1920年, 英国数学家Hardy在研究双重Hilbert级数的收敛时, 建立了经典Hardy算子, 见文献[1].经典的一维Hardy算子的定义为$H(f)(x)=\frac{1}{x}\int^{x}_{0}f(t)dt, \ x\neq 0, $其中$1<p<+\infty, $ $f$是$(0, \infty)$上的非负局部可积函数.经典的一维Hardy不等式的形式为

$ \begin{equation*}\label{eh03} (\int^{\infty}_{0}(H(f)(x))^{p}dx)^{\frac{1}{p}}<\frac{p}{p-1}(\int^{\infty}_{0}f^{p}(x)dx)^{\frac{1}{p}}, \end{equation*} $

其中$1<p<\infty, $且$\frac{p}{p-1}$是最佳常数(见文献[2]).借助于简单的变量替换, Hardy算子$H$可以表示为$ H(f)(x)=\int^{1}_{0}f(tx)dt, \ x\neq 0.$

1984年, Carton-Lebrun和Fosset在文献[3]中首先定义了如下的加权Hardy算子:设$\omega: [0, 1]\rightarrow[0, \infty)$是一个函数, $f$是$\mathbb{R}^{n}$上的复值可测函数, $ H_{\omega}f(x)=\int^{1}_{0}f(tx)\omega(t)dt, \ x\in\mathbb{R}^{n}.$当$\omega(t)\equiv1$及$n=1$时, $H_{\omega}$是经典的一维Hardy算子$H.$ 2001年, Xiao在文献[4]中得到了加权Hardy算子在$L^p(\mathbb{R}^{n})$空间中的最佳常数.

定理A^[4] 设$p\in[1, \infty], $ $\omega: [0, 1]\rightarrow[0, \infty)$一个函数, 则$H_{\omega}$在$L^{p}(\mathbb{R}^{n})$上有界当且仅当$ C_{1}=\int_{0}^{1}t^{-\frac{n}{p}}\omega(t)dt<\infty, $且$ \|H_{\omega}f\|_{L^{p}(\mathbb{R}^{n})\rightarrow L^{p}(\mathbb{R}^{n})}=C_{1}. $同时, Xiao在文献[4]中研究了加权Hardy算子的共轭算子, 加权Cesàro算子, 其定义如下.设$\omega: [0, 1]\rightarrow[0, \infty)$是一个函数, $f$是$\mathbb{R}^{n}$上的复值可测函数.加权Cesàro算子的定义为$ V_{\omega}f(x)=\int^{1}_{0}f({x}/{t})t^{-n}\omega(t)dt, \ x\in\mathbb{R}^{n}. $当$\omega(t)\equiv1$及$n=1$时, $V_{\omega}$是经典的Cesàro算子.加权Hardy算子$H_{\omega}$和加权Cesàro算子$V_{\omega}$是一对共轭算子, 即

$ \int_{\mathbb{R}^{n}}g(x)H_{\omega}f(x)dx=\int_{\mathbb{R}^{n}}f(x)V_{\omega}g(x)dx, $

其中$f\in L^{p}(\mathbb{R}^{n}), \ g\in L^{q}(\mathbb{R}^{n}), \ 1<p<\infty, \ {1}/{p}+{1}/{q}=1.$因此$H_{\omega}$和$V_{\omega}$满足交换律: $H_{\omega}V_{\omega}=V_{\omega}H_{\omega}$.

受多线性算子理论的影响, 2015年, 傅等在文献[5]中将加权Hardy算子推广到多线性情形.设$\omega:[0, 1]\times\cdots\times[0, 1]\to [0, \infty)$是一个可积函数, $f_{1}, \cdots, f_{m}$是$\mathbb{R}^{n}$上的复值可测函数, 加权多线性Hardy算子定义为

$ H_{\omega}^{m}f(x)=\int^{1}_{0}\cdots\int^{1}_{0}f_{1}(t_{1}x)\cdots f_{m}(t_{m}x)\omega(t_{1}, \cdots, t_{m})dt_{1}\cdots t_{m}, \ x\in\mathbb{R}^{n}. $

当$m=2$时, $H_{\omega}^{2}$称为加权双线性Hardy算子.傅等在文献[5]得到了加权多线性Hardy算子$H_{\omega}^{m}$在$L^{p}(\mathbb{R}^{n})$空间中的有界性和最佳常数.由于$m\geq3$和$m=2$的情形并没有本质不同, 为此仅叙述双线性的结果.

