1 引言
设 f R n P Q
P f ( x ) = 1 | x | n ∫ | y | ≤ | x | f ( y ) d y , Q f ( x ) = ∫ | y | ≥ | x | f ( y ) | y | n d y , x ∈ R n .
高维 Hardy 算子是 Faris[1 ] 研究量子力学问题时定义的, 在数学物理许多问题的研究中起着重要作用. Chirst 与 Grafakos[2 ] 证明了若 1 < p < ∞ P : L p ( R n ) → L p ( R n ) Q : L p ( R n ) → L p ( R n )
设 b R n b ∈ B M O ( R n )
‖
其中上确界是对 \mathbb{R}^n B |B| B b_{B}=\frac{1}{|B|}\int_{B}b(x){\rm d}x .
设 1\leq q<\infty b\in L^q_{Loc}(\mathbb{R}^n) b\in CMO^q(\mathbb{R}^n)
\begin{align*} \|b\|_{CMO^q}=\sup_{t>0}\bigg(\frac{1}{|B(0,t)|}\int_{B(0,t)}|b(x)-b_{B(0,t)}|^q{\rm d}x\bigg)^{1/q}<\infty. \end{align*}
其中 B(0,t)=\{x\in \mathbb{R}^n: |x|\leq t\} . 注意到对 1\leq p<q<\infty CMO^q(\mathbb{R}^n)\subsetneqq CMO^p(\mathbb{R}^n) . CMO^p BMO [3 ] 定义的. 易见对 1\leq p<\infty BMO(\mathbb{R}^n)\subsetneqq CMO^p(\mathbb{R}^n) .
设 b \mathbb{R}^n P Q b
\begin{align*} P_bf(x)=\frac{1}{|x|^n}\int_{|y|\leq|x|}(b(x)-b(y))f(y){\rm d}y, \\ Q_bf(x)=\int_{|y|\geq|x|}\frac{(b(x)-b(y))f(y)}{|y|^{n}}{\rm d}y. \end{align*}
Fu 等[4 ] 给出了高维 Hardy 算子交换子的特征, Zhao 等[5 ] 得到了高维 Hardy 算子及其交换子的端点估计, 关于高维 Hardy 算子及其交换子的其它结果参见文献[6 ⇓ -8 ].
Duoandikoetxea, Martín-Reyes 与 Ombrosi[9 ] 为了研究高维 Hardy 算子 P 1<p<\infty A_{p,0} . 权函数 w\in A_{p,0}
\begin{eqnarray*} \sup_{t>0}\bigg(\frac{1}{|B(0,t)|}\int_{B(0,t)}w(y){\rm d}y\bigg)\bigg(\frac{1}{|B(0,t)|}\int_{B(0,t)}w(y)^{-p'/p}{\rm d}y\bigg)^{p/p'}<\infty, \end{eqnarray*}
并证明了 w\in A_{p,0} P Q L^p(w) P+Q: L^p(w)\rightarrow L^p(w)
设 1<p<\infty w [10 ] 定义的 A_p w\in A_p
\begin{align*} \sup_B\bigg(\frac{1}{|B|}\int_B w(x){\rm d}x\bigg)\bigg(\frac{1}{|B|}\int_B w(x)^{-p'/p}{\rm d}x\bigg)^{p/p'}<\infty, \end{align*}
其中上确界是对 \mathbb{R}^n B A_p\subset A_{p,0} .
关于高维 Hardy 算子的双权有界性, Drábek 等[11 ] 证明了高维 Hardy 算子 P L^p(v)\rightarrow L^p(u) (u,v) M_p 1\leq p<\infty
\begin{align*} \sup_{r>0}\bigg(\int_{|x|> r}|x|^{-np}u(x){\rm d}x\bigg)^{1/p}\bigg(\int_{|x|\leq r}v(x)^{1-p'}{\rm d}x\bigg)^{1/p'}<\infty. \end{align*}
Zhao 等[12 ] 对高维双线性 Hardy 算子也证明了类似的加权不等式.
