海森堡型群上的 Qr1,r2函数空间The Qr1,r2 Spaces on the H-Type Groups

DOI: 10.12677/PM.2021.114082, PDF, HTML, XML, 下载: 14  浏览: 46

Abstract: Function spaces play an important role in harmonic analysis. In this paper, we study the Q-type space Qr1,r2(N) on the H-type group N. We show the characterization of Qr1,r2(N) by the Carleson measure on the Siegel type domain N×R+. Furthermore, the embedding result between Qr1,r2(N) and BMOβ(N) is founded.

1. 引言

${‖f‖}_{{Q}_{P}}={\left(\underset{w\in D}{\mathrm{sup}}{\int }_{D}{|{f}^{\prime }\left(z\right)|}^{2}{g}^{p}\left(z,w\right)\text{d}A\left(z\right)\right)}^{\frac{1}{2}}<+\infty$

ii) 若有 ${r}_{2}<0$${r}_{1}>\frac{d+2}{d}$，则 ${Q}_{{r}_{1},{r}_{2}}\left(N\right)$ 仅包含常数函数；

iii) 若任意的 $\beta >0$，有 ${r}_{1}<1$$\beta +1={r}_{1}+{r}_{2}$，那么 ${Q}_{{r}_{1},{r}_{2}}\left(N\right)=BM{O}^{\beta }\left(N\right)$

${\int }_{S\left(I\right)}{|\nabla F\left(z\right)|}^{2}{a}^{d\left(1-{r}_{1}\right)+1}\text{d}z\le C{|I|}^{{r}_{2}}$

2. 预备知识

2.1. 海森堡型群 $N$

$G$ 为二阶幂零李代数，其上有内积 $〈\cdot ,\cdot 〉$，中心表示为 $z$。我们称 $G$ 为海森堡型李代数，若 $\left[{z}^{\perp },{z}^{\perp }\right]=z$ 且对任意的 $t\in z$，映射 ${J}_{t}:{z}^{\perp }\to {z}^{\perp }$ 表示为

$〈{\text{J}}_{\text{t}}\text{u},\text{w}〉:=〈t,\left[u,w\right]〉$$u,w\in z$

$|t|=1$ 时，Jt为正交映射。海森堡型群为单连通的二阶幂零李群，其李代数是海森堡型李代数。

$〈A\left(\rho \right)u,w〉=\rho \left(\left[u,w\right]\right)=〈{J}_{{Z}_{\rho }}u,w〉$

$A\left(\rho \right){E}_{i}\left(\rho \right)=|{Z}_{\rho }|{J}_{\frac{{Z}_{\rho }}{|{Z}_{\rho }|}}{E}_{i}\left(\rho \right)=|\rho |{\stackrel{¯}{E}}_{i}\left(\rho \right),A\left(\rho \right){\stackrel{¯}{E}}_{i}\left(\rho \right)=-|\rho |{E}_{i}\left(\rho \right)$

$\mathrm{dim}z=m$$\left\{{\epsilon }_{1},\cdot \cdot \cdot ,{\epsilon }_{m}\right\}$$z$ 的一组标准正交基，满足 $\rho \left({\epsilon }_{1}\right)=|\rho |$$\rho \left({\epsilon }_{j}\right)=0\left(1\le j\le m\right)$，则

$\left(z,t\right)=\left(x,y,t\right)=\underset{i=1}{\overset{n}{\sum }}\left({x}_{i}{E}_{i}+{y}_{i}{\stackrel{¯}{E}}_{i}\right)+\underset{j=1}{\overset{m}{\sum }}{t}_{j}{\epsilon }_{j}$

$\left(z,t\right)\left({z}^{\prime },{t}^{\prime }\right)=\left(z+{z}^{\prime },t+{t}^{\prime }+\frac{1}{2}\left[z,{z}^{\prime }\right]\right)$

$N$ 上可以定义一组等价的范数

$|w|={\left(\frac{1}{16}{|{x}^{\prime }|}^{4}+{|{u}^{\prime }|}^{2}\right)}^{\frac{1}{4}}$