定理B^[5] 令$1< p<\infty, 1<p_{1}, p_2<\infty$, $1/p=1/p_{1}+1/p_{2}.$则加权双线性Hardy算子$H^{2}_{\omega}$是从Lebesgue乘积空间$L^{p_{1}}(\mathbb{R}^{n})\times L^{p_{2}}(\mathbb{R}^{n})$到$L^{p}(\mathbb{R}^{n})$有界当且仅当

$ C_{2}=\int^{1}_{0}\int^{1}_{0}t_{1}^{-\frac{n}{p_{1}}} t_{2}^{-\frac{n}{p_{2}}}\omega(t_{1}, t_{2})dt_{1} dt_{2}<\infty $

且

$ \|H^{2}_{\omega}\|_{L^{p_{1}}(\mathbb{R}^{n})\times L^{p_{2}}(\mathbb{R}^{n})\to{L^{p}(\mathbb{R}^{n})}}=C_{2}. $

另一方面, 刘和江在文献[6]定义了加权双线性Cesàro算子.设$\omega:[0, 1]\times[0, 1]\rightarrow[0, \infty)$是一个函数, $f_{1}, f_{2}$是$\mathbb{R}^{n}$上的复值可测函数, 加权双线性Cesàro算子的定义为

$ V_{\omega}^{2}(f_{1}, f_{2})(x)=\int^{1}_{0}\int^{1}_{0}f_{1}({x}/{t_{1}})f_{2}({x}/{t_{2}})t_{1}^{-n}t_{2}^{-n}\omega(t_{1}, t_{2})dt_{1}t_{2}, \ x\in\mathbb{R}^{n}. $

很显然加权双线性Cesàro算子与加权双线性Hardy算子$H_{\omega}^{2}$并不是一对共轭算子.

众所周知, 加幂权$L^{p}$空间(记为$L^p(\mathbb{R}^n, |x|^{\alpha}dx)$)是一类重要的函数空间.那么加权双线性Hardy算子和加权双线性Cesàro算子在加幂权$L^{p}$空间中的有界性以及最佳常数是否可以得到呢?其最佳常数与底空间的维数$n$、幂权指标$\alpha$存在何种关系?研究结果将回答这些问题.

定理1 令$1\leq p<\infty, $ $1<p_{1}, p_{2}<\infty, $ $1/p=1/p_{1}+1/p_{2}, $ $\alpha=\alpha_{1}+\alpha_{2}.$则加权双线性Hardy算子$H^{2}_{\omega}$是从Lebesgue乘积空间$L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)$到$L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)$有界当且仅当

$ \begin{equation}\label{e1} C_{3}=\int^{1}_{0}\int^{1}_{0}t_{1}^{-\frac{n}{p_{1}}-\frac{\alpha_{1}}{p}}t_{2}^{-\frac{n}{p_{2}}-\frac{\alpha_{2}}{p}}\omega(t_{1}, t_{2})dt_{1}dt_{2}<\infty \end{equation} $

(2.1)

且

$ \|H^{2}_{\omega}\|_{L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}=C_{3}. $

注 在定理$1$中, 若取$\alpha_1=\alpha_2=\alpha=0, $则可以得到定理B.

证 假设(2.1)成立.由Minkowski不等式有

$ \begin{align*} \|H_{\omega}^{2}(f_{1}, f_{2})\|_{L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)} &=(\int_{\mathbb{R}^{n}}|\int_{0}^{1}\int_{0}^{1}f_{1}(t_{1}x)f_{2}(t_{2}x) \omega(t_{1}, t_{2})dt_{1}dt_{2}|^{p}|x|^{\alpha}dx)^{\frac{1}{p}}\\ &\leq\int_{0}^{1}\int_{0}^{1}(\int_{\mathbb{R}^{n}}|f_{1}(t_{1}x)f_{2}(t_{2}x)|^{p}|x|^{\alpha}dx)^{\frac{1}{p}}\omega(t_{1}, t_{2})dt_{1}dt_{2}. \end{align*} $

注意$\alpha=\alpha_1+\alpha_2, $利用Hölder不等式($1/p=1/p_{1}+1/p_{2}$)和变量替换$y_{i}=t_{i}x, \ i=1, 2, $得