本文利用 Long 等[13 ] , Li 等[14 ] 研究经典 Hardy 算子的方法,给出高维 Hardy 算子 {P} Q A_p
2 高维 Hardy 算子的双权不等式
本节我们给出高维 Hardy 算子 P Q
定理2.1 设 1<p<\infty r>1 (u,v) t>0
\begin{eqnarray*} \bigg(\frac{1}{|B(0,t)|}\int_{B(0,t)}u(y){\rm d}y\bigg)\bigg(\frac{1}{|B(0,t)|}\int_{B(0,t)}v(y)^{-rp'/p}{\rm d}y\bigg)^{p/rp'}\leq C<\infty, \end{eqnarray*}
则存在常数 C>0 f
\begin{eqnarray*} \int_{\mathbb{R}^n}|Pf(x)|^pu(x){\rm d}x\leq C \int_{\mathbb{R}^n}|f(x)|^pv(x){\rm d}x. \end{eqnarray*}
\begin{eqnarray*} &&\int_{\mathbb{R}^n}|{P}f(x)|^pu(x){\rm d}x \\ &\leq&\sum_{j=-\infty}^{\infty}\int_{{2^j}\leq |x|<{2^{j+1}}}\bigg(\frac{1}{2^{jn}}\int_{|y|\leq 2^{j+1}}|f(y)|{\rm d}y\bigg)^pu(x){\rm d}x\\ &=&\sum_{j=-\infty}^{\infty}\int_{{2^j}\leq |x|<{2^{j+1}}}\bigg(\frac{1}{2^{jn}}\sum_{k=-\infty}^{j}\int_{2^k\leq |y|< 2^{k+1}}|f(y)|{\rm d}y\bigg)^pu(x){\rm d}x\\ &\leq&\sum_{j=-\infty}^{\infty}\int_{{2^j}\leq|x|<{2^{j+1}}}u(x){\rm d}x\\ &&\times\bigg(\frac{1}{2^{jn}}\sum_{k=-\infty}^{j}\bigg(\int_{2^k\leq|y|< 2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg)^{1/p}\bigg(\int_{2^k\leq|y|< 2^{k+1}}v(y)^{-p'/p}{\rm d}y\bigg)^{1/p'}\bigg)^p\\ &\leq&\sum_{j=-\infty}^{\infty}\int_{|x|<{2^{j+1}}}u(x){\rm d}x\bigg(\frac{1}{2^{jn}}\sum_{k=-\infty}^{j}\bigg(\int_{2^k\leq|y|< 2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg)^{1/p}\\ &&\times\bigg(\int_{|y|< 2^{j+1}}v(y)^{-rp'/p}{\rm d}y\bigg)^{1/{rp'}}\bigg(\int_{2^k\leq|y|<2^{k+1}}1{\rm d}y\bigg)^{1/{r'p'}}\bigg)^p\\ &\leq& C\sum_{j=-\infty}^{\infty}\bigg(\sum_{k=-\infty}^j2^\frac{(k-j)n}{r'p'}\bigg(\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg)^{1/p}\bigg)^p\\ &\leq& C\sum_{j=-\infty}^{\infty}\bigg(\sum_{k=-\infty}^j2^\frac{(k-j)n}{2r'}\bigg)^{p/p'} \bigg(\sum_{k=-\infty}^j2^\frac{(k-j)pn}{2r'p'}\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg)\\ &\leq& C\sum_{j=-\infty}^{\infty}\sum_{k=-\infty}^j2^{\frac{(k-j)pn}{2r'p'}}\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\\ &=&C\sum_{k=-\infty}^{\infty}\sum_{j=k}^{\infty}2^{\frac{(k-j)pn}{2r'p'}}\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\\ &\leq& C\sum_{k=-\infty}^{\infty}\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\\ &=&C\int_{\mathbb{R}^n}|f(y)|^pv(y){\rm d}y. \end{eqnarray*}
定理2.2 设 1<p<\infty r>1 (u,v) t>0
\begin{eqnarray*} \bigg(\frac{1}{|B(0,t)|}\int_{B(0,t)}u(y)^r{\rm d}y\bigg)^{1/r}\bigg(\frac{1}{|B(0,t)|}\int_{B(0,t)}v(y)^{-p'/p}{\rm d}y\bigg)^{p/p'}\leq C<\infty. \end{eqnarray*}
则存在常数 C>0 f
