ \lambda $满足$ \eta $ -弱逆倍条件, $ \eta \in (0, \alpha p) $,由文献[8,定理3.6和定理3.8]知

$ \widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu) = \widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu)\subset \dot{K}_q^{\alpha, p}(\mu). $

定义3.1^[18]  设函数$ K $是$ {\cal X}\times{\cal X}\setminus \left\{ (x, x): x \in {\cal X}\right\} $上的局部可积函数且满足下列条件:

$ \rm(i) $存在正常数$ C $,使得对于任意的$ x, y \in {\cal X}, x \neq y, $有

(3.1) $ \begin{eqnarray} \left| K(x, y)\right| \leqslant C\frac{d(x, y)}{\lambda(x, d(x, y))}; \end{eqnarray} $

$ \rm(ii) $对于任意的$ x, \tilde{x}, y\in {\cal X}, 2d(x, \tilde{x}) \leqslant d(x, y), 0<\delta\leqslant1, $有

(3.2) $ \begin{eqnarray} \left| K(x, y) - K(\tilde{x}, y) \right| + \left| K(y, x) - K(y, \tilde{x}) \right| \leqslant C \frac{[d(x, \tilde{x})]^{ \delta}}{[d(x, y)]^{\delta-1}\lambda(x, d(x, y))}. \end{eqnarray} $

关于核$ K(x, y) $的Marcinkiewicz积分算子定义为

(3.3) $ \begin{eqnarray} {\cal M}(f)(x): = \left(\int_{0}^{\infty}\left|\int_{d(x, y)<t}K(x, y)f(y)\, {\rm d}{\mu(y)}\right|^2\, \frac{{\rm d}t}{t^3}\right)^{1/2}, \; x \in {\cal X}. \end{eqnarray} $

显然(3.3)式所定义的Marcinkiewicz积分算子满足文献[19]中有关Marcinkiewicz积分算子的所有结论.

定义3.2^[7]  令$ \rho\in (1, \infty) $.如果对于所有的球$ B\subset {\cal X} $,存在一个依赖于$ \rho $但不依赖于$ B $的正常数$ \widetilde{C}_1 $,使得$ N_{B, \widetilde{B}^\rho}^{(\rho)}\leq \widetilde{C}_1, $则称$ \mu $满足$ \rho $ -弱双倍条件.

定义3.3^[7]  令$ \alpha \in (0, \infty), \eta \in (1, \infty), \rho\in (\eta, \infty) $.若$f \in L_{{\text{loc}}}^1(\mu )$满足

$ \begin{eqnarray*} \|f\|_{{\cal E}^{\alpha, q}_{\rho, \eta, \gamma}(\mu)}&: = & \sup\limits_{B}\left\{\frac{1}{\mu(\eta B)} \frac{1}{[\lambda(c_B, r_B)]^{\alpha q}}\int_{B}|f(y)-f_{\widetilde{B}^\rho}|^q \, {\rm d}\mu(y) \right\}^{1/q}\\ &&+ \sup\limits_{B\subset S: B, S \, (\rho), \, \beta_\rho \mbox双倍的} \frac{|f_B-f_S|}{[\lambda(c_S, r_S)]^{\alpha }[\widetilde{K}_{B, S}^{(\rho), 1/(\alpha+1)}]^\gamma}< \infty, \end{eqnarray*} $

则称$ f $是属于Campanato空间$ {\cal E}^{\alpha, q}_{\rho, \eta, \gamma}(\mu) $的.

作者已证明了当$ \mu $满足$ \rho $ -弱双倍条件时, $ {\cal E}^{\alpha, q}_{\rho, \eta, \gamma} $与$ q, \rho, \eta, \gamma $取值无关^[7],因此,为方便,我们将其简记为$ {\cal E}_{\alpha}^{\rho}(\mu) $,并将$ \|\cdot\|_ {{\cal E}^{\alpha, q}_{\rho, \eta, \gamma}(\mu)} $简记为$ \|\cdot\|_{{\cal E}^{\alpha}_{\rho}(\mu)} $,当$ f\in {\cal E}_{\alpha}^{\rho}(\mu) $时,作者证明了

$ \left(\frac{1}{\mu(\eta B)}\int_{B}|f(x)-f_{\widetilde{B}_k^\rho}|^q\, {\rm d}\mu(x)\right) ^{1/q}\leq C \|f\|_{{\cal E}^{\alpha}_{\rho}(\mu)} [\lambda(c_B, r_{B})]^{\alpha}, $

其中$ f_B: = \frac{1}{\mu( B)}\int_{B}f(x)\, {\rm d}\mu(x) $.

设函数$ b\in {\cal E}_{\alpha}^{\rho}(\mu) $,相应的Marcinkiewicz积分算子与$ b $生成的交换子定义为

$ \left[b, {\cal M}\right](f)(x): = b(x){\cal M}f(x)-{\cal M}(bf)(x). $

首先看一个重要引理.

引理3.1^[19]  假设$ ({\cal X}, d , \mu) $是一个非齐度量测度空间,令$ {\cal M} $是一个Marcinkiewicz积分算子,则以下几种情况是等价的:

$ \rm(i) $对于某个$ p_0\in (1, \infty) $, $ {\cal M} $在$ L^{p_0}(\mu) $上是有界的;

$ \rm(ii) $$ {\cal M} $是$ L^1(\mu) $到$ L^{1, \infty}(\mu) $有界的;

$ \rm(iii) $对于$ q > 1 $, $ {\cal M} $在$ L^q(\mu) $上是有界的;

$ \rm(iv) $$ {\cal M} $是$ H^1(\mu) $到$ L^1(\mu) $有界的.

本文我们主要获得了Marcinkiewicz积分算子在非齐度量空间上的Herz空间以及Herz型Hardy空间上的有界性.

定理3.1  设$ ({\cal X}, d , \mu) $为一个非齐度量测度空间. Marcinkiewicz积分算子$ {\cal M} $由(3.3)式所定义,令$ 0 < p < \infty, $$ 1 < q < \infty $, $ 0<\alpha < 1-\frac{1}{q} $,并且$ \lambda $满足$ \eta $ -弱逆倍条件, $ \eta \in(0, \min \{\frac{\alpha p}2, \frac{\alpha p'}2, \frac{(1-1/q-\alpha)p}2, \frac{(1-1/q-\alpha)p'}2 \}) $.若$ {\cal M} $在$ L^2(\mu) $上有界,则$ {\cal M} $在$ \dot{K}_q^{\alpha, p}(\mu) $上有界.