${|w|}_{\infty }=\mathrm{max}\left\{\frac{1}{2}|{x}_{1}|,\cdots ,\frac{1}{2}|{x}_{p}|,\sqrt{|{u}_{1}|},\cdots ,\sqrt{|{u}_{q}|}\right\}$

$I=\left\{w:{|{n}^{-1}w|}_{\infty }\le \frac{l\left(I\right)}{2}\right\}$

2.2. Carleson测度与Poisson积分

$S\left(I\right)=\left\{z=\left({x}^{\prime },{u}^{\prime },a\right)=\left(w,a\right)\in S:{|{n}^{-1}w|}_{\infty }\le \frac{l\left(I\right)}{2},a\le l\left(I\right)\right\}$

$S$ 上的一正Borel测度 $\mu$ 称为p-Carleson测度，如果对p > 0，存在常数M > 0，使得

$\mu \left(S\left(I\right)\right)\le M{|I|}^{p}$

$\begin{array}{l}{X}_{j}=\frac{\partial }{\partial {x}_{j}}+\frac{1}{2}{x}_{j}{C}_{ji}^{l}\frac{\partial }{\partial {u}_{l}},1\le j,i\le p,1\le l\le q;\\ {T}_{k}=\frac{\partial }{\partial {u}_{k}},1\le k\le q\end{array}$

$N$ 上任意的光滑函数f，f的梯度定义为

$\nabla f\left(w\right)=\left({X}_{1}f\left(w\right),\cdot \cdot \cdot ,{X}_{p}f\left(w\right),{T}_{1}f\left(w\right),\cdot \cdot \cdot ,{T}_{q}f\left(w\right)\right)$

$\nabla f\left(w\right)=\left({X}_{1}f\left(w\right),\cdot \cdot \cdot ,{X}_{p}f\left(w\right)\right)$

$R=\frac{\partial }{\partial a}$，类似地，对 $z\in S$$S$ 上函数的梯度及它的度量定义为

$\stackrel{˜}{\nabla }F\left(z\right)=\left({X}_{1}F\left(z\right),\cdot \cdot \cdot ,{X}_{p}F\left(z\right),{T}_{1}F\left(z\right),\cdot \cdot \cdot ,{T}_{q}F\left(z\right),RF\left(z\right)\right)$

${|\stackrel{˜}{\nabla }F\left(z\right)|}^{2}=\underset{k=1}{\overset{p}{\sum }}{|{X}_{k}F\left(z\right)|}^{2}+{|RF\left(z\right)|}^{2}$

$S$ 上的Poisson核 [19]，由 ${P}_{a}\left(\cdot \right)$ 表示，定义为

$P\left(z\right)=P\left(w,a\right)={P}_{a}\left(w\right)=\frac{c{a}^{d}}{{\left[{\left({a}^{2}+\frac{1}{4}{|{x}^{\prime }|}^{2}\right)}^{2}+{|{u}^{\prime }|}^{2}\right]}^{\frac{d}{2}}}$

$F\left(z\right)=f\ast P\left(z\right)={\int }_{N}f\left(n\right){P}_{a}\left({n}^{-1}w\right)\text{d}n$

ii) 记Z为 ${X}_{j}\left(j=1,\cdot \cdot \cdot ,p\right)$ 中的任意一个，则

$|Z{P}_{a}\left(w\right)|\le \left\{\begin{array}{l}C{a}^{-d-1},a\ge |w|\\ C{|w|}^{-d-1},a<|w|\end{array},|RP\left(z\right)|\le \left\{\begin{array}{l}C{a}^{-d-1},a\ge |w|\\ C{|w|}^{-d-1},a<|w|\end{array}$