\begin{eqnarray*} \int_{\mathbb{R}^n}|Qf(x)|^pu(x){\rm d}x\leq C \int_{\mathbb{R}^n}|f(x)|^pv(x){\rm d}x. \end{eqnarray*}
\begin{eqnarray*} &&\int_{\mathbb{R}^n}|Qf(x)|^pu(x){\rm d}x \\ &\leq&\sum_{j=-\infty}^{\infty}\int_{{2^j}\leq|x|<{2^{j+1}}}\bigg(\sum_{k=j}^{\infty}\frac{1}{2^{kn}}\int_{2^k\leq|y|<2^{k+1}}f(y){\rm d}y\bigg)^pu(x){\rm d}x\\ &\leq&\sum_{j=-\infty}^{\infty}\int_{{2^j}\leq|x|<{2^{j+1}}}u(x){\rm d}x\\ &&\times\bigg(\sum_{k=j}^{\infty}\frac{1}{2^{kn}}\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg]^{1/p}\bigg[\int_{2^k\leq|y|<2^{k+1}}v(y)^{-p/p'}{\rm d}y\bigg]^{1/p'}\bigg)^p\\ &\leq&C\sum_{j=-\infty}^{\infty}2^{\frac{jn}{r'}}\bigg(\int_{|x|<{2^{j+1}}}u(x)^r{\rm d}x\bigg)^{1/r}\\ &&\times\bigg(\sum_{k=j}^{\infty}\frac{1}{2^{kn}}\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg]^{1/p}\bigg[\int_{|y|<2^{k+1}}v(y)^{-p/p'}{\rm d}y\bigg]^{1/p'}\bigg)^p\\ &\leq& C\sum_{j=-\infty}^{\infty}\bigg(\sum_{k=j}^{\infty}2^\frac{(j-k)n}{r'p}\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg]^{1/p}\bigg)^p\\ &\leq& C\sum_{j=-\infty}^{\infty}\bigg(\sum_{k=j}^{\infty}2^\frac{(j-k)np'}{2r'p}\bigg)^{p/p'} \bigg(\sum_{k=j}^{\infty}2^\frac{(j-k)n}{2r'}\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg)\\ &\leq& C\sum_{j=-\infty}^{\infty}\sum_{k=j}^{\infty}2^{\frac{(j-k)n}{2r'}}\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\\ &=& C\sum_{k=-\infty}^{\infty}\sum_{j=-\infty}^{k}2^{\frac{(j-k)n}{2r'}}\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\\ &\leq& C\sum_{k=-\infty}^{\infty}\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\\ &=& C\int_{\mathbb{R}^n}|f(y)|^pv(y){\rm d}y. \end{eqnarray*}
3 高维 Hardy 算子交换子的双权有界性
本节我们给出高维 Hardy 算子 P Q CMO P_b Q_b
引理3.1 [4 ] 设 b\in{CMO}^{1}(\mathbb{R}^n) j, k\in \mathbb{Z}
\begin{align*} |b(x)-b_{B(0,2^{j})}|\leq|b(x)-b_{B(0,2^{k})}|+C|j-k|\|b\|_{{\rm CMO}^1}. \end{align*}
定理3.1 设 1<p<\infty b\in {CMO}^{r'\max\{p,p'\}} r>1 (u,v) t>0
\begin{eqnarray*} \bigg(\frac{1}{|B(0,t)|}\int_{B(0,t)}u(y)^rdy\bigg)^{1/r}\bigg(\frac{1}{|B(0,t)|}\int_{B(0,t)}v(y)^{-rp'/p}{\rm d}y\bigg)^{p/rp'}\leq C, \end{eqnarray*}
则存在常数 C>0 f
\begin{eqnarray*} \int_{\mathbb{R}^n}|P_bf(x)|^pu(x){\rm d}x\leq C\int_{\mathbb{R}^n}|f(x)|^pv(x){\rm d}x, \\ \int_{\mathbb{R}^n}|Q_bf(x)|^pu(x){\rm d}x\leq C\int_{\mathbb{R}^n}|f(x)|^pv(x){\rm d}x. \end{eqnarray*}