证  对于任意的$ f \in \dot{K}_q^{\alpha, p}(\mu) $,由引理2.3知, $ f(x) = \sum\limits_{j = -\infty}^{+\infty}{\lambda_j b_j(x)} $,其中每个$ b_j $都是中心$ (\alpha, q) $ -块, supp($ b_j) \subset B_j $,且$ \left \|f \right \|_{\dot{K}_q^{\alpha, p}(\mu)} \sim \inf \left\{ {\sum\limits_{j = -\infty}^{+\infty}{{\left| \lambda_j \right|}^p}}\right\}^{\frac{1}{p}}. $因此

$ \begin{eqnarray*} \left \|{\cal M}f \right \|_{\dot{K}_q^{\alpha, p}(\mu)}^p & = & \sum\limits_{l = -\infty}^{+\infty} [{\lambda(x_0, 2^l)}]^{\alpha p} \left \|({\cal M}f) \chi_l \right \|_{L^q(\mu)}^p\\ &\leqslant& \sum\limits_{l = -\infty}^{+\infty}[{\lambda(x_0, 2^l)}]^{\alpha p} \left\{\sum\limits_{j = -\infty}^{l-2} \left| \lambda_j \right| \left \|({\cal M}b_j)\chi_l\right \|_{L^q(\mu)}\right\}^p\\ &&+\sum\limits_{l = -\infty}^{+\infty}[{\lambda(x_0, 2^l)}]^{\alpha p} \left\{\sum\limits_{j = l-1}^{\infty} \left| \lambda_j \right| \left \|({\cal M}b_j)\chi_l\right \|_{L^q(\mu)}\right\}^p\\ & = :&F_1 + F_2. \end{eqnarray*} $

对于$ F_2 $,分为以下两种情况进行讨论.

${\text{ (}}{{\text{i}}_{\text{1}}}{\text{) }}$当0 $ < p \leqslant 1 $时,由引理3.1, (2.1)式, $ b_j $的大小条件和$ \eta $ -弱逆倍条件,有

$ \begin{eqnarray*} F_2 &\leqslant& C \sum\limits_{l = -\infty}^{+\infty} [{\lambda(x_0, 2^l)}]^{\alpha p} \sum\limits_{j = l-1}^{\infty} \left| \lambda_j \right|^p \left \|b_j\right \|_{L^q(\mu)}^p\\ &\leqslant& C \sum\limits_{j = -\infty}^{+\infty} { \left |\lambda_j \right|}^p \sum\limits_{l = -\infty}^{j+1} \left[ \frac{\lambda(x_0, 2^l)}{\lambda(x_0, 2^j)} \right]^{\alpha p} \leqslant C \sum\limits_{j = -\infty}^{+\infty} { \left |\lambda_j \right|}^p. \end{eqnarray*} $

${\text{(}}{{\text{i}}_{\text{2}}}{\text{)}} $当1 $ < p < \infty $时,由引理3.1, (2.1)式, Hölder不等式和$ \eta $ -弱逆倍条件,可得

$ \begin{eqnarray*} F_2 &\leqslant& \sum\limits_{l = -\infty}^{+\infty}[{\lambda(x_0, 2^l)}]^{\alpha p} \left\{\sum\limits_{j = l-1}^{\infty} \left| \lambda_j \right| \left \|b_j\right \|_{L^q(\mu)}\right\}^p \\ & \leqslant& C \sum\limits_{l = -\infty}^{+\infty} [{\lambda(x_0, 2^l)}]^{\alpha p}\left\{ \sum\limits_{j = l-1}^{\infty} \left| \lambda_j \right|^p [\lambda(x_0, 2^j)]^{-{}\frac{\alpha p}{2}} \right\} \left\{\sum\limits_{j = l-1}^{\infty} [\lambda(x_0, 2^j)]^{-\frac{\alpha p'}{2}} \right\} ^{\frac{p}{p'}} \\ &\leqslant& C \sum\limits_{l = -\infty}^{+\infty} \sum\limits_{j = l-1}^{\infty} { \left |\lambda_j \right|}^p \left[ \frac{\lambda(x_0, 2^l)}{\lambda(x_0, 2^j)} \right]^{\frac{\alpha p}{2}} \leqslant C \sum\limits_{j = -\infty}^{+\infty} { \left |\lambda_j \right|}^p. \end{eqnarray*} $

对于$ F_1 $,注意到$ j \leqslant l-2, x \in C_l, y \in B_j $,则$ x \in {\cal X}\setminus 2B_j $意味着$ d(x, y)\sim d(x, x_0) $且$ \lambda(x, d(x, y)) \sim \lambda(x_0, d(x, x_0)) $.因此,由定义3.1, Hölder不等式和定义2.7,知

$ \begin{eqnarray*} \left \|({\cal M}b_j) \chi_l \right \|_{L^q(\mu)} &\leqslant & C \left\{ \int_{C_l}\left|\int_{0}^{\infty} \left(\int_{d(x, y)<t}\frac{\left|b_j(y) \right|d(x, y)}{\lambda(x, d(x, y))}\, {\rm d}\mu(y)\right)^2\, \frac{{\rm d}t}{t^3}\right|^{\frac{q}{2}}\, {\rm d}\mu(x) \right\}^{\frac{1}{q}}\\ &\leqslant &C \left\{ \int_{C_l} \left(\int_{c d(x, x_0)}^{\infty}\, \frac{{\rm d}t}{t^3}\right)^{\frac{q}{2}}\left( \int_{B_j} \frac{\left|b_j(y) \right| d(x, x_0)}{\lambda(x_0, d(x, x_0))}\, {\rm d}\mu(y)\right)^q \, {\rm d}\mu(x) \right\}^{\frac{1}{q}} \\ &\leqslant &C \left\{ \int_{C_l} \frac{1}{\left[\lambda(x_0, d(x, x_0))\right]^q} \, {\rm d}{\mu(x)} \right\}^{\frac{1}{q}} \int_{B_j}\left|b_j(y) \right|\, {\rm d}{\mu(y)} \\ &\leqslant& C \frac{[\mu(B_l)]^{\frac{1}{q}}\left \|b_j(y) \right \|_{L^q(\mu)} \left[\mu(B_j)\right]^{\frac{1}{q'}}}{\lambda(x_0, 2^l)} \leqslant C \frac{\left[\lambda(x_0, 2^j)\right]^{-\alpha } \left[\lambda(x_0, 2^j)\right]^{1-\frac{1}{q}}}{[\lambda(x_0, 2^l)]^{1-\frac{1}{q}}} . \end{eqnarray*} $

根据0 $ < p \leqslant 1 $和$ 1 < p < \infty $,将$ F_1 $分为以下两种情况进行讨论.