3. ${Q}_{{r}_{1},{r}_{2}}$ 空间结构

${‖f\left(w\right)‖}_{BM{O}^{\beta }}^{2}=\underset{I}{\mathrm{sup}}\frac{1}{{|I|}^{\beta }}{\int }_{I}{|f\left(w\right)-f\left(I\right)|}^{2}\text{d}w<+\infty$

i) $\frac{1}{{|I|}^{\beta }}{\int }_{I}{|f\left(w\right)-c|}^{2}\text{d}w=\frac{1}{{|I|}^{\beta }}{\int }_{I}{|f\left(w\right)-f\left(I\right)|}^{2}\text{d}w+{|f\left(I\right)-c|}^{2}\cdot {|I|}^{1-\beta },c\in C$

ii) ${\int }_{I}{|f\left(w\right)-f\left(I\right)|}^{2}\text{d}w=\underset{c}{\mathrm{inf}}{\int }_{I}{|f\left(w\right)-c|}^{2}\text{d}w$

iii) $\frac{1}{{|I|}^{1+\beta }}{\int }_{I}{\int }_{I}{|f\left(w\right)-f\left(n\right)|}^{2}\text{d}w\text{d}n=\frac{2}{{|I|}^{\beta }}{\int }_{I}{|f\left(w\right)-f\left(I\right)|}^{2}\text{d}w$

iv) ${\int }_{I}{|f\left(w\right)-f\left(I\right)|}^{2}\text{d}w\le {\int }_{J}{|f\left(w\right)-f\left(J\right)|}^{2}\text{d}w$

$\begin{array}{c}{|f\left(w\right)-c|}^{2}={|f\left(w\right)-f\left(I\right)|}^{2}+\left[f\left(w\right)-f\left(I\right)\right]\stackrel{¯}{\left[f\left(I\right)-c\right]}\\ \text{\hspace{0.17em}}+\stackrel{¯}{\left[f\left(w\right)-f\left(I\right)\right]}\left[f\left(I\right)-c\right]+{|f\left(I\right)-c|}^{2}\end{array}$ (3.1)

${\int }_{I}\left[f\left(w\right)-f\left(I\right)\right]\text{d}w={\int }_{I}\stackrel{¯}{f\left(w\right)-f\left(I\right)}\text{ }\text{d}w=0$ (3.2)

$\frac{\text{1}}{{|I|}^{1+\beta }}{\int }_{I}{\int }_{I}{|f\left(w\right)-f\left(n\right)|}^{2}\text{d}w\text{d}n=\frac{\text{1}}{{|I|}^{\beta }}{\int }_{I}\text{d}n\left(\frac{\text{1}}{|I|}{\int }_{I}{|f\left(w\right)-f\left(n\right)|}^{2}\text{d}w\right)=\frac{\text{2}}{{|I|}^{\beta }}{\int }_{I}{|f\left(w\right)-f\left(I\right)|}^{2}\text{d}w$

${\int }_{I}{|f\left(w\right)-f\left(I\right)|}^{2}\text{d}w\le {\int }_{I}{|f\left(w\right)-f\left(J\right)|}^{2}\text{d}w\le {\int }_{J}{|f\left(w\right)-f\left(J\right)|}^{2}\text{d}w$

${‖f‖}_{{Q}_{{r}_{1},{r}_{2}}}^{2}=\underset{I}{\mathrm{sup}}\frac{1}{{|I|}^{{r}_{2}}}{\int }_{I}{\int }_{I}\frac{{|f\left(w\right)-f\left(n\right)|}^{2}}{{|{n}^{-1}w|}^{d{r}_{1}}}\text{d}w\text{d}n<+\infty$

ii) ${Q}_{\text{0},\text{2}}\left(N\right)=BM{O}^{1}\left(N\right)$

iii) ${Q}_{{r}_{1},{r}_{2}}\left(N\right)$$N$ 伸缩、平移、旋转下不变。

$\underset{I}{\mathrm{sup}}\frac{1}{{|I|}^{{r}_{2}}}{\int }_{{|n|}_{\infty }

$\begin{array}{c}{\int }_{I}{\int }_{I}\frac{{|f\left(w\right)-f\left(n\right)|}^{2}}{{|{n}^{-1}w|}^{d{r}_{1}}}\text{d}w\text{d}n={\int }_{I}{\int }_{I}\frac{{|f\left(w\right)-f\left(n\right)|}^{2}}{{|{n}^{-1}w|}^{d{{r}^{\prime }}_{1}}}\cdot {|{n}^{-1}w|}^{d\left({{r}^{\prime }}_{1}-{r}_{1}\right)}\text{d}w\text{d}n\\ \le C{|I|}^{{r}_{1}}{‖f‖}_{{Q}_{{{r}^{\prime }}_{1}+{{r}^{\prime }}_{2}}}^{2}\end{array}$

