分数次极大算子在广义加权变指标Morrey空间上的有界性

doi:10.12677/PM.2021.1111211

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分数次极大算子在广义加权变指标Morrey空间上的有界性
Boundedness of Fractional Maximal Operator on Generalized Weighted Morrey Spaces with Variable Exponents

DOI: 10.12677/PM.2021.1111211, PDF, HTML, XML, 下载: 331 浏览: 2,270
作者: 徐博：西北师范大学数学与统计学院，甘肃兰州
关键词: 分数次极大算子；交换子；广义加权变指标Morrey空间；BMO空间；Fractional Maximal Operator； Commutator； Generalized Weighted Morrey Space with Variable Exponent； BMO Space

摘要: 利用A_{p(⋅),q(⋅)}权函数的性质以及调和分析的实方法，得到了分数次极大算子在广义加权变指标Morrey空间上的有界性，同时也给出了交换子的相应结果。

Abstract: By applying the properties of A_{p(⋅),q(⋅)} weighted functions and real-variable methods of harmonic analysis, the boundedness of the fractional maximal operator is obtained on generalized weighted Morrey spaces with variable exponent. Meanwhile, the corresponding result of its commutator is also given.

文章引用：徐博. 分数次极大算子在广义加权变指标Morrey空间上的有界性[J]. 理论数学, 2021, 11(11): 1888-1896. https://doi.org/10.12677/PM.2021.1111211

1. 引言及主要结果

设 $0 < α < n$ ，分数次极大算子和分数次积分算子分别定义为：

$M_{α} f (x) = \sup_{t > 0} \frac{1}{{| B (x, t) |}^{1 - \frac{α}{n}}} \int_{B (x, t)} | f (y) | d y, I_{α} f (x) = \int_{ℝ^{n}} \frac{f (y)}{{| x - y |}^{n - α}} d y$ .

给定可测函数b，相应的交换子可定义如下：

$[b, M_{α}] f = M_{α} (b f) - b M_{α} (f), [b, I_{α}] f = I_{α} (b f) - b I_{α} (f)$ ,

同时给出定义：

$M_{b, α} f (x) = \sup_{t > 0} \frac{1}{{| B (x, t) |}^{1 - \frac{α}{n}}} \int_{B (x, t)} | b (x) - b (y) | | f (y) | d y$ ,

$[b, I_{α}] f (x) = \int_{ℝ^{n}} [b (x) - b (y)] f (y) {| x - y |}^{α - n} d y$ .

当 $b (x) \geq 0$ 时，对于任意的局部可积函数f，有 $| [b, M_{α}] f (x) | \leq M_{b, α} f (x)$ 。

设 $Ω \subset ℝ^{n}$ 为无界开集， $χ_{E} (x)$ 表示 $E \subset ℝ^{n}$ 上的特征函数， $B (x, r) = {y \in ℝ^{n} : | x - y | < r}$ ， $\tilde{B} (x, r) = B (x, r) \cap Ω$ ，本文中 $φ (x, r), φ_{1} (x, r), φ_{2} (x, r)$ 均为 $Ω \times (0, \infty)$ 上的非负可测函数。

给定可测函数 $p (\cdot) : Ω \to (1, \infty)$ ，变指标Lebesgue空间 $L^{p (\cdot)}$ 定义为：

$L^{p (\cdot)} (Ω) = {f : \exists λ > 0, st . {\int_{Ω} | f (x) |}^{p (x)} d x < \infty}$ ,

其上的Luxemburg-Nakano范数为：

${‖ f ‖}_{L^{p (\cdot)} (Ω)} = \inf {λ > 0 : \int_{Ω} {(\frac{| f (x) |}{λ})}^{p (x)} d x \leq 1}$ .

定义1.1 [1] 设 $ω > 0$ 为 $Ω$ 上的一个局部可积函数，它表示权函数。加权变指标Lebesgue空间 $L_{ω}^{p (\cdot)} (Ω)$ 定义为：

$L_{ω}^{p (\cdot)} (Ω) = {f : {‖ f ‖}_{L_{ω}^{p (\cdot)} (Ω)} \equiv {‖ f ω ‖}_{L^{p (\cdot)} (Ω)} < \infty}$ .