$ (\text{ii}_1) $当0 $ < p \leqslant 1 $时,由Jensen不定式, $ \eta $ -弱逆倍条件,有

$ \begin{eqnarray*} F_1 \leqslant C \sum\limits_{l = -\infty}^{+\infty} \sum\limits_{j = -\infty}^{l-2} { \left |\lambda_j \right|}^p \left[ \frac{\lambda(x_0, 2^j)}{\lambda(x_0, 2^l)} \right]^{(1-\frac{1}{q}-\alpha)p} \leqslant C \sum\limits_{j = -\infty}^{+\infty} { \left |\lambda_j \right|}^p. \end{eqnarray*} $

$ (\text{ii}_2) $当1 $ < p < \infty $时,由Hölder不等式和$ \eta $ -弱逆倍条件可得

$ \begin{eqnarray*} F_1 &\leqslant& C \sum\limits_{l = -\infty}^{+\infty} \left\{\sum\limits_{j = -\infty}^{l-2} \left| \lambda_j \right| \left[\frac{\lambda(x_0, 2^j)}{\lambda(x_0, 2^l)} \right]^{(1-\frac{1}{q}-\alpha )} \right\}^p \\ &\leqslant& C \sum\limits_{l = -\infty}^{+\infty} \left\{\sum\limits_{j = -\infty}^{l-2} \left| \lambda_j \right|^p \left[\frac{\lambda(x_0, 2^j)}{\lambda(x_0, 2^l)} \right]^{(1-\frac{1}{q}-\alpha)\frac{p}{2} } \right\} \left\{\sum\limits_{j = -\infty}^{l-2} \left[\frac{\lambda(x_0, 2^j)}{\lambda(x_0, 2^l)} \right]^{(1-\frac{1}{q}-\alpha)\frac{p'}{2} } \right\}^{\frac{p}{p'}} \\ & = &C \sum\limits_{l = -\infty}^{+\infty} \sum\limits_{j = -\infty}^{l-2} { \left |\lambda_j \right|}^p \left[ \frac{\lambda(x_0, 2^j)}{\lambda(x_0, 2^l)} \right]^{(1-\frac{1}{q}-\alpha)\frac{ p}{2}}\leqslant C \sum\limits_{j = -\infty}^{+\infty} { \left |\lambda_j \right|}^p. \end{eqnarray*} $

因此

$ \left \|{\cal M}f \right \|_{\dot{K}_q^{\alpha, p}(\mu)}^p \leqslant C \sum\limits_{j = -\infty}^{+\infty} { \left |\lambda_j \right|}^p \leqslant C \left \|f \right \|_{\dot{K}_q^{\alpha, p}(\mu)}^p. $

定理3.1证毕.

注3.1  由注2.1和定理3.1, Marcinkiewicz积分算子$ {\cal M} $也是从$ {\widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu)} $到$ \dot{K}_{q}^{\alpha, {p}}(\mu) $,或从$ \widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu) $到$ \dot{K}_{q}^{\alpha, {p}}(\mu) $有界的.

下面,我们主要研究非齐度量测度空间上的Marcinkiewicz积分算子$ {\cal M} $从$ \widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu) $到$ \widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, \epsilon}(\mu) $的有界性.若对于所有的$ h\in L_b^{\infty}(\mu) $, $ \int_{{\cal X}} h(y)\, {\rm d}\mu(y) = 0, $有

$ \int_{{\cal X}} {\cal M}h(y)\, {\rm d}\mu(y) = 0, $

则称$ {\cal M} $满足$ {\cal M}^{*}1 = 0 $.

定理3.2  假设$ ({\cal X}, d, \mu) $是一个非齐度量测度空间. $ {\cal M} $由(3.3)式所定义,令$ 0 < p \leqslant 1 \leqslant q \leqslant \infty, p \neq q $, $ \alpha \in (0, \delta/v) $, $ \rho\in(1, \infty) $, $ \gamma\in[1, \infty) $,且$ \epsilon\in(0, \infty). $若$ {\cal M} $在$ L^2(\mu) $上有界,且$ {\cal M}^*1 = 0 $,则$ {\cal M} $是从$ \widetilde{H}\dot{K}_{atb, q, \rho}^{\alpha, p, \gamma}(\mu) $到$ \widetilde{H}\dot{K}_{mb, q, \rho}^{\alpha, p, \gamma, (\delta-v\alpha)/2}(\mu) $有界的.

证  只需证明对于任意的($ \alpha, p, q, 2, 2 $)$ _{\lambda} $ -原子块$ b $, $ {\cal M}b $是一个($ \alpha, p, q, 1, $$ (\delta-v\alpha)/2, 2 $)$ _{\lambda} $ -分子块,且

$ \left|{\cal M}b \right|_{\widetilde{H}\dot{K}_{mb, q, 2}^{\alpha, p, 1, (\delta-v\alpha)/2}(\mu)} \leqslant C \left| b \right|_{\widetilde{H}\dot{K}_{atb, q, 2}^{\alpha, p, 2}(\mu)}. $

事实上,对于任意的($ \alpha, p, q, 2, 2)_{\lambda} $ -原子块$ b $, $ b $ = $ \sum\limits_{j = 1}^{2} \lambda_j a_j $, supp ($ a_j $) $ \subset B_j \subset B $,且