$ \begin{align*} &\|H_{\omega}^{2}(f_{1}, f_{2})\|_{L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}\\ \leq&\int_{0}^{1}\int_{0}^{1}(\int_{\mathbb{R}^{n}}|f_{1}(t_{1}x)|^{p_{1}}|x|^{\frac{\alpha_{1}p_{1}}{p}}dx)^{\frac{1}{p_{1}}}(\int_{\mathbb{R}^{n}}|f_{2}(t_{2}x)|^{p_{2}}|x|^{\frac{\alpha_{2}p_{2}}{p}}dx)^{\frac{1}{p_{2}}}\omega(t_{1}, t_{2})dt_{1}dt_{2}\\ =&\int_{0}^{1}\int_{0}^{1}(\prod\limits_{i=1}^{2}(\int_{\mathbb{R}^{n}}|f_{i}(y_{i})|^{p_{i}}|y_{i}|^{\frac{\alpha_{i}p_{i}}{p}}dy_{i})^{\frac{1}{p_{i}}}t_{i}^{-(\frac{n}{p_{i}}+\frac{\alpha_{i}}{p})})\omega(t_{1}, t_{2})dt_{1}dt_{2}\\ =&(\prod\limits_{i=1}^{2}\|f_{i}\|_{L^{p_{i}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{i}p_{i}}{p}}dx)})\int_{0}^{1}\int_{0}^{1}(\prod\limits_{i=1}^{2}t_{i}^{-(\frac{n}{p_{i}}+\frac{\alpha_{i}}{p})})\omega(t_{1}, t_{2})dt_{1}dt_{2}. \end{align*} $

因此$H_{\omega}^{2}$是从Lebesgue乘积空间$L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)$到$L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)$上的有界算子, 且

$ \|H^{2}_{\omega}\|_{L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}\leq C_{3}. $

必要性 为了得到合适的下界, 需要选取特殊的函数.对于充分小的$\epsilon\in(0, 1), $取

$ \begin{eqnarray*}\label{e222} f_{1}^{\epsilon}(x)= \begin{cases} 0\ &|x| \leq 1, \\ |x|^{-\frac{n}{p_{1}}-\frac{\alpha_{1}}{p}-\frac{p_{2}}{p_{1}}\epsilon}\ &|x|>1 \end{cases}\quad f_{2}^{\epsilon}(x)= \begin{cases} 0\ &|x| \leq 1, \\ |x|^{-\frac{n}{p_{2}}-\frac{\alpha_{2}}{p}-\epsilon}\ &|x|>1. \end{cases} \end{eqnarray*} $

经过简单的计算得

$ \begin{equation*} \|f_{1}^{\epsilon}\|_{L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)}^{p_{1}}=\|f_{2}^{\epsilon}\|_{L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)}^{p_{2}}=\frac{|\mathbb{S}^{n-1}|}{p_{2}\epsilon}, \end{equation*} $

其中$|\mathbb{S}^{n-1}|$表示$n$维单位球面的面积, 有

$ \begin{align*} &\|H_{\omega}^{2}(f_{1}^{\epsilon}, f_{2}^{\epsilon})\|_{L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}\\ =&(\int_{\mathbb{R}^{n}}|\int_{0}^{1}\int_{0}^{1}f_{1}^{\epsilon}(t_{1}x)f_{2}^{\epsilon}(t_{2}x) \omega(t_{1}, t_{2})dt_{1}dt_{2}|^{p}|x|^{\alpha}dx)^{\frac{1}{p}}\\ =&(\int_{\mathbb{R}^{n}}|x|^{-n-p_{2}\epsilon}(\int_{\frac{1}{|x|}}^{1}\int_{\frac{1}{|x|}}^{1}t_{1}^{-\frac{n}{p_{1}}-\frac{\alpha_{1}}{p} -\frac{p_{2}\epsilon}{p_{1}}}t_{2}^{-\frac{n}{p_{2}}-\frac{\alpha_{2}}{p}-\epsilon}\omega(t_{1}, t_{2})dt_{1}dt_{2})^{p}dx)^{\frac{1}{p}}. \end{align*} $