\begin{eqnarray*} &&\int_{\mathbb{R}^n}|P_bf(x)|^pu(x){\rm d}x \\ &=&\int_{\mathbb{R}^n}\bigg|\frac{1}{|x|^n}\int_{|y|\leq|x|}(b(x)-b(y))f(y){\rm d}y\bigg|^pu(x){\rm d}x\\ &\leq&\sum_{j=-\infty}^{\infty}\int_{{2^j}\leq|x|<{2^{j+1}}}\bigg(\frac{1}{|x|^n}\int_{|y|\leq|x|}|b(x)-b(y)||f(y)|{\rm d}y\bigg)^pu(x){\rm d}x\\ &\leq&\sum_{j=-\infty}^{\infty}\int_{{2^j}\leq|x|<{2^{j+1}}}\bigg(\frac{1}{2^{jn}}\sum_{k=-\infty}^j\int_{2^k\leq|y|<2^{k+1}}|b(x)-b(y)||f(y)|{\rm d}y\bigg)^pu(x){\rm d}x\\ &\leq&\sum_{j=-\infty}^{\infty}\int_{{2^j}\leq|x|<{2^{j+1}}} \bigg(\frac{1}{2^{jn}}\sum_{k=-\infty}^j\int_{2^k\leq|y|<2^{k+1}}\Big[|b(x)-b_{B(0,2^{j+1}}||f(y)|\\ &&+|b(y)-b_{B(0,2^{j+1})}||f(y)|\Big]{\rm d}y\bigg)^pu(x){\rm d}x\\ &\leq&2^{p/p'}\sum_{j=-\infty}^{\infty}\int_{{2^j}\leq|x|<{2^{j+1}}}\bigg(\frac{1}{2^{jn}}\sum_{k=-\infty}^j\int_{2^k\leq|y|<2^{k+1}}|b(x)-b_{B(0,2^{j+1})}||f(y)|{\rm d}y\bigg)^pu(x){\rm d}x\\ &&+2^{p/p'}\sum_{j=-\infty}^{\infty}\int_{{2^j}\leq|x|<{2^{j+1}}}\bigg(\frac{1}{2^{jn}}\sum_{k=-\infty}^j\int_{2^k\leq|y|<2^{k+1}}|b(y)-b_{B(0,2^{j+1})}||f(y)|{\rm d}y\bigg)^pu(x){\rm d}x\\ &=&{\rm I}+{\rm II}. \end{eqnarray*}
对于 {\rm I}
\begin{eqnarray*} {\rm I}&=&2^{p/p'}\sum_{j=-\infty}^{\infty}\frac{1}{2^{jpn}}\int_{2^j\leq|x|<2^{j+1}}|b(x)-b_{B(0,2^{j+1})}|^pu(x){\rm d}x\bigg(\sum_{k=-\infty}^j\int_{2^k\leq|y|<2^{k+1}}|f(y)|{\rm d}y\bigg)^p\\ &\leq&2^{p/p'}\sum_{j=-\infty}^{\infty}\frac{1}{2^{jpn}}\bigg(\int_{2^j\leq|x|<2^{j+1}}|b(x)-b_{B(0,2^{j+1})}|^{pr'}{\rm d}x\bigg)^{1/r'} \bigg(\int_{2^j\leq|x|<2^{j+1}}u(x)^r{\rm d}x\bigg)^{1/r}\\ &&\times\bigg(\sum_{k=-\infty}^j\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg]^{1/p} \bigg[\int_{2^k\leq|y|<2^{k+1}}v(y)^{-p'/p}{\rm d}y\bigg]^{1/p'}\bigg)^p\\ &\leq&C\|b\|^p_{{\rm CMO}^{pr'}}\sum_{j=-\infty}^{\infty}2^{jn/r'-jpn} \bigg(\int_{|x|<2^{j+1}}u(x)^r{\rm d}x\bigg)^{1/r}\\ &&\times \bigg(\sum_{k=-\infty}^j\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg]^{1/p}\\ &&\times\bigg[\int_{2^k\leq|y|<2^{k+1}}v(y)^{-rp'/p}{\rm d}y\bigg]^{1/rp'}\bigg[\int_{2^k\leq|y|<2^{k+1}}1{\rm d}y\bigg]^{1/r'p'}\bigg)^p\\ &\leq& C\|b\|^p_{{\rm CMO}^{pr'}}\sum_{j=-\infty}^{\infty}2^{jn/r'-jpn} \bigg(\int_{|x|<2^{j+1}}u(x)^r{\rm d}x\bigg)^{1/r}\\ &&\times\bigg(\sum_{k=-\infty}^j2^{\frac{kn}{r'p'}}\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg]^{1/p} \bigg[\int_{|y|<2^{j+1}}v(y)^{-rp'/p}{\rm d}y\bigg]^{1/rp'}\bigg)^p\\ &\leq& C\|b\|^p_{{\rm CMO}^{pr'}}\sum_{j=-\infty}^{\infty}\bigg(\sum_{k=-\infty}^j2^{\frac{(k-j)n}{2r'p'}} \cdot2^{\frac{(k-j)n}{2r'p'}}\bigg(\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg)^{1/p}\bigg)^p\\ &\leq& C\|b\|^p_{{\rm CMO}^{pr'}}\sum_{j=-\infty}^{\infty}\bigg(\sum_{k=-\infty}^j2^\frac{(k-j)n}{2r'}\bigg)^{p/p'} \bigg(\sum_{k=-\infty}^j2^{\frac{(k-j)pn}{2r'p'}}\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg)\\ &\leq& C\|b\|^p_{{\rm CMO}^{pr'}}\int_{\mathbb{R}^n}|f(y)|^pv(y){\rm d}y. \end{eqnarray*}