定义1.2 [2] 设 $1 \leq p (x) < \infty$ ， $x \in Ω$ ， $\frac{1}{q (x)} = \frac{1}{p (x)} - \frac{α}{n}$ ，变指标 $A_{p (\cdot)} (Ω)$ 权和变指标 $A_{p (\cdot), q (\cdot)} (Ω)$ 权可分别定义为：

$A_{p (\cdot)} (Ω) = {ω : {[ω]}_{A_{p (\cdot)}} \equiv \sup_{B (x, r)} {| B (x, r) |}^{- 1} {‖ ω ‖}_{L^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω^{- 1} ‖}_{L^{p^{'} (\cdot)} (\tilde{B} (x, r))} < \infty}$

$A_{p (\cdot), q (\cdot)} (Ω) = {ω : {[ω]}_{A_{p (\cdot), q (\cdot)}} \equiv \sup_{B (x, r)} {| B (x, r) |}^{\frac{1}{p (x)} - \frac{1}{q (x)} - 1} {‖ ω ‖}_{L^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω^{- 1} ‖}_{L^{p^{'} (\cdot)} (\tilde{B} (x, r))} < \infty}$ .

定义1.3 [3] 设 $1 \leq p (x) < \infty$ ， $x \in Ω$ ，则广义加权变指标Morrey空间 $M_{ω}^{p (\cdot), φ} (Ω)$ 定义为：

$M_{ω}^{p (\cdot), φ} (Ω) = {f : \sup_{x \in Ω, r > 0} \frac{1}{φ (x, r) {‖ ω ‖}_{L^{p (\cdot)} (\tilde{B} (x, r))}} < \infty}$ .

当 $\frac{1}{p (x)} - \frac{1}{q (x)} = \frac{α}{n}$ 时，C. Capone 在文献 [4] 中证明了分数次极大算子从 $L^{p (\cdot)}$ 到 $L^{q (\cdot)}$ 是有界的，文献 [5] 中证明了带粗糙核的Marcinkiewicz积分在变指标Morrey空间上的有界性，文献 [6] 给出了局部互补广义变指标Morrey空间上几类奇异积分算子的有界性估计，分数次极大算子在广义加权Morrey空间上的有界性估计可参见文献 [7]。最近作者在文献 [8] 中得到了Calderón-Zygmund奇异积分算子在中心Morrey-Orlicz空间上的有界性。受上面研究的启发，本文将研究分数次极大算子及其交换子在广义加权变指标Morrey空间上的有界性。

定义1.4 [9] 给定有界可测函数 $p (x) : Ω \to [1, \infty)$ ，假设 $1 \leq p_{-} \leq p (\cdot) \leq p_{+} < \infty$ 满足局部log-Hölder连续条件：

$| p (x) - p (y) | \leq \frac{C}{- \log | x - y |}, x, y \in Ω, | x - y | \leq \frac{1}{2}$ , (1)

且满足log-Hölder在无穷远处的连续条件：存在 $p_{\infty}$ ，使得

$| p (x) - p_{\infty} | \leq \frac{C}{\log (e + | x |)}, x \in Ω$ , (2)

将满足上述条件的所有 $p (x)$ 构成的集合记为 $P_{\infty}^{\log} (Ω)$ ，其中 $p_{+} = \underset{x \in Ω}{esssup} p (x)$ ， $p_{-} = \underset{x \in Ω}{ess \inf} p (x)$ ， $p_{\infty} = \lim_{x \to \infty} p (x) > 1$ 。

定义1.5 [10] 设 $b \in L_{l o c}^{1} (Ω)$ ，且

${‖ b ‖}_{B M O} = \sup_{x \in ℝ^{n}, r > 0} \frac{1}{| B (x, r) |} \int_{B (x, r)} | b (y) - b_{B (x, r)} | d y < \infty$ ,(3)

其中

$b_{B (x, r)} = \frac{1}{| B (x, r) |} \int_{B (x, r)} b (y) d y$ .

