Holder不等式和Minkowski不等式.《索伯列夫空间引论》

$1<p<\infty$,$1/p+1/p’=1$,$p$与$p’$共轭,$y=x^{p-1}$,$x = y^{\frac{1}{p-1}}=y^{p’-1}$.

在直角坐标系中做出$y=x^{p-1}$的图形,$O$为原点,$A=(x_0,0)$,$B=(0,y_0)$,$E=(x_0,y_0)$,$AE$和曲线交于点$C$,$BE$和曲线交于点$D$.

面积$OAC$+面积$OCDB$$\ge$面积$OAEB$.于是有

\begin{gather*}\int_{0}^{x_0}{x^{p-1}dx} + \int_{0}^{y_0}{y^{p’-1}dy} \ge x_0y_0 \\\frac{x_0^p}{p} + \frac{y_0^{p’}}{p’} \ge x_0y_0\end{gather*}

定理(带权Holder不等式)

设$1<p<\infty$,$p$,$p’$是一对共轭指数,若

\[\int_{\Omega}{} < \infty, \int_{\Omega}{}<\infty,\]

则

\[\vert{\int_{\Omega}{F(x)G(x)P(x)}}\vert \le [\int_{\Omega}{\vert{F(x)}\vert^pP(x)dx}]^{1/p}[\int_{\Omega}{|G(x)|^{p’}P(x)dx}]^{1/p’}.\]

令

\[f(x)=\frac{|F(x)|}{[\int_{\Omega}{\vert{F(x)}\vert^pP(x)dx}]^{1/p}},g(x)=\frac{|G(x)|}{[\int_{\Omega}{|G(x)|^{p’}P(x)dx}]^{1/p’}},\]

注意到

\[\int_{\Omega}{|f(x)g(x)|P(x)dx} \le 1,\]

即可得到上述不等式.

这里$f(x)$,$g(x)$的构造原则,

\[\int_{\Omega}{|f(x)|^pP(x)dx}=1, \int_{\Omega}{|g(x)|^{p’}P(x)dx}=1,\]

系:设$p_1,p_2,\cdots,p_n>0$,且$p_1+p_2+\cdots+p_n=1$,如果有

\[\int_{\Omega}{|F_i(x)|^{1/p_i}P(x)dx}<\infty,i=1,2,\cdots,n,\]

则

\[|\int_{\Omega}{F_1F_2\cdots{}F_nPdx}| \le [\int_{\Omega}{|F_1|^{1/p_1}Pdx}]^{p_1}\cdots[\int_{\Omega}{|F_n|^{1/p_n}Pdx}]^{p_n}.\]

用归纳法:$p=\frac{1}{p_m}$,$p’=\frac{1}{p_1+\cdots+p_{m-1}}$.

当$p=1$时,$p’ \to \infty$,Holder不等式变为

\[|\int_{\Omega}{F(x)G(x)P(x)dx}| \le \text{esssup}_{x \in \Omega}{|G(x)|\int_{\Omega}{|F(x)|P(x)dx}}.\]

定理(带权Minkowski不等式)

设$1 \le p <\infty$,则

\[[\int_{\Omega}{F(x)+G(x)||^pP(x)dx}]^{1/p} \le [\int_{\Omega}{|F(x)|^pP(x)dx}]^{1/p}+[\int_{\Omega}{|G(x)|^pP(x)dx}]^{1/p}.\]

$p=1$较明显.