借助简单的计算, 得到

$ \begin{align*} &\|H_{\omega}^{2}(f_{1}^{\epsilon}, f_{2}^{\epsilon})\|_{L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}^{p}\\ \geq&\int_{|x|>\frac{1}{\epsilon}}|x|^{-n-p_{2}\epsilon}(\int_{\epsilon}^{1}\int_{\epsilon}^{1}t_{1}^{-\frac{n}{p_{1}}-\frac{\alpha_{1}}{p}-\frac{p_{2}\epsilon}{p_{1}}}t_{2}^{-\frac{n}{p_{2}}-\frac{\alpha_{2}}{p}-\epsilon}\omega(t_{1}, t_{2})dt_{1}dt_{2})^{p}dx\\ =&\frac{\epsilon^{p_{2}\epsilon}|\mathbb{S}^{n-1}|}{p_{2}\epsilon}(\int_{\epsilon}^{1}\int_{\epsilon}^{1}t_{1}^{-\frac{n}{p_{1}}-\frac{\alpha_{1}}{p}-\frac{p_{2}\epsilon}{p_{1}}}t_{2}^{-\frac{n}{p_{2}}-\frac{\alpha_{2}}{p}-\epsilon}\omega(t_{1}, t_{2})dt_{1}dt_{2})^{p}\\ =&\epsilon^{p_{2}\epsilon}\prod\limits_{i=1}^{2}\|f_{i}^{\epsilon}\|_{L^{p_{i}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{i}p_{i}}{p}}dx)}^{p}\times(\int_{\epsilon}^{1}\int_{\epsilon}^{1}t_{1}^{-\frac{n}{p_{1}}-\frac{\alpha_{1}}{p}-\frac{p_{2}\epsilon}{p_{1}}}t_{2}^{-\frac{n}{p_{2}}-\frac{\alpha_{2}}{p}-\epsilon}\omega(t_{1}, t_{2})dt_{1}dt_{2})^{p}. \end{align*} $

因此

$ \begin{align*}&\|H^{2}_{\omega}\|_{L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}\\ \quad\geq&\epsilon^{\frac{p_{2}\epsilon}{p}}\int_{\epsilon}^{1}\int_{\epsilon}^{1}t_{1}^{-\frac{n}{p_{1}}-\frac{\alpha_{1}}{p}-\frac{p_{2}\epsilon}{p_{1}}}t_{2}^{-\frac{n}{p_{2}}-\frac{\alpha_{2}}{p}-\epsilon}\omega(t_{1}, t_{2})dt_{1}dt_{2}. \end{align*} $

令$\epsilon\to 0^{+}$时, 则$\epsilon^{\frac{p_{2}\epsilon}{p}}\to 1.$从而$\|H^{2}_{\omega}\|_{L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}\geq C_{3}.$

综上所证, 定理$1$得证.

定理2 令$1\leq p<\infty$, $1<p_{1}, p_{2}<\infty$, $1/p=1/p_{1}+1/p_{2}, $ $\alpha=\alpha_{1}+\alpha_{2}.$则加权双线性Ces$\grave{a}$ro算子$V^{2}_{\omega}$是从Lebesgue乘积空间$L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)$到$L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)$有界当且仅当

$ \begin{equation}\label{e2} C_{4}=\int^{1}_{0}\int^{1}_{0}t_{1}^{\frac{n(1-p_{1})}{p_{1}}+\frac{\alpha_{1}}{p}} t_{2}^{\frac{n(1-p_{2})}{p_{2}}+\frac{\alpha_{2}}{p}}\omega(t_{1}, t_{2})dt_{1}dt_{2}<\infty \end{equation} $

(2.2)

且

$ \|V^{2}_{\omega}\|_{L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}=C_{4}. $

证 定理$2$的证明类似于定理$1$的证明.假设(2.2)成立.由Minkowski不等式有

$ \begin{align*} &\|V_{\omega}^{2}(f_{1}, f_{2})\|_{L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}\\ =&(\int_{\mathbb{R}^{n}}|\int_{0}^{1}\int_{0}^{1}f_{1}({x}/{t_{1}})f_{2}({x}/{t_{2}})t_{1}^{-n}t_{2}^{-n}\omega(t_{1}, t_{2})dt_{1}dt_{2}|^{p}|x|^{\alpha}dx)^{\frac{1}{p}}\\ \leq&\int_{0}^{1}\int_{0}^{1}(\int_{\mathbb{R}^{n}}|f_{1}({x}/{t_{1}})f_{2}({x}/{t_{2}})|^{p} |x|^{\alpha}dx)^{\frac{1}{p}}t_{1}^{-n}t_{2}^{-n}\omega(t_{1}, t_{2})dt_{1}dt_{2}. \end{align*} $