ii) 考虑 $f\in {Q}_{{r}_{\text{1}},{r}_{2}}$，显然当 ${r}_{2}<0$ 时有 $f\equiv C$。对于 ${r}_{1}>\frac{d+2}{d}$，我们利用反证法。假设 $f\in {Q}_{{r}_{\text{1}},{r}_{2}}\left(N\right)\cap {C}^{1}\left(N\right)$ 是实值函数且不恒为常数。那么，存在一个包含 ${X}_{\text{0}}$ 的锥体 $V\subset g$，使得 $|{n}^{-1}w|<\epsilon$ 时，任意的单位向量 $X\in V$$Xf\left(w\right)\ge \delta >0$。记 $D=\left\{\eta =\mathrm{exp}X:|\eta |<\epsilon ,X\in V\right\}$。若 $w,n\in I$，且 ${n}^{-1}w\in D$，则有 $f\left(w\right)-f\left(n\right)\ge \delta |{n}^{-1}w|$，以及有

${\int }_{{|n|}_{\infty }

iii) 由于 $\beta >0$$\beta +1={r}_{1}+{r}_{2}$，根据(i)，若 $0\le {r}_{1}<1$，从而 ${Q}_{{r}_{1},{r}_{2}}\left(N\right)\subset {Q}_{\text{0},\beta +1}\left(N\right)=BM{O}^{\beta }\left(N\right)$。反过来，若 $f\in BM{O}^{\beta }\left(N\right)$，对任意的方体 $I\subset N$$n\in N$ 且有 ${|n|}_{\infty }，我们可以得到

$\frac{1}{{|I|}^{{r}_{2}}}{\int }_{{|n|}_{\infty }

${r}_{1}<\text{0}$ 时，同样地由(i)，得到 $BM{O}^{\beta }\left(N\right)={Q}_{\text{0},\beta +1}\left(N\right)\subset {Q}_{{r}_{1},{r}_{2}}\left(N\right)$。那么，反过来，若 $f\in {Q}_{{r}_{\text{1}},{r}_{2}}\left(N\right)$，对任意方体 $I\subset N$，令 $E=\left\{\xi \in I|\mathrm{min}\left\{|{\xi }^{-1}w|,|{\xi }^{-1}n|\right\}>\frac{1}{8}l\left(I\right)\right\}$ 以及 $K\left(w,n,\xi \right)=\mathrm{min}{\left\{|{\xi }^{-1}w|,|{\xi }^{-1}n|\right\}}^{-d{r}_{1}}$，此时我们得到

$\begin{array}{c}{|I|}^{-\beta }{\int }_{I}{|f\left(w\right)-f\left(I\right)|}^{2}\text{d}w\le C{|I|}^{-1-\beta }{\int }_{I}{\int }_{I}{|f\left(w\right)-f\left(n\right)|}^{2}\text{d}w\text{d}n\\ \le C{|I|}^{-2-\beta +{r}_{1}}{\int }_{I}{\int }_{I}{\int }_{I}K\left(w,n,\xi \right){|f\left(w\right)-f\left(n\right)|}^{2}\text{d}w\text{d}n\text{d}\xi \\ \le C{|I|}^{-{r}_{2}}{\int }_{I}{\int }_{I}\frac{{|f\left(w\right)-f\left(\xi \right)|}^{2}}{{|{\xi }^{-1}w|}^{d{r}_{1}}}\text{d}w\text{d}\xi \\ \le C{‖f‖}_{{Q}_{{r}_{1},{r}_{2}}}^{2}\end{array}$

${‖{f}_{k}\to f‖}_{{Q}_{{r}_{1},{r}_{2}}}\le \underset{j\to +\infty }{\mathrm{lim}}\mathrm{sup}{‖{f}_{j}\to {f}_{k}‖}_{{Q}_{{r}_{1},{r}_{2}}}$