则 $B M O (Ω)$ 空间可定义为：

$B M O (Ω) = {b \in L_{l o c}^{1} (Ω) : {‖ b ‖}_{B M O} < \infty}$ .

本文的主要结果如下：

定理1 设 $Ω \subset ℝ^{n}$ 为无界开集， $0 < α < n$ ， $p \in P_{\infty}^{\log} (Ω)$ ， $p_{+} < \frac{n}{α}$ ， $\frac{1}{q (x)} = \frac{1}{p (x)} - \frac{α}{n}$ ， $ω \in A_{p (\cdot), q (\cdot)} (Ω)$ ，

则对任意 $f \in L_{ω}^{p (\cdot)} (Ω)$ ，有

${‖ M_{α} f ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \leq C {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}^{- 1}$ , (4)

其中C与 $f, x \in Ω$ 和r均无关。

定理2 设 $Ω \subset ℝ^{n}$ 为无界开集， $0 < α < n$ ， $p \in P_{\infty}^{\log} (Ω)$ ， $p_{+} < \frac{n}{α}$ ， $\frac{1}{q (x)} = \frac{1}{p (x)} - \frac{α}{n}$ ， $ω \in A_{p (\cdot), q (\cdot)} (Ω)$ ，且函数 $φ_{1} (x, r)$ 和 $φ_{2} (x, r)$ 满足条件

$\sup_{r > t} \frac{\underset{r < s < \infty}{ess \inf} φ_{1} (x, s) {‖ ω ‖}_{L^{p (\cdot)} (\tilde{B} (x, s))}}{{‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))}} \leq C φ_{2} (x, t)$ , (5)

其中C与 $x \in Ω$ 和r均无关，则 $M_{α}$ 从 $M_{ω}^{p (\cdot), φ_{1}} (Ω)$ 到 $M_{ω}^{q (\cdot), φ_{2}} (Ω)$ 上有界。

定理3设 $Ω \subset ℝ^{n}$ 为无界开集， $0 < α < n$ ， $p \in P_{\infty}^{\log} (Ω)$ ， $p_{+} < \frac{n}{α}$ ， $\frac{1}{q (x)} = \frac{1}{p (x)} - \frac{α}{n}$ ， $ω \in A_{p (\cdot), q (\cdot)} (Ω)$ ， $b \in B M O (Ω)$ ，则对任意的 $f \in L_{ω}^{p (\cdot)} (Ω)$ 有

${‖ M_{b, α} f ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \leq C {‖ b ‖}_{B M O} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} (1 + \ln \frac{r}{t}) {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}^{- 1}$ , (6)

其中C与 $f, x \in Ω$ 和r均无关。

定理4设 $Ω \subset ℝ^{n}$ 为无界开集， $0 < α < n$ ， $p \in P_{\infty}^{\log} (Ω)$ ， $p_{+} < \frac{n}{α}$ ， $\frac{1}{q (x)} = \frac{1}{p (x)} - \frac{α}{n}$ ， $ω \in A_{p (\cdot), q (\cdot)} (Ω)$ ， $b \in B M O (Ω)$ ，且函数 $φ_{1} (x, r)$ 和 $φ_{2} (x, r)$ 满足条件

$\sup_{r > t} (1 + \ln \frac{r}{t}) \frac{\underset{r < s < \infty}{ess \inf} φ_{1} (x, s) {‖ ω ‖}_{L^{p (\cdot)} (\tilde{B} (x, s))}}{{‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))}} \leq C φ_{2} (x, t)$ (7)

其中C与 $x \in Ω$ 和r均无关，则 $M_{b, α}$ 从 $M_{ω}^{p (\cdot), φ_{1}} (Ω)$ 到 $M_{ω}^{q (\cdot), φ_{2}} (Ω)$ 上有界。