由Hölder不等式和变量替换$y_{1}=x/t_{1}, y_{2}=x/t_{2}, $得

$ \begin{align*} &\|V_{\omega}^{2}(f_{1}, f_{2})\|_{L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}\\ \leq&\int_{0}^{1}\int_{0}^{1}(\int_{\mathbb{R}^{n}}|f_{1}({x}/{t_{1}})|^{p_{1}} |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)^{\frac{1}{p_{1}}}(\int_{\mathbb{R}^{n}}|f_{2}({x}/{t_{2}})|^{p_{2}} |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)^{\frac{1}{p_{2}}}t_{1}^{-n}t_{2}^{-n}\omega(t_{1}, t_{2})dt_{1}dt_{2}\\ =&\int_{0}^{1}\int_{0}^{1}(\prod\limits_{i=1}^{2}(\int_{\mathbb{R}^{n}}|f_{i}(y_{i})|^{p_{i}}|y_{i}|^{\frac{\alpha_{i}p_{i}}{p}} dy_{i})^{\frac{1}{p_{i}}}t_{i}^{\frac{n(1-p_{i})}{p_{i}}+\frac{\alpha_{i}}{p}})\omega(t_{1}, t_{2})dt_{1}dt_{2}\\ =&(\prod\limits_{i=1}^{2}\|f_{i}\|_{L^{p_{i}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{i}p_{i}}{p}}dx)})\int_{0}^{1}\int_{0}^{1}(\prod\limits_{i=1}^{2}t_{i}^{\frac{n(1-p_{i})}{p_{i}}+\frac{\alpha_{i}}{p}})\omega(t_{1}, t_{2})dt_{1}dt_{2}. \end{align*} $

因此$V_{\omega}^{2}$是从$L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)$到$L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)$上的有界算子, 且

$ \|V^{2}_{\omega}\|_{L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}\leq C_{4}. $

必要性 对充分小的$\epsilon\in(0, 1), $取$f_{1}^{\epsilon}, \ f_{2}^{\epsilon}$为定理$1$中如(2.2)所表示的函数, 则

$ \begin{align*} &\|V_{\omega}^{2}(f_{1}^{\epsilon}, f_{2}^{\epsilon})\|_{L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}\\ =&(\int_{\mathbb{R}^{n}}|\int_{0}^{1}\int_{0}^{1}f_{1}^{\epsilon}({x}/{t_{1}})f_{2}^{\epsilon}({x}/{t_{2}})t_{1}^{-n}t_{2}^{-n}\omega(t_{1}, t_{2})dt_{1}dt_{2}|^{p}|x|^{\alpha}dx)^{\frac{1}{p}}\\ =&(\int_{\mathbb{R}^{n}}|x|^{-n-p_{2}\epsilon}(\int_{0}^{|x|}\int_{0}^{|x|}t_{1}^{\frac{\alpha_{1}}{p}+\frac{n(1-p_{1})}{p_{1}}+\frac{p_{2}\epsilon}{p_{1}}}t_{2}^{\frac{\alpha_{2}}{p}+\frac{n(1-p_{2})}{p_{2}}+\epsilon}\omega(t_{1}, t_{2})dt_{1}dt_{2})^{p}dx)^{\frac{1}{p}}. \end{align*} $

注意到当$0<\epsilon<1$时, 若$|x|>1/{\epsilon}$时, 进一步计算得到

$ \begin{align*} &\|V_{\omega}^{2}(f_{1}^{\epsilon}, f_{2}^{\epsilon})\|_{L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}^{p}\\ \geq&\int_{|x|>\frac{1}{\epsilon}}|x|^{-n-p_{2}\epsilon}(\int_{0}^{1}\int_{0}^{1}t_{1}^{\frac{\alpha_{1}}{p}+\frac{n(1-p_{1})}{p_{1}}+\frac{p_{2}\epsilon}{p_{1}}}t_{2}^{\frac{\alpha_{2}}{p}+\frac{n(1-p_{2})}{p_{2}}+\epsilon}\omega(t_{1}, t_{2})dt_{1}dt_{2})^{p}dx\\ =&\frac{\epsilon^{p_{2}\epsilon}|\mathbb{S}^{n-1}|}{p_{2}\epsilon}(\int_{0}^{1}\int_{0}^{1}t_{1}^{\frac{\alpha_{1}}{p}+\frac{n(1-p_{1})}{p_{1}}+\frac{p_{2}\epsilon}{p_{1}}}t_{2}^{\frac{\alpha_{2}}{p}+\frac{n(1-p_{2})}{p_{2}}+\epsilon}\omega(t_{1}, t_{2})dt_{1}dt_{2})^{p}\\ =&\epsilon^{p_{2}\epsilon}\prod\limits_{i=1}^{2}\|f_{i}^{\epsilon}\|_{L^{p_{i}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{i}p_{i}}{p}}dx)}^{p}\times(\int_{0}^{1}\int_{0}^{1}t_{1}^{\frac{\alpha_{1}}{p}+\frac{n(1-p_{1})}{p_{1}}+\frac{p_{2}\epsilon}{p_{1}}}t_{2}^{\frac{\alpha_{2}}{p}+\frac{n(1-p_{2})}{p_{2}}+\epsilon}\omega(t_{1}, t_{2})dt_{1}dt_{2})^{p}. \end{align*} $