\begin{eqnarray*} {\rm II}&\leq&2^{2p/p'}\sum_{j=-\infty}^{\infty}\frac{1}{2^{jpn}}\int_{2^j\leq|x|<2^{j+1}}\bigg( \sum_{k=-\infty}^j\int_{2^k\leq|y|<2^{k+1}}|b(y)-b_{B(0,2^{k+1})}||f(y)|{\rm d}y\bigg)^pu(x){\rm d}x\\ &&+C\sum_{j=-\infty}^{\infty}\frac{1}{2^{jpn}}\int_{2^j\leq|x|<2^{j+1}}\bigg(\sum_{k=-\infty}^j\int_{2^k\leq|y|<2^{k+1}}(j-k)\|b\|_{{\rm CMO}^{1}}|f(y)|{\rm d}y\bigg)^pu(x){\rm d}x\\ &= &{\rm II_1+II_2}. \end{eqnarray*}
对于 {\rm II_1}
\begin{eqnarray*} {\rm II_1} &=&2^{2p/p'}\sum_{j=-\infty}^{\infty}\frac{1}{2^{jpn}}\int_{2^j\leq|x|<2^{j+1}}u(x){\rm d}x \bigg(\sum_{k=-\infty}^j\int_{2^k\leq|y|<2^{k+1}}|b(y)-b_{B(0,2^{k+1})}||f(y)|{\rm d}y\bigg)^p\\ &=&2^{2p/p'}\sum_{j=-\infty}^{\infty}\frac{1}{2^{jpn}}\int_{2^j\leq|x|<2^{j+1}}u(x){\rm d}x \bigg(\sum_{k=-\infty}^j\int_{2^k\leq|y|<2^{k+1}}|f(y)|v(y)^{1/p}\\ &&\times|b(y)-b_{B(0,2^{k+1})}|v(y)^{-1/p}{\rm d}y\bigg)^p\\ &\leq&2^{2p/p'}\sum_{j=-\infty}^{\infty}\frac{1}{2^{jpn}}\int_{|x|<2^{j+1}}u(x){\rm d}x\bigg(\sum_{k=-\infty}^j\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^{p}v(y){\rm d}y\bigg]^{1/p}\\ &&\times\bigg[\int_{2^k\leq|y|<2^{k+1}}|b(y)-b_{B(0,2^{k+1})}|^{r'p'}{\rm d}y\bigg]^{1/r'p'} \bigg[\int_{|y|<2^{k+1}}v(y)^{-rp'/p}{\rm d}y\bigg]^{1/rp'}\bigg)^p\\ &\leq& C\|b\|^p_{{\rm CMO}^{p'r'}}\sum_{j=-\infty}^{\infty}\frac{1}{2^{jpn}}\int_{|x|<2^{j+1}}u(x){\rm d}x\\ &&\times\bigg(\sum_{k=-\infty}^j2^{\frac{kn}{r'p'}}\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^{p}v(y){\rm d}y\bigg]^{1/p} \bigg[\int_{|y|<2^{j+1}}v(y)^{-rp'/p}{\rm d}y\bigg]^{1/rp'}\bigg)^p\\ &\leq& C\|b\|^p_{{\rm CMO}^{p'r'}}\sum_{j=-\infty}^{\infty} \bigg(\sum_{k=-\infty}^j2^{\frac{(k-j)n}{r'p'}}\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^pv(y){\rm d}y\bigg]^{1/p}\bigg)^p\\ &\leq& C\|b\|^p_{{\rm CMO}^{p'r'}}\int_{\mathbb{R}^n}|f(y)|^pv(y){\rm d}y. \end{eqnarray*}