2. 预备知识

引理2.1 [11] 设 $Ω \subset ℝ^{n}$ 为无界开集， $0 < α < n$ ， $p \in P_{\infty}^{\log} (Ω)$ ， $p_{+} < \frac{n}{α}$ ， $\frac{1}{q (x)} = \frac{1}{p (x)} - \frac{α}{n}$ ， $ω \in A_{p (\cdot), q (\cdot)} (Ω)$ ，则 $M_{α}$ 从 $L_{ω}^{p (\cdot)} (Ω)$ 到 $L_{ω}^{q (\cdot)} (Ω)$ 上有界。

引理2.2 [12] 设 $Ω \subset ℝ^{n}$ 为无界开集， $0 < α < n$ ， $p \in P_{\infty}^{\log} (Ω)$ ， $p_{+} < \frac{n}{α}$ ， $\frac{1}{q (x)} = \frac{1}{p (x)} - \frac{α}{n}$ ， $ω \in A_{p (\cdot), q (\cdot)} (Ω)$ ， $b \in B M O (Ω)$ ，则 $[b, I_{α}]$ 从 $L_{ω}^{p (\cdot)} (Ω)$ 到 $L_{ω}^{q (\cdot)} (Ω)$ 上有界。

引理2.3 [12] 设 $1 < p (x) < \infty$ ， $\frac{1}{q (x)} = \frac{1}{p (x)} - \frac{α}{n}$ ，且 $ω \in A_{p (\cdot), q (\cdot)} (Ω)$ ，则 $ω^{- 1} \in A_{q^{'} (\cdot), p^{'} (\cdot)} (Ω)$ 。

引理2.4 设 $Ω \subset ℝ^{n}$ 为无界开集， $0 < α < n$ ， $p \in P_{\infty}^{\log} (Ω)$ ， $p_{+} < \frac{n}{α}$ ， $\frac{1}{q (x)} = \frac{1}{p (x)} - \frac{α}{n}$ ， $ω \in A_{p (\cdot), q (\cdot)} (Ω)$ ，则下列条件等价：

(i) $b \in B M O (Ω)$ ；

(ii) $M_{b, α}$ 从 $L_{ω}^{p (\cdot)} (Ω)$ 到 $L_{ω}^{q (\cdot)} (Ω)$ 上有界。

证明 (i) $\Rightarrow$ (ii) 设 $f \in L_{ω}^{p (\cdot)} (Ω)$ 且 $b \in B M O (Ω)$ ，则由引理2.2可知

${‖ M_{b, α} f ‖}_{L_{ω}^{q (\cdot)} (Ω)} \leq C {‖ [b, I_{α}] f ‖}_{L_{ω}^{q (\cdot)} (Ω)} \leq C {‖ b ‖}_{B M O} {‖ f ‖}_{L_{ω}^{p (\cdot)} (Ω)}$ 。

(ii) $\Rightarrow$ (i) 设 $M_{b, α}$ 从 $L_{ω}^{p (\cdot)} (Ω)$ 到 $L_{ω}^{q (\cdot)} (Ω)$ 上有界，则由广义Hölder不等式和变指标 $A_{p (\cdot), q (\cdot)}$ 权的性质可得

$\begin{array}{l} \frac{1}{| B (x, t) |} \int_{\tilde{B} (x, t)} | b (z) - b_{B (x, t)} | d z \\ = \frac{1}{| B (x, t) |} \int_{\tilde{B} (x, t)} | b (z) - \frac{1}{| B (x, t) |} \int_{B (x, t)} b (y) d y | d z \\ = \frac{1}{| B (x, t) |} \int_{\tilde{B} (x, t)} \frac{1}{| B (x, t) |} | \int_{B (x, t)} b (z) - b (y) d y | d z \\ \leq \frac{1}{{| B (x, t) |}^{1 + \frac{α}{n}}} \int_{\tilde{B} (x, t)} \frac{1}{{| B (x, t) |}^{1 - \frac{α}{n}}} \int_{B (x, t)} | b (z) - b (y) | d y d z \end{array}$