因此

$ \begin{align*} &\|V^{2}_{\omega}\|_{L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}\\ \quad \geq&\epsilon^{\frac{p_{2}\epsilon}{p}}\int_{0}^{1}\int_{0}^{1}t_{1}^{\frac{\alpha_{1}}{p}+\frac{n(1-p_{1})}{p_{1}}+\frac{p_{2}\epsilon}{p_{1}}}t_{2}^{\frac{\alpha_{2}}{p}+\frac{n(1-p_{2})}{p_{2}}+\epsilon}\omega(t_{1}, t_{2})dt_{1}dt_{2}. \end{align*} $

令$\epsilon\rightarrow0^{+}, $则$ \|V^{2}_{\omega}\|_{L^{p_{1}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}^{n}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}^{n}, |x|^{\alpha}dx)}\geq C_{4}. $

定理$2$得证.

下面首先给出两个常见的算子.设$0<\beta<1, $$f$是$[0, +\infty)$上一个局部可积函数, 则Riemann-Liouville分数积分算子$R_{\beta}$的定义为$R_{\beta}f(x)=\frac{1}{\Gamma(\beta)}\int_{0}^{x}f(t)(x-t)^{\beta-1}dt.$若$p>1, \ \beta>0, $则不等式(见文献[2, 定理3.29])

$ \begin{eqnarray*} (\int_{0}^{\infty}(\frac{R_{\beta}f(x)}{x^{\beta}})^{p}dx)^{1/p}<\frac{\Gamma(1-1/p)}{\Gamma(1+\beta-1/p)}(\int_{0}^{\infty}f^{p}(x)dx)^{1/p} \end{eqnarray*} $

成立, 其中$\Gamma(1-1/p)/(\Gamma(1+\beta-1/p))$是最佳常数.

设$0<\beta<1, $$f$是$[0, +\infty)$上一个局部可积函数, 则Weyl分数积分算子$W_{\beta}$的定义为

$ \begin{eqnarray*} W_{\beta}f(x)=\frac{1}{\Gamma(\beta)}\int_{x}^{\infty}f(t)(t-x)^{\beta-1}dt. \end{eqnarray*} $

若$p>1, \ \beta>0, $则不等式(见文献[2, 定理3.29])

$ (\int_{0}^{\infty}(W_{\beta}f(x))^{p}dx)^{1/p}<\frac{\Gamma(1/p)}{\Gamma(\beta+1/p)}(\int_{0}^{\infty}(x^{\beta}f(x))^{p}dx)^{1/p} $

成立, 其中$\Gamma(1/p)/\Gamma(\beta+1/p)$是最佳常数.

Weyl算子$W_{\beta}$和Riemann-Liouville算子$R_{\beta}$是一对共轭算子, 见文献[7].即

$ \int_{0}^{\infty}f(x)W_{\beta}g(x)dx=\int_{0}^{\infty}R_{\beta}f(x)g(x)dx, $

其中$f, g$都是$[0, +\infty)$上非负函数.

作为应用, 当$H^{2}_{\omega}$和$V^{2}_{\omega}$中的$\omega$取特殊的函数, 可以得到某些双线性平均算子.