\begin{eqnarray*} {\rm II_2} &\leq& C\|b\|^p_{{\rm CMO}^{1}}\sum_{j=-\infty}^{\infty}\frac{1}{2^{jpn}}\int_{2^j\leq|x|<2^{j+1}}u(x){\rm d}x\\ &&\times\bigg(\sum_{k=-\infty}^j(j-k)\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^{p}v(y){\rm d}y\bigg]^{1/p} \bigg[\int_{2^k\leq|y|<2^{k+1}}v(y)^{-rp'/p}{\rm d}y\bigg]^{1/rp'}\\ &&\times\bigg[\int_{2^k\leq|y|<2^{k+1}}1{\rm d}y\bigg]^{1/r'p'}\bigg)^p\\ &\leq& C\|b\|^p_{{\rm CMO}^{1}}\sum_{j=-\infty}^{\infty}\frac{1}{2^{jpn}}\int_{|x|<2^{j+1}}u(x){\rm d}x \bigg(\sum_{k=-\infty}^j(j-k)2^\frac{kn}{r'p'}\\ &&\times\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^{p}v(y){\rm d}y\bigg]^{1/p}\bigg[\int_{|y|<2^{k+1}}v(y)^{-rp'/p}{\rm d}y\bigg]^{1/rp'}\bigg)^p\\ &\leq& C\|b\|^p_{{\rm CMO}^{p'r'}}\sum_{j=-\infty}^{\infty}\frac{1}{2^{jpn}}\int_{|x|<2^{j+1}}u(x){\rm d}x\bigg(\int_{|y|<2^{j+1}}v(y)^{-rp'/p}{\rm d}y\bigg)^{p/rp'}\\ &&\times \bigg(\sum_{k=-\infty}^j(j-k)2^\frac{kn}{r'p'}\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^{p}v(y){\rm d}y\bigg]^{1/p}\bigg)^p\\ &\leq& C\|b\|^p_{{\rm CMO}^{p'r'}}\sum_{j=-\infty}^{\infty}\bigg(\sum_{k=-\infty}^j(j-k)2^\frac{(k-j)n}{r'p'}\bigg[\int_{2^k\leq|y|<2^{k+1}}|f(y)|^{p}v(y){\rm d}y\bigg]^{1/p}\bigg)^p\\ &\leq& C\|b\|^p_{{\rm CMO}^{p'r'}}\sum_{k=-\infty}^{\infty}\sum_{j=k}^{\infty}2^\frac{(k-j)np}{2r'p'}\int_{2^k\leq|y|<2^{k+1}}|f(y)|^{p}v(y){\rm d}y\\ &\leq& C\|b\|^p_{{\rm CMO}^{p'r'}}\sum_{k=-\infty}^{\infty}\int_{2^k\leq|y|<2^{k+1}}|f(y)|^{p}v(y){\rm d}y\\ &\leq& C\|b\|^p_{{\rm CMO}^{p'r'}}\int_{\mathbb{R}^n}|f(y)|^{p}v(y){\rm d}y. \end{eqnarray*}
用类似的方法可证明 Q_b
当 u=v=w\in A_p A_p
推论3.1 1<p<\infty b\in {CMO}^{\max\{p,p'\}} w\in A_p
\begin{eqnarray*} \int_{\mathbb{R}^n}|P_bf(x)|^pw(x){\rm d}x\leq C\int_{\mathbb{R}^n}|f(x)|^pw(x){\rm d}x,\\ \int_{\mathbb{R}^n}|Q_bf(x)|^pw(x){\rm d}x\leq C\int_{\mathbb{R}^n}|f(x)|^pw(x){\rm d}x. \end{eqnarray*}
注意, 当 1<p<\infty A_p\subsetneqq A_{p,0} A_{p,0} w\in A_{p,0} P_b Q_b L^p(w)
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... 高维 Hardy 算子是 Faris[1 ] 研究量子力学问题时定义的, 在数学物理许多问题的研究中起着重要作用. Chirst 与 Grafakos[2 ] 证明了若 1<p<\infty P: L^p(\mathbb{R}^n)\rightarrow L^p(\mathbb{R}^n) Q: L^p(\mathbb{R}^n)\rightarrow L^p(\mathbb{R}^n)
Best constants for two non-convolution inequalities
1
1995
... 高维 Hardy 算子是 Faris[1 ] 研究量子力学问题时定义的, 在数学物理许多问题的研究中起着重要作用. Chirst 与 Grafakos[2 ] 证明了若 1<p<\infty P: L^p(\mathbb{R}^n)\rightarrow L^p(\mathbb{R}^n) Q: L^p(\mathbb{R}^n)\rightarrow L^p(\mathbb{R}^n)
The central BMO
1
1995
... 其中 B(0,t)=\{x\in \mathbb{R}^n: |x|\leq t\} . 注意到对 1\leq p<q<\infty CMO^q(\mathbb{R}^n)\subsetneqq CMO^p(\mathbb{R}^n) . CMO^p BMO [3 ] 定义的. 易见对 1\leq p<\infty BMO(\mathbb{R}^n)\subsetneqq CMO^p(\mathbb{R}^n) . ...