引理2.5 [12] 设 $b \in B M O (Ω)$ ， $p \in P_{\infty}^{\log} (Ω)$ ，且 $ω \in A_{p (\cdot)} (Ω)$ ，则 $M_{b}$ 在 $L_{ω}^{p (\cdot)} (Ω)$ 上有界。

引理2.6 [13] 设 $b \in B M O (Ω)$ ，则存在常数 $C > 0$ ，使得

$| b_{\tilde{B} (x, r)} - b_{\tilde{B} (x, t)} | \leq C {‖ b ‖}_{B M O} \ln \frac{t}{r}, 0 < 2 r < t$ , (8)

其中C与 $b, x, r$ 和t均无关。

引理2.7 [14] 设 $Ω \subset ℝ^{n}$ 为无界开集， $p \in P_{\infty}^{\log} (Ω)$ ，且 $ω$ 为Lebesgue可测函数。若 $ω \in A_{p (\cdot)} (Ω)$ ，则范数 ${‖ \cdot ‖}_{B M O}$ 与范数 ${‖ \cdot ‖}_{B M O_{p (\cdot), ω}}$ 等价，其中对于任意的局部可积函数f，有

${‖ f ‖}_{B M O_{p (\cdot), ω}} = \sup_{x \in Ω, r > 0} \frac{{‖ (f (\cdot) - f_{\tilde{B} (x, r)}) χ_{\tilde{B} (x, r)} ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))}}{{‖ χ_{\tilde{B} (x, r)} ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))}}$ .

引理2.8 [15] 设 $p \in P_{\infty}^{\log} (Ω)$ ，若 $ω \in A_{p (\cdot), q (\cdot)} (Ω)$ ，则 $ω {(\cdot)}^{q (\cdot)} \in A_{\infty}$ 。

3. 主要结果的证明

定理1的证明设 $f \in L_{ω}^{p (\cdot)} (Ω)$ ，分解 $f = f_{1} + f_{2}$ ，其中 $f_{1} (y) = f (y) χ_{\tilde{B} (x, 2 t)} (y)$ ， $f_{2} (y) = f (y) χ_{Ω \ \tilde{B} (x, 2 t)} (y)$ ， $t > 0$ ，则有

${‖ M_{α} f ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \leq {‖ M_{α} f_{1} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} + {‖ M_{α} f_{2} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))}$ 。 (9)

由引理2.1，有

${‖ M_{α} f_{1} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \leq {‖ M_{α} f_{1} ‖}_{L_{ω}^{q (\cdot)} (Ω)} \leq C {‖ f_{1} ‖}_{L_{ω}^{p (\cdot)} (Ω)} = C {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, 2 t))}$ 。

另一方面，

$\begin{matrix} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, 2 t))} \leq C {| B (x, t) |}^{1 - \frac{α}{n}} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, 2 t))} \sup_{r > 2 t} {| B (x, r) |}^{\frac{α}{n} - 1} \\ \leq C {| B (x, t) |}^{1 - \frac{α}{n}} \sup_{r > 2 t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {| B (x, r) |}^{\frac{α}{n} - 1} \\ \leq C {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} {‖ ω^{- 1} ‖}_{L^{p^{'} (\cdot)} (\tilde{B} (x, t))} \sup_{r > 2 t} {| B (x, r) |}^{\frac{α}{n} - 1} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} \\ \leq C {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > 2 t} {| B (x, r) |}^{\frac{α}{n} - 1} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω^{- 1} ‖}_{L^{p^{'} (\cdot)} (\tilde{B} (x, r))} \\ \leq C {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} {| B (x, r) |}^{\frac{α}{n} - 1} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω^{- 1} ‖}_{L^{p^{'} (\cdot)} (\tilde{B} (x, r))} \\ \leq C {[ω]}_{A_{p (\cdot), q (\cdot)}} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{_{L^{q (\cdot)} (\tilde{B} (x, r))}}^{- 1} \end{matrix}$

所以

${‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, 2 t))} \leq C {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{_{L^{q (\cdot)} (\tilde{B} (x, r))}}^{- 1}$ ， (10)