(1) 令$n=1, \ 0\leq t_{1}, t_2<1, \ 0<\beta<2, \ |(1-t_{1}, 1-t_{2})|=\sqrt{(1-t_{1})^2+(1-t_{2})^2}, $取$ \omega(t_{1}, t_{2})=\frac{1}{\Gamma(\beta)|(1-t_{1}, 1-t_{2})|^{2-\beta}}, $有

$ \begin{equation*} H_{\omega}^{2}(f_{1}, f_{2})(x)=x^{-\beta}\textbf{R}_{\beta}^{2}(f_{1}, f_{2})(x), \ x>0, \end{equation*} $

这里$\textbf{R}_{\beta}^{2}$表示双线性Riemann-Liouvile算子, 定义如下

$ \begin{equation*} \textbf{R}_{\beta}^{2}(f_{1}, f_{2})(x)=\frac{1}{\Gamma(\beta)}\int_{0}^{x}\int_{0}^{x}\frac{f_{1}(t_{1})f_{2}(t_{2})}{|(x-t_{1}, x-t_{2})|^{2-\beta}}dt_{1}dt_{2}. \end{equation*} $

由定理$1$可得推论1.

推论1 若$1\leq p< \infty, 1<p_{1}, p_{2}<\infty, 1/p=1/p_{1}+1/p_{2}, \alpha=\alpha_{1}+\alpha_{2}.$则算子$\textbf{R}_{\beta}^{2}$是从Lebesgue乘积空间$L^{p_{1}}(\mathbb{R}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)$到$L^{p}(\mathbb{R}, |x|^{\alpha-p\beta}dx)$有界当且仅当

$ C_{5}=\frac{1}{\Gamma(\beta)}\int^{1}_{0}\int^{1}_{0}t_{1}^{-\frac{1}{p_{1}}-\frac{\alpha_{1}}{p}}t_{2}^{-\frac{1}{p_{2}}-\frac{\alpha_{2}}{p}}|(1-t_{1}, 1-t_{2})|^{\beta-2}dt_{1}dt_{2}<\infty, $

且$\|\textbf{R}_{\beta}^{2}\|_{L^{p_{1}}(\mathbb{R}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}, |x|^{\alpha-p\beta}dx)} =C_{5}.$

(2) 令$n=1, \ 0< t_{i}<1, \ i=1, 2, \ 0<\beta<2, $取$ \omega(t_{1}, t_{2})=\frac{1}{\Gamma(\beta)t_{1}t_{2}|(\frac{1}{t_{1}}-1, \frac{1}{t_{2}}-1)|^{2-\beta}}, $则得到

$ \begin{equation*} V_{\omega}^{2}(f_{1}, f_{2})(x)=x^{-\beta}W_{\beta}^{2}(f_{1}, f_{2})(x), \ x>0, \end{equation*} $

其中$W_{\beta}^{2}$表示双线性Weyl算子, 定义如下

$ \begin{equation*} W_{\beta}^{2}(f_{1}, f_{2})(x)=\frac{1}{\Gamma(\beta)}\int_{x}^{\infty}\int_{x}^{\infty}\frac{f_{1}(x_{1})f_{2}(x_{2})}{|(t_{1}-x, t_{2}-x)|^{2-\beta}}dt_{1}dt_{2}. \end{equation*} $

由定理$2, $可得推论2.

推论2 若$1\leq p< \infty, 1<p_{1}, p_{2}<\infty, 1/p=1/p_{1}+1/p_{2}, \alpha=\alpha_{1}+\alpha_{2}.$则算子$\textbf{W}_{\beta}^{2}$是从Lebesgue乘积空间$L^{p_{1}}(\mathbb{R}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)$到$L^{p}(\mathbb{R}, |x|^{\alpha-p\beta}dx)$有界当且仅当

$ C_{6}=\frac{1}{\Gamma(\beta)}\int^{1}_{0}\int^{1}_{0}t_{1}^{\frac{\alpha_{1}}{p}+\frac{1}{p_{1}}-2}t_{2}^{\frac{\alpha_{2}}{p}+\frac{1}{p_{2}}-2}|(\frac{1}{t_{1}}-1, \frac{1}{t_{2}}-1)|^{\beta-2}dt_{1}dt_{2}<\infty $

且$ \|\textbf{W}_{\beta}^{2}\|_{L^{p_{1}}(\mathbb{R}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}, |x|^{\alpha-p\beta}dx)} =C_{6}. $

(3) 令$n=1, \ 0\leq t_{1}, t_2<1, \ 0<\beta_{1}, \beta_{2}<1, \ $取$\omega(t_{i})=\frac{1}{\Gamma(\beta_{i})(1-t_{i})^{1-\beta_{i}}}, \ i=1, 2.$令$\omega(t_{1}, t_{2})=\omega(t_{1})\omega(t_{2}), $有