${f}_{k}\to f\in {Q}_{{r}_{1},{r}_{2}}\left(N\right)$

4. ${Q}_{{r}_{1},{r}_{2}}\left(N\right)$ 的刻画

${\int }_{S\left(I\right)}{|\nabla F\left(z\right)|}^{2}{a}^{d\left(1-{r}_{1}\right)+1}\text{d}z\le C{\int }_{J}{\int }_{J}\frac{|f\left(w\right)-f\left(n\right)|}{{|{n}^{-1}w|}^{d{r}_{1}}}\text{d}w\text{d}n+C{\left[l\left(I\right)\right]}^{2\left(d+1\right)-d{r}_{1}}{\left({\int }_{N\\frac{2}{3}J}\frac{|f\left(w\right)-f\left(J\right)|}{{|{w}_{0}^{-1}w|}^{d+1}}\text{d}w\right)}^{2}$

${\int }_{N\\frac{2}{3}J}\frac{f\left(w\right)-f\left(J\right)}{{|w|}^{d+1}}\text{d}w\le \underset{k=0}{\overset{+\infty }{\sum }}{\int }_{{3}^{k}J\{3}^{k-1}J}\frac{f\left(w\right)-f\left({3}^{k}J\right)}{{|w|}^{d+1}}\text{d}w+\underset{k=0}{\overset{+\infty }{\sum }}{\int }_{{3}^{k}J\{3}^{k-1}J}\frac{f\left({3}^{k}J\right)-f\left(J\right)}{{|w|}^{d+1}}\text{d}w\le C{\left[l\left(I\right)\right]}^{-1+\frac{\beta -1}{2}d}{‖f‖}_{BM{O}^{\beta }}$

${\int }_{S\left(I\right)}{|\nabla F\left(z\right)|}^{2}{a}^{d\left(1-{r}_{1}\right)+1}\text{d}z\le C{‖f‖}_{{Q}_{{r}_{1},{r}_{2}}}^{2}{|J|}^{{r}_{2}}+C{\left[l\left(I\right)\right]}^{d+\beta d-d{r}_{1}}{‖f‖}_{BM{O}^{\beta }}^{2}\le C{‖f‖}_{{Q}_{{r}_{1},{r}_{2}}}^{2}{|I|}^{{r}_{2}}\le C{|I|}^{{r}_{2}}$

${\int }_{{|n|}_{\infty }

$\begin{array}{c}|f\left(wn\right)-f\left(w\right)|\le |f\left(wn\right)-F\left(wn,|n|\right)|+|F\left(wn,|n|\right)-F\left(w,|n|\right)|+|F\left(w,|n|\right)-f\left(w\right)|\\ ={A}_{1}+{A}_{2}+{A}_{3}\end{array}$

$F\left(w,|n|\right)-f\left(w\right)={\int }_{0}^{|n|}\frac{\partial f\left(w,a\right)}{\partial a}\text{d}a={\int }_{0}^{|n|}\left(RF\right)\text{d}a$

${\int }_{{|n|}_{\infty }

${\int }_{I}{|{A}_{1}|}^{2}\text{d}w={\int }_{nI}{|{A}_{3}|}^{2}\text{d}w\le {\int }_{3I}{|{A}_{3}|}^{2}\text{d}w$

${\int }_{{|n|}_{\infty }

${A}_{2}\le C{\int }_{0}^{|n|}|\nabla F\left(wr{\sigma }_{n},|n|\right)|\text{d}r,{\sigma }_{n}\in B$

$\begin{array}{c}{\left({\int }_{I}{|{A}_{2}|}^{2}\text{d}w\right)}^{\frac{1}{2}}\le C{\int }_{0}^{|n|}{\left[{\int }_{I}{|\nabla F\left(wr{\sigma }_{n},|n|\right)|}^{2}\text{d}w\right]}^{\frac{1}{2}}\text{d}r\\ \le C|n|{\left[{\int }_{3I}{|\nabla F\left(w,|n|\right)|}^{2}\text{d}w\right]}^{\frac{1}{2}}\end{array}$

${\int }_{{|n|}_{\infty }

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