(3.4) $ \begin{eqnarray} \left \| a_j \right \|_{L^q(\mu)} \leqslant[\lambda(x_0, r_B)]^{-\alpha}[\widetilde{K}_{B_j, B}^{(2), p}]^{-2}. \end{eqnarray} $

令$ B_0 = 8B, $进一步分解

$ {\cal M}b = ({\cal M}b)\chi_{B_0} + \sum\limits_{k = 1}^{\infty}({\cal M}b)\chi_{2^k B_0 \setminus {2^{k-1}B_0}} = : W_1 + W_2. $

对于$ W_1 $,由于$ B_j\subset B $,显然$ 3B_j \subset 8B = B_0 $.令$ N_j : = N_{2B_j, B_0}^{(2)}\geqslant -1 $.不失一般性,假设$ N_j\geqslant 3. $由于$ 2B_j \subset B_0 \subset 2^5 B_j $,对于其它情况,即当$ N_j \in[-1, 3) $时可以转化为$ N_j \geqslant 3 $.因此

$ \begin{eqnarray*} W_1 & = &\sum\limits_{j = 1}^{2}\lambda_j({\cal M}a_j)\chi_{2B_j} + \sum\limits_{j = 1}^{2}\sum\limits_{i = 1}^{N_j-2}\lambda_j({\cal M}a_j)\chi_{2^{i+1}B_j\setminus{2^iB_j}}+ \sum\limits_{j = 1}^{2}\lambda_j({\cal M}a_j)\chi_{B_0\setminus{2^{N_j-1}B_j}} \\ & = &:W_{1, 1}+W_{1, 2}+W_{1, 3}. \end{eqnarray*} $

对于$ W_{1, 1} $,由引理3.1, (3.4)式, (2.1)式,引理2.2,和$ \widetilde{K}_{2B_j, B_0}^{(2), p}\geqslant 1 $,对于任意的$ j = 1, 2 $,有

$ \begin{eqnarray*} \left \| ({\cal M}a_j)\chi_{2B_j}\right \|_{L^q(\mu)} &\leqslant& C \left \| a_j\right \|_{L^q(\mu)}\\ & \leqslant& C [\lambda(x_0, r_B)]^{-\alpha}[\widetilde{K}_{B_j, B}^{(2), p}]^{-2} \leqslant C [\lambda(x_0, 4r_{B_0})]^{-\alpha}[\widetilde{K}_{2B_j, 8B}^{(2), p}]^{-2} \\ &\leqslant& c_{11} [\lambda(x_0, 4r_{B_0})]^{-\alpha}[\widetilde{K}_{2B_j, 4B_0}^{(2), p}]^{-1}, \end{eqnarray*} $

其中$ c_{11} $是不依赖于$ a_j $和$ j $的正常数.令$ \sigma_{j, 1}: = c_{11} \lambda_j, \tau_{j, 1}: = c_{11}^{-1} ({\cal M}a_j)\chi_{2B_j} $.则$ W_{1.1} = \sum\limits_{j = 1}^{2}\sigma_{j, 1}\tau_{j, 1} $, supp$ (\tau_{j, 1})\subset 2B_j\subset B_0 $,且

$ \left \| \tau_{j, 1}\right \|_{L^q(\mu)} \leqslant [\lambda(x_0, 4r_{B_0})]^{-\alpha}[\widetilde{K}_{2B_j, 4B_0}^{(2), p}]^{-1}. $

对于$ W_{1, 3} $,由于$ B_0 \subset 2^{N_j+3}B_j $,有$ r_{B_0}\sim r_{2^{N_j-1}B_j} $.由(3.1), (2.2), (3.4)和(2.1)式, Hölder不等式,引理2.2,并且$ \widetilde{K}_{B_j, B}^{(2), p}\geqslant 1 $,同时注意到$ d(x, y)\sim d(x, c_{B_j}) $且$ \lambda(x, d(x, y)) \sim \lambda(c_{B_j}, d(x, c_{B_j})) $,可知对于任意的$ j = 1, 2, $有

$ \begin{eqnarray*} &&\left \| ({\cal M}a_j)\chi_{B_0\setminus{2^{N_j-1}B_j}}\right \|_{L^q(\mu)}\\ & = & \left\{ \int_{8B\setminus{2^{N_j-1}B_j}} \left| {\cal M}a_j(x)\right|^q \, {\rm d}\mu(x) \right\}^{\frac{1}{q}} \\ &\leqslant & C \left\{ \int_{8B\setminus{2^{N_j-1}B_j}}\left|\int_{0}^{\infty} \left(\int_{d(x, y)<t}\frac{\left|a_j(y) \right|d(x, y)}{\lambda(x, d(x, y))}\, {\rm d}\mu(y)\right)^2\, \frac{{\rm d}t}{t^3}\right|^{\frac{q}{2}}\, {\rm d}\mu(x) \right\}^{\frac{1}{q}} \\ &\leqslant& C \left\{\int_{8B\setminus{2^{N_j-1}B_j}} \left(\int_{d(x, c_{B_j})}^{\infty}\, \frac{{\rm d}t}{t^3}\right)^{\frac{q}{2}}\left( \int_{B_j} \frac{\left|a_j(y) \right| d(x, c_{B_j})}{\lambda(c_{B_j}, d(x, c_{B_j}))}\, {\rm d}\mu(y)\right)^q \, {\rm d}\mu(x) \right\}^{\frac{1}{q}} \\ &\leqslant& C \left\{\int_{8B\setminus{2^{N_j-1}B_j}} \frac{1}{[\lambda(c_{B_j}, d(x, c_{B_j}))]^q} \, {\rm d}\mu(x) \right\}^{\frac{1}{q}} \int_{B_j} |a_j(y)|\, {\rm d}\mu(y) \\ &\leqslant& C \frac{[\mu(8B\setminus{2^{N_j-1}B_j})]^{1/q}}{\lambda(c_{B_j}, 2^{N_j-1}r_{B_j})} [\mu(B_j)]^{1/q'} \left \| a_j\right \|_{L^q(\mu)} \leqslant c_{12} [\lambda(x_0, r_{4B_0})]^{-\alpha} [\widetilde{K}_{2B_0, 4B_0}^{(2), p}]^{-1}, \end{eqnarray*} $