则有

${‖ M_{α} f_{1} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \leq C {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{_{L^{q (\cdot)} (\tilde{B} (x, r))}}^{- 1}$ 。 (11)

设对于任意 $z \in B (x, r)$ ，若 $B (z, r) \cap {(B (x, 2 t))}^{c} \neq \emptyset$ ，则 $r > t$ 。事实上，若 $y \in B (z, r) \cap {(B (x, 2 t))}^{c}$ ，则 $r > | y - z | \geq | x - y | - | x - z | \geq 2 t - r > t$ 。另一方面，当 $y \in B (z, r) \cap {(B (x, 2 t))}^{c}$ ，有 $| x - y | \leq | y - z | + | x - z | < t + r < 2 r$ 。因此， $B (z, r) \cap {(B (x, 2 t))}^{c} \subset B (x, 2 r)$ 。则由广义Hölder不等式和变指数 $A_{p (\cdot), q (\cdot)}$ 权的定义可得，

$\begin{matrix} M_{α} f_{2} (z) = \sup_{r > 0} \frac{1}{{| B (z, r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (z, r)} | f_{2} (y) | d y \\ = \sup_{r > 0} \frac{1}{{| B (z, r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (z, r) \cap {(\tilde{B} (x, 2 t))}^{c}} | f (y) | d y \\ \leq \sup_{r > t} \frac{1}{{| B (z, r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (x, 2 r)} | f (y) | d y \\ = \sup_{r > t} \frac{{| B (x, 2 r) |}^{1 - \frac{α}{n}}}{{| B (z, r) |}^{1 - \frac{α}{n}}} \frac{1}{{| B (x, 2 r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (x, 2 r)} | f (y) | d y \end{matrix}$

$\begin{array}{l} \leq C \sup_{r > t} \frac{1}{{| B (x, 2 r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (x, 2 r)} | f (y) | d y \\ \leq C \sup_{r > 2 t} \frac{1}{{| B (x, r) |}^{1 - \frac{α}{n}}} \int_{B (x, 2 r)} | f (y) | d y \\ \leq C \sup_{r > t} \frac{1}{{| B (x, r) |}^{1 - \frac{α}{n}}} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{_{L^{q (\cdot)} (\tilde{B} (x, r))}}^{- 1} \end{array}$

因此，

${‖ M_{α} f_{2} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \leq C {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{_{L^{q (\cdot)} (\tilde{B} (x, r))}}^{- 1}$ 。 (12)

由式(9)，(11)和(12)即可得到定理1。

定理2的证明设 $f \in M_{ω}^{p (\cdot), φ_{1}} (Ω)$ ，则由定理1和式(5)可得

$\begin{matrix} {‖ M_{α} f ‖}_{M_{ω}^{q (\cdot), φ_{2}} (Ω)} = \sup_{x \in Ω, t > 0} \frac{1}{φ_{2} (x, t) {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}} {‖ M_{α} f ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \\ \leq C \sup_{x \in Ω, t > 0} \frac{1}{φ_{2} (x, t)} \sup_{r > t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{_{L^{q (\cdot)} (\tilde{B} (x, r))}}^{- 1} \\ \leq C {‖ f ‖}_{M_{ω}^{p (\cdot), φ_{1}} (Ω)} \sup_{x \in Ω, t > 0} \frac{1}{φ_{2} (x, t)} \sup_{r > t} \frac{φ_{1} (x, r) {‖ ω ‖}_{L^{p (\cdot)} (\tilde{B} (x, r))}}{{‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}} \\ \leq C {‖ f ‖}_{M_{ω}^{p (\cdot), φ_{1}} ( Ω )} \end{matrix}$

定理3的证明设 $b \in B M O (Ω)$ ， $f \in L_{ω}^{p (\cdot)} (\tilde{B} (x, t))$ ，由定理1的证明，记 $f = f_{1} + f_{2}$ ，则有

${‖ M_{b, α} f ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \leq {‖ M_{b, α} f_{1} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} + {‖ M_{b, α} f_{2} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))}$ 。 (13)