Characterization for commutators of n
2
2007
... Fu 等[4 ] 给出了高维 Hardy 算子交换子的特征, Zhao 等[5 ] 得到了高维 Hardy 算子及其交换子的端点估计, 关于高维 Hardy 算子及其交换子的其它结果参见文献[6 ⇓ -8 ]. ...
... 引理3.1 [4 ] 设 b\in{CMO}^{1}(\mathbb{R}^n) j, k\in \mathbb{Z}
Endpoint estimates for n-dimensional Hardy operators and their commutators
1
2012
... Fu 等[4 ] 给出了高维 Hardy 算子交换子的特征, Zhao 等[5 ] 得到了高维 Hardy 算子及其交换子的端点估计, 关于高维 Hardy 算子及其交换子的其它结果参见文献[6 ⇓ -8 ]. ...
Some recent progress of n
1
2013
... Fu 等[4 ] 给出了高维 Hardy 算子交换子的特征, Zhao 等[5 ] 得到了高维 Hardy 算子及其交换子的端点估计, 关于高维 Hardy 算子及其交换子的其它结果参见文献[6 ⇓ -8 ]. ...
Sharp bounds for m
1
2012
... Fu 等[4 ] 给出了高维 Hardy 算子交换子的特征, Zhao 等[5 ] 得到了高维 Hardy 算子及其交换子的端点估计, 关于高维 Hardy 算子及其交换子的其它结果参见文献[6 ⇓ -8 ]. ...
Hardy's integral inequality for commutators of Hardy operators
1
2006
... Fu 等[4 ] 给出了高维 Hardy 算子交换子的特征, Zhao 等[5 ] 得到了高维 Hardy 算子及其交换子的端点估计, 关于高维 Hardy 算子及其交换子的其它结果参见文献[6 ⇓ -8 ]. ...
Calderòn weights as Muckenhoupt weights
1
2013
... Duoandikoetxea, Martín-Reyes 与 Ombrosi[9 ] 为了研究高维 Hardy 算子 P 1<p<\infty A_{p,0} . 权函数 w\in A_{p,0}
Weighted norm inequalities for the Hardy maximal function
1
1972
... 设 1<p<\infty w [10 ] 定义的 A_p w\in A_p
Higher dimensional Hardy inequality
1
1997
... 关于高维 Hardy 算子的双权有界性, Drábek 等[11 ] 证明了高维 Hardy 算子 P L^p(v)\rightarrow L^p(u) (u,v) M_p 1\leq p<\infty
M_p \mathbb{R}^n
1
2014
... Zhao 等[12 ] 对高维双线性 Hardy 算子也证明了类似的加权不等式. ...
Commutators of Hardy operators
1
2002
... 本文利用 Long 等[13 ] , Li 等[14 ] 研究经典 Hardy 算子的方法,给出高维 Hardy 算子 {P} Q A_p
Two-Weight inequalities for Hardy operator and commutators
1
2015
... 本文利用 Long 等[13 ] , Li 等[14 ] 研究经典 Hardy 算子的方法,给出高维 Hardy 算子 {P} Q A_p