其中$ c_{12} $是不依赖于$ a_j $和$ j $的正常数.令$ \sigma_{j, 3}: = c_{12} \lambda_j, \tau_{j, 3}: = c_{12}^{-1} (Ma_j)\chi_{B_0\setminus{2^{N_j-1}B_j}} $,则$ W_{1.3} = \sum\limits_{j = 1}^{2}\sigma_{j, 3}\tau_{j, 3} $, supp($ \tau_{j, 3})\subset 16B = 2B_0 $,且$ \left \| \tau_{j, 3}\right \|_{L^q(\mu)} \leqslant [\lambda(x_0, 4r_{B_0})]^{-\alpha} [\widetilde{K}_{2B_0, 4B_0}^{(2), p}]^{-1}. $$ W_{1, 2} $与$ W_{1, 3} $类似有

$ \begin{eqnarray*} \left \| ({\cal M}a_j)\chi_{2^{i+1}B_j\setminus{2^iB_j}}\right \|_{L^q(\mu)} \leqslant c_{13} \frac{\mu(2^{i+1}B_j)}{\lambda(c_{B_j}, 2^{i}r_{B_j})} [\widetilde{K}_{B_j, B}^{(2), p}]^{-1} [\lambda(x_0, r_{4B_0})]^{-\alpha}[\widetilde{K}_{2^{i+2}B_j, 4B_0}^{(2), p}]^{-1}, \end{eqnarray*} $

其中$ c_{13} $是不依赖于$ a_j $, $ j $和$ i $的正常数.令

$ \sigma_{j, 2}: = c_{13} \lambda_j\frac{\mu(2^{i+1}B_j)}{\lambda(c_{B_j}, 2^{i}r_{B_j})} [\widetilde{K}_{B_j, B}^{(2), p}]^{-1}, $

$ \tau_{j, 2}: = \left[c_{13} \frac{\mu(2^{i+1}B_j)}{\lambda(c_{B_j}, 2^{i}r_{B_j})} \right]^{-1} \widetilde{K}_{B_j, B}^{(2), p}(Ma_j)\chi_{2^{i+1}B_j\setminus{2^iB_j}}, $

则$ W_{1, 2} = \sum\limits_{j = 1}^{2}\sum\limits_{i = 1}^{N_j-2}\sigma_{j, 2}^{(i)} \tau_{j, 2}^{(i)} $, supp($ \tau_{j, 2})\subset 2^{i+2}B_j\subset 2B_0 $,且

$ \big\| \tau_{j, 2}^{(i)}\big \|_{L^q(\mu)} \leqslant[\lambda(x_0, 4r_{B_0})]^{-\alpha}[\widetilde{K}_{2^{i+2}B_j, 4B_0}^{(2), p}]^{-1} . $

下面估计$ W_2 $.由几何双倍条件知道对于任意的$ k\in {\Bbb N} $,存在一个球覆盖$ \left\{B_{k, j} \right\}_{j = 1}^{M_0} $,且它们的势$ M_0 \leqslant N_0 8^n $,其中这些球的半径都是$ 2^{k-3}r_{B_0} $, $ \widetilde{U}_k(B_0): = 2^kB_0\setminus{2^{k-1}B_0} $.不失一般性,假设这些球的中心都属于$ \widetilde{U}_k(B_0) $.令$ C_{k, 1}: = B_{k, 1}, C_{k, l}: = B_{k, l}\setminus \bigcup\limits_{m = 1}^{l-1}B_{k, m} $, $ l = 2, \ldots, M_0 $且对于所有的$ l = 1, \ldots, M_0 $，$ D_{k, l}: = C_{k, l}\cap \widetilde{U}_k(B_0) $,则$ \left\{ D_{k, l} \right\}_{l = 1}^{M_0} $是互不相交的, $ \widetilde{U}_k(B_0) = \bigcup\limits_{l = 1}^{M_0}D_{k, l} $,且对于任意的$ l = 1, \ldots, M_0 $, $ D_{k, l}\subset 2B_{k, l} \subset U_k(B_0): = 2^{k+2}B_0\setminus 2^{k-2}B_0. $因此

$ W_2 = \sum\limits_{k = 0}^{\infty}Mb\sum\limits_{l = 0}^{M_0}\chi_{D_{k, l}} = \sum\limits_{k = 0}^{\infty}\sum\limits_{l = 0}^{M_0}(Mb)\chi_{D_{k, l}}. $

因为$ \int_{{\cal X}} b(y)\, {\rm d}\mu(y) = 0 $,由(3.2), (2.2), (2.1)和(3.4)式,应用Hölder不等式和引理2.2,同时注意到$ 4B_{k, l}\subset 2^{k+1}B_0 $和$ \widetilde{K}_{B_j, B}^{(2), p} \geqslant 1 $,可以得到

$ \begin{eqnarray*} &&\left \| ({\cal M}b)\chi_{D_{k, l}}\right \|_{L^q(\mu)} \\ &\leqslant & \left\{ \int_{D_{k, l}}\left|\int_{0}^{\infty} \left(\int_{d(x, y)<t}|K(x, y)-K(x, c_B)|\left|b(y) \right|\, {\rm d}\mu(y)\right)^2\, \frac{{\rm d}t}{t^3}\right|^{\frac{q}{2}}\, {\rm d}\mu(x) \right\}^{\frac{1}{q}} \\ &\leqslant& C \left\{\int_{D_{k, l}} \left(\int_{c d(x, c_B)}^{\infty}\, \frac{{\rm d}t}{t^3}\right)^{q/2} \left|\int_{B}\frac{[d(y, c_B)]^{ \delta}}{[d(x, c_B)]^{\delta-1}\lambda(c_B, d(x, c_B))}\left|b(y) \right|\, {\rm d}\mu(y) \right|^{q} \, {\rm d}\mu(x) \right\}^{\frac{1}{q}} \\ &\leqslant& C \frac{r_B^{\delta}[\mu(D_{k, l})]^{1/q}}{\lambda(c_B, r_{2^{k-1}B_0})(r_{2^{k-1}B_0})^{\delta}} \int_{B}|b(y)| \, {\rm d}\mu(y) \\ &\leqslant& C 2^{-k\delta}[\mu(2^{k+1}B_0)]^{1/q-1} \sum\limits_{j = 1}^{2}|\lambda_j| [\mu(B_j)]^{1/q'} \left \| a_j\right \|_{L^q(\mu)} \\ &\leqslant& C 2^{-k(\delta-v\alpha)}[\mu(2^{k+1}B_0)]^{1/q-1} \sum\limits_{j = 1}^{2}|\lambda_j| [\mu(2^{k+1}B_0)]^{1/q'}[\lambda(x_0, r_{2^{k+2}B_0})]^{-\alpha} \\ &\leqslant& c_{14} 2^{-\frac{k(\delta-v\alpha)}{2}} 2^{-\frac{k(\delta-v\alpha)}{2}} \sum\limits_{j = 1}^{2}|\lambda_j| [\lambda(x_0, r_{2^{k+2}B_0})]^{-\alpha} [\widetilde{K}_{2B_{k, l}, 2^{k+2}B_0}^{(2), p}]^{-1}, \end{eqnarray*} $