由引理2.4，有

${‖ M_{b, α} f_{1} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \leq {‖ M_{b, α} f_{1} ‖}_{L_{ω}^{q (\cdot)} (Ω)} \leq C {‖ b ‖}_{B M O} {‖ f_{1} ‖}_{L_{ω}^{p (\cdot)} (Ω)} = C {‖ b ‖}_{B M O} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, 2 t))}$ ，

则由式(10)有

${‖ M_{b, α} f_{1} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \leq C {‖ b ‖}_{B M O} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{_{L^{q (\cdot)} (\tilde{B} (x, r))}}^{- 1}$ 。(14)

设对任意 $z \in B (x, r)$ ，若 $B (z, r) \cap {(B (x, 2 t))}^{c} \neq \emptyset$ ，则 $r > t$ 。事实上，若 $y \in B (z, r) \cap {(B (x, 2 t))}^{c}$ ，则 $r > | y - z | \geq | x - y | - | x - z | \geq 2 t - r > t$ 。另一方面，当 $y \in B (z, r) \cap {(B (x, 2 t))}^{c}$ ，有 $| x - y | \leq | y - z | + | x - z | < t + r < 2 r$ 。因此， $B (z, r) \cap {(B (x, 2 t))}^{c} \subset B (x, 2 r)$ 。

$\begin{matrix} M_{b, α} f_{2} (z) = \sup_{r > 0} \frac{1}{{| B (z, r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (z, r)} | b (y) - b (z) | | f_{2} (y) | d y \\ = \sup_{r > 0} \frac{1}{{| B (z, r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (z, r) \cap {(\tilde{B} (x, 2 t))}^{c} \neq \emptyset} | b (y) - b (z) | | f (y) | d y \\ \leq \sup_{r > t} \frac{1}{{| B (z, r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (x, 2 r)} | b (y) - b (z) | | f (y) | d y \\ = \sup_{r > t} \frac{{| B (x, 2 r) |}^{1 - \frac{α}{n}}}{{| B (z, r) |}^{1 - \frac{α}{n}}} \frac{1}{{| B (x, 2 r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (x, 2 r)} | b (y) - b (z) | | f (y) | d y \\ \leq C \sup_{r > t} \frac{1}{{| B (x, 2 r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (x, 2 r)} | b (y) - b (z) | | f (y) | d y \end{matrix}$

$\begin{array}{l} \leq C \sup_{r > 2 t} \frac{1}{{| B (x, r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (x, r)} | b (y) - b (z) | | f (y) | d y \\ \leq C \sup_{r > t} \frac{1}{{| B (x, r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (x, r)} | b (y) - b_{\tilde{B} (x, r)} | | f (y) | d y \\ + C \sup_{r > t} \frac{1}{{| B (x, r) |}^{1 - \frac{α}{n}}} \int_{\tilde{B} (x, r)} | b (z) - b_{\tilde{B} (x, r)} | | f (y) | d y \\ = : I_{1} + I_{2}, \end{array}$

对于 $I_{1}$ ，由广义Hölder不等式，定义1.2，引理2.7和变指数 $A_{p (\cdot), q (\cdot)}$ 权的定义可知，

$\begin{matrix} I_{1} \leq C \sup_{r > t} {| B (x, r) |}^{- 1 + \frac{α}{n}} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ b (\cdot) - b_{\tilde{B} (x, r)} ‖}_{L_{ω^{- 1}}^{p^{'} (\cdot)} (\tilde{B} (x, r))} \\ \leq C {‖ b ‖}_{B M O} \sup_{r > t} {| B (x, r) |}^{- 1 + \frac{α}{n}} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ χ_{\tilde{B} (x, r)} ‖}_{L_{ω^{- 1}}^{p^{'} (\cdot)} (\tilde{B} (x, r))} \\ = C {‖ b ‖}_{B M O} \sup_{r > t} {| B (x, r) |}^{- 1 + \frac{α}{n}} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω^{- 1} ‖}_{L^{p^{'} (\cdot)} (\tilde{B} (x, r))} \\ \leq C {‖ b ‖}_{B M O} \sup_{r > t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}^{- 1} \end{matrix}$