$ H_{\omega}^{2}(f_{1}, f_{2})(x)=x^{-(\beta_{1}+\beta_{2})}R_{\beta_{1}}(f_{1})(x)\cdot R_{\beta_{2}}(f_{2})(x), \ x>0, $

其中算子$R_{\beta_{i}}$的定义如下

$ R_{\beta_{i}}(f)(x)=\frac{1}{\Gamma(\beta_{i})}\int_{0}^{x}\frac{f(t_{i})}{(x-t_{i})^{1-\beta_{i}}}dt_{i}, \ x>0, \ i=1, 2. $

由定理$1, $可得推论3.

推论3 若$1\leq p< \infty, 1<p_{1}, p_{2}<\infty, 1/p=1/p_{1}+1/p_{2}, \alpha=\alpha_{1}+\alpha_{2}, \alpha_{i}<\frac{p(p_{i}-1)}{p_{i}}, i=1, 2.$则算子$R_{\beta_{1}}\cdot R_{\beta_{2}}$是从Lebesgue乘积空间$L^{p_{1}}(\mathbb{R}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)$到$L^{p}(\mathbb{R}, |x|^{\alpha-p(\beta_{1}+\beta_{2})}dx)$有界且

$ \begin{align*} &\|R_{\beta_{1}}\cdot R_{\beta_{2}}\|_{L^{p_{1}}(\mathbb{R}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}, |x|^{\alpha-p(\beta_{1}+\beta_{2})}dx)} =\prod\limits_{i=1}^{2}\frac{\Gamma(1-\frac{1}{p_{i}}-\frac{\alpha_{i}}{p})}{\Gamma(\beta_{i}+1-\frac{1}{p_{i}}-\frac{\alpha_{i}}{p})}. \end{align*} $

(4) 令$n=1, \ 0< t_{1}, t_2<1, \ 0<\beta_{1}, \beta_{2}<1.$取$\omega(t_{i})=\frac{1}{\Gamma(\beta_{i})t_{i}(\frac{1}{t_{i}}-1)^{1-\beta_{i}}}, \ i=1, 2.$令$\omega(t_{1}, t_{2})=\omega(t_{1})\omega(t_{2}), $有

$ V_{\omega}^{2}(f_{1}, f_{2})(x)=x^{-(\beta_{1}+\beta_{2})}W_{\beta_{1}}(f_{1})(x)\cdot W_{\beta_{2}}(f_{2})(x), \ x>0, $

其中算子$W_{\beta_{i}}$的定义为

$ W_{\beta_{i}}(f)(x)=\frac{1}{\Gamma(\beta_{i})}\int_{0}^{x}\frac{f(x_{i})}{(x-t_{i})^{1-\beta_{i}}}dt_{i}, \ x>0, \ i=1, 2. $

由定理$2, $可得推论4.

推论4 若$1\leq p < \infty, 1<p_{1}, p_{2}<\infty, 1/p=1/p_{1}+1/p_{2}, \alpha=\alpha_{1}+\alpha_{2}, \ \alpha_{i}>\frac{p(p_{i}\beta_{i}-1)}{p_{i}}, \ i=1, 2.$则算子$W_{\beta_{1}}\cdot W_{\beta_{2}}$是从Lebesgue乘积空间$L^{p_{1}}(\mathbb{R}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)$到$L^{p}(\mathbb{R}, |x|^{\alpha-p(\beta_{1}+\beta_{2})}dx)$有界且

$ \|W_{\beta_{1}}\cdot W_{\beta_{2}}\|_{L^{p_{1}}(\mathbb{R}, |x|^{\frac{\alpha_{1}p_{1}}{p}}dx)\times L^{p_{2}}(\mathbb{R}, |x|^{\frac{\alpha_{2}p_{2}}{p}}dx)\rightarrow L^{p}(\mathbb{R}, |x|^{\alpha-p(\beta_{1}+\beta_{2})}dx)} =\prod\limits_{i=1}^{2}\frac{\Gamma(\frac{1}{p_{i}}+\frac{\alpha_{i}}{p}-\beta_{i})}{\Gamma(\frac{1}{p_{i}}+\frac{\alpha_{i}}{p})}. $

值得注意的是, 推论$3$和推论$4$中的常数可以由Hölder不等式得到上界, 但是说明这两个常数是最佳的似乎并不是很显然的事情.