其中$ c_{14} $是不依赖于$ b $和$ k $的正常数.令$ \lambda_{k, l}: = c_{14} 2^{-\frac{k(\delta-v\alpha)}{2}} \sum\limits_{j = 1}^{2}|\lambda_j| , m_{k, l}: = \lambda_{k, l}^{-1} (Mb)\chi_{D_{k, l}}. $则$ W_{2} = \sum\limits_{k = 1}^{\infty}\sum\limits_{l = 1}^{M_0}\lambda_{k, l}m_{k, l} $, supp($ m_{k, l})\subset 2B_{k, l}\subset U_k(B_0) $,且

$ \left \|m_{k, l} \right \|_{L^q(\mu)} \leqslant2^{-\frac{k(\delta-v\alpha)}{2}} [\lambda(x_0, 2^{k+2}r_{B_0})]^{-\alpha} [\widetilde{K}_{2B_{k, l}, 2^{k+2}B_0}^{(2), p}]^{-1} . $

由$ W_1 $和$ W_2 $的估计,得到$ {\cal M}b $是一个($ \alpha, p, q, 1, \delta, 2 $)$ _{\lambda} $ -分子块,且

$ \begin{eqnarray*} \left| {\cal M}b \right|_{\widetilde{H}\dot{K}_{mb, q, 2}^{\alpha, p, 1, (\delta-v\alpha)/2}(\mu)}^p & = &\sum\limits_{j = 1}^{2}|\sigma_{j, 1}|^p + \sum\limits_{j = 1}^{2}\sum\limits_{i = 1}^{N_j-1}|\sigma_{j, 2}^{(i)}|^p + \sum\limits_{j = 1}^{2}|\sigma_{j, 3}|^p + \sum\limits_{k = 1}^{\infty}\sum\limits_{l = 1}^{M_0} |\lambda_{k, l}|^p \\ &\leqslant &C \left\{ \sum\limits_{j = 1}^{2}|\lambda_j|^p + \sum\limits_{j = 1}^{2}\sum\limits_{i = 1}^{N_j-2}|\lambda_j|^p\left[\frac{\mu(2^{i+3}B_j)}{\lambda(c_{B_j}, 2^{i}r_{B_j})}\right]^p [\widetilde{K}_{B_j, B}^{(2), p}]^{-p} \right.\\ &&\left. + \sum\limits_{k = 1}^{\infty}\sum\limits_{l = 1}^{M_0}2^{-\frac{k(\delta-v\alpha)}{2}} \sum\limits_{j = 1}^{2}|\lambda_j|^p \right\} \\ &\leqslant &C\left\{ \sum\limits_{j = 1}^{2}|\lambda_j|^p + \sum\limits_{k = 1}^{\infty}2^{-\frac{k(\delta-v\alpha)}{2}} M_0 \sum\limits_{j = 1}^{2}|\lambda_j|^p \right\} \\ &\leqslant& C \sum\limits_{j = 1}^{2}|\lambda_j|^p \leqslant C \left| b \right|_{\widetilde{H}\dot{K}_{atb, q, 2}^{\alpha, p, 2}(\mu)}^p. \end{eqnarray*} $

定理3.2得证.

下面证明Marcinkiewicz积分交换子$ [b, {\cal M}] $从$ \widetilde{H}\dot{K}_{atb, q_1, \rho}^{\alpha_1, p, \gamma}(\mu) $到$ \dot{K}_{q_2}^{\alpha_2, p}(\mu) $的有界性.

定理3.3  假设$ ({\cal X}, d, \mu) $是一个非齐度量测度空间. $ {\cal M} $由(3.3)式所定义,令$ 0 < p \leqslant 1 <q, $$ q_1, q_2< \infty $, $ \frac{1}{q_2} = \frac{1}{q_1}+\frac{1}{q}, \alpha \in (0, \infty), \alpha_1 = 1-\frac{1}{q}, \alpha_1 = \alpha_2+\alpha+\frac{1}{q} $, $ \rho\in(1, \infty) $, $ \gamma\in[1, \infty) $, $ \alpha_2\in(0, 1-1/q_2 +\delta/\nu) $, $ \alpha_1\in(0, 1-1/q_1 +\delta/\nu) $, $ \lambda $满足$ \eta $ -弱逆倍条件, $ \eta \in(0, \alpha_1) $,且$ b\in {\cal E}^{\alpha}_{\rho}(\mu). $则交换子$ [b, {\cal M}] $是从$ \widetilde{H}\dot{K}_{atb, q_1, \rho}^{\alpha_1, p, \gamma}(\mu) $到$ \dot{K}_{q_2}^{\alpha_2, p}(\mu) $有界的.