另一方面，对于 $I_{2}$ ，由引理2.6和广义Hölder不等式，有

$\begin{matrix} I_{2} \leq C \sup_{r > t} {| B (x, r) |}^{- 1 + \frac{α}{n}} | b (z) - b_{\tilde{B} (x, t)} | \int_{\tilde{B} (x, r)} | f (y) | d y \\ + C \sup_{r > t} {| B (x, r) |}^{- 1 + \frac{α}{n}} | b_{\tilde{B} (x, t)} - b_{\tilde{B} (x, r)} | \int_{\tilde{B} (x, r)} | f (y) | d y \\ \leq C \sup_{r > t} {| B (x, r) |}^{- 1 + \frac{α}{n}} | b (z) - b_{\tilde{B} (x, t)} | {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω^{- 1} ‖}_{L^{p^{'} (\cdot)} (\tilde{B} (x, r))} \\ + C {‖ b ‖}_{B M O} \sup_{r > t} {| B (x, r) |}^{- 1 + \frac{α}{n}} \ln \frac{r}{t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω^{- 1} ‖}_{L^{p^{'} (\cdot)} (\tilde{B} (x, r))} \\ \leq C M_{b} χ_{B (x, t)} (z) \sup_{r > t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}^{- 1} \\ + C {‖ b ‖}_{B M O} \sup_{r > t} \ln \frac{r}{t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}^{- 1} \end{matrix}$

则由引理2.5，有

$\begin{matrix} {‖ M_{b, α} f_{2} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \leq {‖ I_{1} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} + {‖ I_{2} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \\ \leq C {‖ b ‖}_{B M O} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}^{- 1} \\ + C {‖ M_{b} χ_{B (x, t)} ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}^{- 1} \\ + C {‖ b ‖}_{B M O} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} \ln \frac{r}{t} {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}^{- 1} \\ \leq C {‖ b ‖}_{B M O} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))} \sup_{r > t} (1 + \ln \frac{r}{t}) {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}^{- 1} \end{matrix}$

由式(13)和式(14)即可得到定理3。

定理4的证明设 $f \in M_{ω}^{p (\cdot), φ_{1}} (Ω)$ ，则由定理3和式(7)可得

$\begin{matrix} {‖ M_{b, α} f ‖}_{M_{ω}^{q (\cdot), φ_{2}} (Ω)} = \sup_{x \in Ω, t > 0} \frac{1}{φ_{2} (x, t) {‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, t))}} {‖ M_{b, α} f ‖}_{L_{ω}^{q (\cdot)} (\tilde{B} (x, t))} \\ \leq C {‖ b ‖}_{B M O} \sup_{x \in Ω, t > 0} \frac{1}{φ_{2} (x, t)} \sup_{r > t} (1 + \ln \frac{r}{t}) {‖ f ‖}_{L_{ω}^{p (\cdot)} (\tilde{B} (x, r))} {‖ ω ‖}_{_{L^{q (\cdot)} (\tilde{B} (x, r))}}^{- 1} \\ \leq C {‖ b ‖}_{B M O} {‖ f ‖}_{M_{ω}^{p (\cdot), φ_{1}} (Ω)} \sup_{x \in Ω, t > 0} \frac{1}{φ_{2} (x, t)} \sup_{r > t} (1 + \ln \frac{r}{t}) \frac{φ_{1} (x, r) {‖ ω ‖}_{L^{p (\cdot)} (\tilde{B} (x, r))}}{{‖ ω ‖}_{L^{q (\cdot)} (\tilde{B} (x, r))}} \\ \leq C {‖ b ‖}_{B M O} {‖ f ‖}_{M_{ω}^{p (\cdot), φ_{1}} (Ω)} \\ \leq C {‖ f ‖}_{M_{ω}^{p (\cdot), φ_{1}} ( Ω )} \end{matrix}$

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