$1<p<\infty$的情形,用Holder不等式:

\[\begin{aligned}&\int_{\Omega}{|F(x)+G(x)|^pP(x)dx} \le \int_{\Omega}{|F(x)+G(x)|^{p-1}|F(x)|P(x)dx} \\&\ + \int_{\Omega}{|F(x)+G(x)|^{p-1}|G(x)|P(x)dx} \\&\le[\int_{\Omega}{|F(x)+G(x)|^{(p-1)p’}P(x)dx}]^{1/p’}[\int_{\Omega}{|F(x)|^pP(x)dx}]^{1/p}\\&\ + [\int_{\Omega}{|F(x)+G(x)|^{(p-1)p’}P(x)dx}]^{1/p’}[\int_{\Omega}{|G(x)|^pP(x)dx}]^{1/p}\\&=[\int_{\Omega}{|F(x)+G(x)|^pP(x)dx}]^{1/p’}\{[\int_{\Omega}{|F(x)|^pP(x)dx}]^{1/p}+[\int_{\Omega}{|G(x)|^pP(x)dx}]^{1/p}\}.\end{aligned}\]

系:设$1 \le p<\infty$,则成立

\[[\int_{\Omega}{|F_1(x)+\cdots+F_n(x)|^pP(x)dx}]^{1/p} \le [\int_{\Omega}{|F_1(x)|^pP(x)dx}]^{1/p} + \cdots + [\int_{\Omega}{|F_n(x)|^pP(x)dx}]^{1/p}.\]

用归纳法.

定理(带权Holder逆不等式)

设$0<p<1$,因而$p’=\frac{p}{p-1}<0$,若

\[\int_{\Omega}{|F(x)|^pP(x)dx} < \infty, 0<\int_{\Omega}{|G(x)|^{p’}P(x)dx}<\infty,\]

则

\[\int_{\Omega}{|F(x)G(x)|P(x)dx} \ge [\int_{\Omega}{|F(x)|^pP(x)dx}]^{1/p}[\int_{\Omega}{|G(x)|^{p’}P(x)dx}]^{1/p’}.\]

令

\[g(x)=|G(x)|^{-p},f(x)=|F(x)G(x)|^p,f(x)g(x)=|F(x)|^p,(f(x))^q=|F(x)G(x)|,\]

这里$q=1/p$.用$q’$表示$q$的共轭指数.$p’=-pq’$,因而

\[\begin{aligned}\int_{\Omega}{(f(x))^qP(x)dx} &= \int_{\Omega}{|F(x)G(x)|P(x)dx}<\infty \\\int_{\Omega}{(g(x))^{q’}P(x)dx} &= \int_{\Omega}{|G(x)|^{-pq’}P(x)dx}\\&=\int_{\Omega}{|G(x)|^{p’}P(x)dx}<\infty \\\int_{\Omega}{|F(x)|^pP(x)dx} &= \int_{\Omega}{f(x)g(x)P(x)dx}\\&\le[\int_{\Omega}{(f(x))^qP(x)dx}]^{1/q}[\int_{\Omega}{(g(x))^{q’}P(x)dx}]^{1/q’}\\&=[\int_{\Omega}{|F(x)G(x)|P(x)dx}]^{1/q}[\int_{\Omega}{|G(x)|^{p’}P(x)dx}]^{1/q’}\end{aligned}\]

除以$\int_{\Omega}{|G(x)|^{p’}P(x)dx}$,开$p=1/q$次方.

定理(带权Minkowski逆不等式)

设$0<p<1$,则

\[[\int_{\Omega}{(|F(x)|+|G(x)|)^{p}P(x)dx}]^{1/p} \ge [\int_{\Omega}{|F(x)|^{p}P(x)dx}]^{1/p} + [\int_{\Omega}{|G(x)|^{p}P(x)dx}]^{1/p}.\]

\[\begin{aligned}&\int_{\Omega}{(|F(x)|+|G(x)|)^pP(x)dx}\\&=\int_{\Omega}{(|F(x)|+|G(x)|)^{p-1}|F(x)|P(x)dx} \\&\ + \int_{\Omega}{(|F(x)|+|G(x)|)^{p-1}|G(x)|P(x)dx} \\&\ge[\int_{\Omega}{(|F(x)|+|G(x)|)^{(p-1)p’}P(x)dx}]^{1/p’}\\&\ \{[\int_{\Omega}{|F(x)|^pP(x)dx}]^{1/p}+[\int_{\Omega}{|G(x)|^pP(x)dx}]^{1/p}\}\end{aligned}\]