证  对任意的$ f \in \widetilde{H}\dot{K}_{atb, q_1, \rho}^{\alpha_1, p, \gamma}(\mu) $,存在($ \alpha_1, p, q_1, \gamma, \rho $)$ _{\lambda} $ -原子块$ b_j, $$ b_j = \sum\limits_{i = 1}^{2}\lambda_{j, i}a_{j, i} $, supp($ a_{j, i} $)$ \subset B_{j, i} \subset B_j $,其中$ B_j = B(x_0, 2^j) $,使得$ f = \sum\limits_{j = -\infty}^{\infty}b_j = \sum\limits_{j = -\infty}^{\infty} \sum\limits_{i = 1}^{2}\lambda_{j, i}a_{j, i} $,则

$ \begin{eqnarray*} \left \|[b, {\cal M}]f \right \|_{\dot{K}_{q_2}^{\alpha_2, p}(\mu)} & = &\left\{\sum\limits_{k = -\infty}^{+\infty}[{\lambda(x_0, 2^k)}]^{\alpha_2 p} \left \|[b, {\cal M}]f \chi_k\right \|_{L^{q_2}(\mu)}^p\right\}^{1/p}\qquad\\ &\leqslant& \left\{\sum\limits_{k = -\infty}^{+\infty}[{\lambda(x_0, 2^k)}]^{\alpha_2 p} \left\{\sum\limits_{j = -\infty}^{k-3}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right| \left \|[b, {\cal M}]a_{j, i}\chi_k\right \|_{L^{q_2}(\mu)}\right\}^p\right\}^{1/p} \\ &&+\left\{\sum\limits_{k = -\infty}^{+\infty}[{\lambda(x_0, 2^k)}]^{\alpha_2 p} \left\{\sum\limits_{j = k-2}^{+\infty}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right| \left \|[b, {\cal M}]a_{j, i}\chi_k\right \|_{L^{q_2}(\mu)}\right\}^p\right\}^{1/p} \\ & = :&G_1 + G_2. \end{eqnarray*} $

首先考虑$ G_2 $,进一步分解

$ \begin{eqnarray*} \left \|[b, {\cal M}]a_{j, i}\chi_k\right \|_{L^{q_2}(\mu)} & = &\left\{\int_{C_k}|{\cal M}(b(x)-b(y))a_{j, i}(x)|^{q_2}\, {\rm d}\mu(x)\right\}^{1/q_2} \\ &\leqslant& C \left\{\int_{C_k}|b(x)-b_k|^{q_2}|{\cal M}a_{j, i}(x)|^{q_2}\, {\rm d}\mu(x)\right\}^{\frac{1}{q_2}} \\ && +C \left\{\int_{C_k}|{\cal M}(b_k-b(x))a_{j, i}(x)|^{q_2}\, {\rm d}\mu(x)\right\}^{\frac{1}{q_2}}\\ & = :&G_{2, 1} + G_{2, 2}. \end{eqnarray*} $

由Hölder不等式,引理3.1和原子的大小条件,可得

$ \begin{eqnarray*} G_{2, 1} &\leqslant& C \left\{\int_{C_k}|b(x)-b_k|^{q}\, {\rm d}\mu(x)\right\}^{1/q} \left\{\int_{C_k}|{\cal M}a_{j, i}(x)|^{q_1}\, {\rm d}\mu(x)\right\}^{1/q_1} \\ &\leqslant &C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)}[\lambda(x_0, r_{B_k})]^{\alpha+1/q} [\lambda(x_0, r_{B_j})]^{-\alpha_1}. \\ G_{2, 2} &\leqslant &C \left\{\int_{C_k}|b(x)-b_k|^{q_2}|a_{j, i}|^{q_2}\, {\rm d}\mu(x)\right\}^{1/q_2} \\ &\leqslant &C \left\{\int_{C_k}|b(x)-b_k|^{q}\, {\rm d}\mu(x)\right\}^{1/q} \left\{\int_{C_k}|a_{j, i}(x)|^{q_1}\, {\rm d}\mu(x)\right\}^{1/q_1} \\ &\leqslant &C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)}[\lambda(x_0, r_{B_k})]^{\alpha+1/q} [\lambda(x_0, r_{B_j})]^{-\alpha_1}. \end{eqnarray*} $

因此,由$ \eta $ -弱逆倍条件,得

$ \begin{eqnarray*} G_{2} &\leqslant &C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)} \left\{\sum\limits_{k = -\infty}^{+\infty} \left\{\sum\limits_{j = k-2}^{+\infty}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right| \left[\frac{\lambda(x_0, 2^k)}{\lambda(x_0, 2^j)}\right]^{\alpha_1} \right\}^p\right\}^{1/p} \\ &\leqslant& C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)} \left\{\sum\limits_{j = -\infty}^{+\infty}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|^p \left(\sum\limits_{k = -\infty}^{j+2}\left[\frac{\lambda(x_0, 2^k)}{\lambda(x_0, 2^j)}\right]^{\alpha_1} \right)^p\right\}^{1/p} \\ &\leqslant& C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)} \left\{\sum\limits_{j = -\infty}^{+\infty}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|^p\right\}^{1/p}. \end{eqnarray*} $

下面估计$ G_{1} $,注意到$ j \leqslant k-3, x \in C_k, y \in B_j $,则

$ \begin{eqnarray*} &&\left \|[b, {\cal M}]a_{j, i}\chi_k\right \|_{L^{q_2}(\mu)}\\ &\leqslant&\left\{\int_{C_k}\left|\int_{0}^{\infty}\left(\int_{d(x, y)<t} (b(x)-b_j)K(x, y)a_{j, i}(y)\, {\rm d}\mu(y)\right)^2\frac{{\rm d}t}{t^3}\right|^{q_2/2}\, {\rm d}\mu(x)\right\}^{1/q_2} \\ &&+ \left\{\int_{C_k}\left|\int_{0}^{\infty}\left(\int_{d(x, y)<t} (b_j-b(y))K(x, y)a_{j, i}(y)\, {\rm d}\mu(y)\right)^2\frac{{\rm d}t}{t^3}\right|^{q_2/2}\, {\rm d}\mu(x)\right\}^{1/q_2} \\ & = :&G_{1, 1} +G_{1, 2}. \end{eqnarray*} $

对于$ G_{1, 1} $,由Hölder不等式和$ a_{j, i} $的消失性条件及大小条件,得

$ \begin{eqnarray*} G_{1, 1}& \leqslant& \left\{\int_{C_k}\!\left\{\int_{0}^{\infty}\!\!\left(\int_{d(x, y)<t} |b(x)-b_j||K(x, y)-K(x, x_0)||a_{j, i}(y)|{\rm d}\mu(y)\right)^2\frac{{\rm d}t}{t^3}\!\right\}^{\frac{q_2}{2}}\!\!{\rm d}\mu(x)\right\}^{\frac{1}{q_2}} \\ &\leqslant& C\left\{\int_{C_k}\left(\int_{c d(x, x_0)}^{\infty}\, \frac{{\rm d}t}{t^3}\right)^{q_2/2} \left\{\int_{B_j}\frac{|b(x)-b_j|[d(y, x_0)]^{\delta }|a_{j, i}(y)|}{[d(x, x_0)]^{\delta-1 }\lambda(x_0, d(x, x_0))}\, {\rm d}\mu(y) \right\}^{q_2}\, {\rm d}\mu(x)\right\}^{1/q_2} \\ &\leqslant& C2^{(j-k)\delta}[\lambda(x_0, 2^k)]^{-1} \left\{\int_{C_k} |b(x)-b_j|^{q_2}\, {\rm d}\mu(x) \right\}^{1/q_2} \int_{B_j} |a_{j, i}(y)|\, {\rm d}\mu(y)\\ &\leqslant& C 2^{(j-k)\delta} \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)}(k-j) [\lambda(x_0, 2^k)]^{\alpha+1/q+1/q_1-1}[\lambda(x_0, 2^j)]^{-\alpha_1+1-1/q_1}.\qquad\qquad\qquad\qquad\end{eqnarray*} $

因此当$ \alpha_1 \in (0, 1-1/q_1) $时,有

$ \begin{eqnarray*} G_{1, 1} &\leqslant& C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)} \left\{\sum\limits_{k = -\infty}^{+\infty} \left\{\sum\limits_{j = -\infty}^{k-3}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|2^{(j-k)\delta} (k-j) \left[\frac{\lambda(x_0, 2^j)}{\lambda(x_0, 2^k)}\right]^{1-1/q_1-\alpha_1} \right\}^p\right\}^{1/p} \\ &\leqslant& C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)} \left\{\sum\limits_{j = -\infty}^{+\infty}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|^p \left(\sum\limits_{k = j+3}^{+\infty} 2^{(j-k)\delta} (k-j) \left[\frac{\lambda(x_0, 2^j)}{\lambda(x_0, 2^k)}\right]^{1-1/q_1-\alpha_1} \right)^p\right\}^{1/p} \\ &\leqslant& C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)} \left\{\sum\limits_{j = -\infty}^{+\infty}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|^p\right\}^{1/p}. \end{eqnarray*} $

当$ \alpha_1 \in [1-1/q_1, 1-1/q_1+\delta/\nu) $时,有

$ \begin{eqnarray*} G_{1, 1} &\leqslant& C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)} \left\{\sum\limits_{k = -\infty}^{+\infty} \left\{\sum\limits_{j = -\infty}^{k-3}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|2^{(j-k)\delta} (k-j) 2^{(j-k)\nu(1-1/q_1-\alpha_1)} \right\}^p\right\}^{1/p} \\ &\leqslant& C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)} \left\{\sum\limits_{j = -\infty}^{+\infty}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|^p \left(\sum\limits_{k = j+3}^{+\infty} 2^{(j-k)(\delta+\nu(1-1/q_1-\alpha_1))} (k-j) \right)^p\right\}^{1/p} \\ &\leqslant &C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)} \left\{\sum\limits_{j = -\infty}^{+\infty}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|^p\right\}^{1/p}. \end{eqnarray*} $

类似于$ G_{1, 1} $,同样可以得到如下估计

$ \begin{eqnarray*} G_{1, 2} &\leqslant& C \left\{\int_{C_k}\left\{\int_{0}^{\infty}\left(\int_{d(x, x_0)<t} \frac{|b_j-b(y)|[d(y, x_0)]^{\delta }|a_{j, i}(y)|}{[d(x, x_0)]^{\delta-1 }\lambda(x_0, d(x, x_0))}{\rm d}\mu(y)\right)^2\frac{{\rm d}t}{t^3}\right\}^{\frac{q_2}{2}}{\rm d}\mu(x)\right\}^{\frac{1}{q_2}} \\ &\leqslant& C 2^{(j-k)\delta}[\lambda(x_0, 2^k)]^{-1} \left\{\int_{C_k} \, {\rm d}\mu(x) \right\}^{\frac{1}{q_2}} \int_{B_j} |b(y)-b_j||a_{j, i}(y)|\, {\rm d}\mu(y) \\ &\leqslant &C 2^{(j-k)\delta}\|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)}[\lambda(x_0, 2^k)]^{-1+1/q_2}[\lambda(x_0, 2^j)]^{-\alpha_2+1-1/q_2}. \end{eqnarray*} $

当$ \alpha_2 \in (0, 1-1/q_2) $时,有

$ \begin{eqnarray*} G_{1, 2} &\leqslant& C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)} \left\{\sum\limits_{k = -\infty}^{+\infty} \left\{\sum\limits_{j = -\infty}^{k-3}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|2^{(j-k)\delta} \left[\frac{\lambda(x_0, 2^j)}{\lambda(x_0, 2^k)}\right]^{-\alpha_2+1-1/q_2} \right\}^p\right\}^{1/p} \\ &\leqslant& C \|b\|_{{\cal E}_{\alpha}^{\rho}(\mu)} \left\{\sum\limits_{j = -\infty}^{+\infty}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|^p\right\}^{1/p}. \end{eqnarray*} $

当$ \alpha_2 \in [1-1/q_2, 1-1/q_2+\delta/\nu) $时,有

$ \begin{eqnarray*} G_{1, 2} &\leqslant& C \|b\|_{{\cal E}^{\alpha}_{\rho}(\mu)} \left\{\sum\limits_{j = -\infty}^{+\infty}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|^p \left(\sum\limits_{k = j+3}^{+\infty} 2^{(j-k)\delta} \left[\frac{\lambda(x_0, 2^j)}{\lambda(x_0, 2^k)}\right]^{-\alpha_2+1-1/q_2} \right)^p\right\}^{1/p} \\ &\leqslant& C \|b\|_{{\cal E}_{\alpha}^{\rho}(\mu)} \left\{\sum\limits_{j = -\infty}^{+\infty}\sum\limits_{i = 1}^{2}\left| \lambda_{j, i} \right|^p\right\}^{1/p}. \end{eqnarray*} $

定理3.3证毕.