Lᵖ双対性を証明する議論(duality argument)について

$x:=(x_1,x_2)$とする．Hölderの不等式より，任意の$v\in L^{p_1^{\prime }}({\mathrm{\Omega }}_1;L^{p_2^{\prime }}({\mathrm{\Omega }}_2))$に対して，

$$\begin{array}{rl}& \underset{\Vert u{\Vert }_{p_1,p_2}=1}{sup}\left|{\int }_{{\mathrm{\Omega }}_1}\left({\int }_{{\mathrm{\Omega }}_2}u(x)v(x)d{x}_2\right)d{x}_1\right|\\ \le & \underset{\Vert u{\Vert }_{p_1,p_2}=1}{sup}{\int }_{{\mathrm{\Omega }}_1}\left|{\int }_{{\mathrm{\Omega }}_2}u(x)v(x)d{x}_2\right|d{x}_1\\ \le & \underset{\Vert u{\Vert }_{p_1,p_2}=1}{sup}{\int }_{{\mathrm{\Omega }}_1}\Vert u(x_1,\cdot ){\Vert }_{p_1}\Vert v(x_1,\cdot ){\Vert }_{p_1^{\prime }}d{x}_1\\ \le & \underset{\Vert u{\Vert }_{p_1,p_2}=1}{sup}\Vert u{\Vert }_{p_1,p_2}\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}=\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}\end{array}$$

だから，あとは

$$\begin{array}{r}\underset{\Vert u{\Vert }_{p_1,p_2}=1}{sup}\left|{\int }_{{\mathrm{\Omega }}_1}\left({\int }_{{\mathrm{\Omega }}_2}u(x)v(x)d{x}_2\right)d{x}_1\right|\ge \Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}\end{array}$$

を示せば[$L^p{L}^q$空間の双対性]が従う．これを

• $p_1=p_2=1$のとき
• $p_1=1$, $p_2\in (1,\mathrm{\infty })$のとき
• $p_1,p_2\in (1,\mathrm{\infty })$のとき

に分けて証明する．

[１]　$p_1=p_2=1$のとき，$p_1^{\prime }=p_2^{\prime }=\mathrm{\infty }$である．

任意の$ϵ>0$, $i\in \{1,2\}$に対して，ある可測集合${\mathrm{\Omega }}_i^{\prime }\subset \mathrm{\Omega }$で$0<|{\mathrm{\Omega }}_i^{\prime }|<\mathrm{\infty }$を満たすものが存在して，$x\in {\mathrm{\Omega }}_1^{\prime }\times {\mathrm{\Omega }}_2$が成り立つなら

$$\begin{array}{r}|v(x)|>\Vert v(x_1,\cdot ){\Vert }_{L^{\mathrm{\infty }}({\mathrm{\Omega }}_2)}-ϵ>\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}-2ϵ\end{array}$$

が成り立つ．このとき，$w:{\mathrm{\Omega }}_1\times {\mathrm{\Omega }}_2\to \mathbb{C}$を

$$\begin{array}{r}w(x):=\frac{\overline{v(x)}}{|{\mathrm{\Omega }}_1^{\prime }||{\mathrm{\Omega }}_2^{\prime }|v(x)}{\mathbb{I}}_{{\mathrm{\Omega }}_1^{\prime }\times {\mathrm{\Omega }}_2^{\prime }}(x)\end{array}$$

で定めると

$$\begin{array}{rl}\Vert w{\Vert }_{p_1,p_2}=& {\int }_{{\mathrm{\Omega }}_1}\left({\int }_{{\mathrm{\Omega }}_2}|w(x)|d{x}_2\right)d{x}_1\\ =& \frac{1}{|{\mathrm{\Omega }}_1^{\prime }||{\mathrm{\Omega }}_2^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}\left({\int }_{{\mathrm{\Omega }}_2^{\prime }}d{x}_2\right)d{x}_1=1\end{array}$$

だから

$$\begin{array}{rl}& \underset{\Vert u{\Vert }_{p_1,p_2}=1}{sup}\left|{\int }_{{\mathrm{\Omega }}_1}\left({\int }_{{\mathrm{\Omega }}_2}u(x)v(x)d{x}_2\right)d{x}_1\right|\\ \ge & \left|{\int }_{{\mathrm{\Omega }}_1}\left({\int }_{{\mathrm{\Omega }}_2}w(x)v(x)d{x}_2\right)d{x}_1\right|\\ =& \left|\frac{1}{|{\mathrm{\Omega }}_1^{\prime }||{\mathrm{\Omega }}_2^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}\left({\int }_{{\mathrm{\Omega }}_2^{\prime }}\frac{\overline{v(x)}v(x)}{v(x)}d{x}_2\right)d{x}_1\right|\\ =& \frac{1}{|{\mathrm{\Omega }}_1^{\prime }||{\mathrm{\Omega }}_2^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}\left({\int }_{{\mathrm{\Omega }}_2^{\prime }}|v(x)|d{x}_2\right)d{x}_1\\ >& \frac{1}{|{\mathrm{\Omega }}_1^{\prime }||{\mathrm{\Omega }}_2^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}\left({\int }_{{\mathrm{\Omega }}_2^{\prime }}(\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}-2ϵ)d{x}_2\right)d{x}_1\\ =& \Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}-2ϵ\end{array}$$

となって，$ϵ$の任意性から$\underset{\Vert u{\Vert }_{p_1,p_2}=1}{sup}\left|{\displaystyle {\int }_{{\mathrm{\Omega }}_1}\left({\int }_{{\mathrm{\Omega }}_2}u(x)v(x)d{x}_2\right)d{x}_1}\right|\ge \Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}$が従う．

[２]　$p_1=1,p_2\in (1,\mathrm{\infty })$のとき，$p_1^{\prime }=\mathrm{\infty },p_2^{\prime }\in (1,\mathrm{\infty })$である．

任意の$ϵ>0$に対して，ある可測集合${\mathrm{\Omega }}_1^{\prime }\subset {\mathrm{\Omega }}_1$で$0<|{\mathrm{\Omega }}_1^{\prime }|<\mathrm{\infty }$を満たすものが存在して，$x\in {\mathrm{\Omega }}_1^{\prime }\times {\mathrm{\Omega }}_2$が成り立つとき，

$$\begin{array}{r}\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}>\Vert v{\Vert }_{1,p_2^{\prime }}-ϵ\end{array}$$

が成り立つ．このとき，${\theta }_x:=\arg v(x)$とし，$w:{\mathrm{\Omega }}_1\times {\mathrm{\Omega }}_2\to \mathbb{C}$を

$$\begin{array}{r}w(x):=\frac{|v(x){|}^{p_2^{\prime }-1}}{|{\mathrm{\Omega }}_1^{\prime }|\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_2^{\prime }/p_2}}{e}^{-i{\theta }_x}{\mathbb{I}}_{{\mathrm{\Omega }}_1^{\prime }\times {\mathrm{\Omega }}_2}(x)\end{array}$$

で定めると

$$\begin{array}{rl}\Vert w{\Vert }_{p_1,p_2}=& {\int }_{{\mathrm{\Omega }}_1}{\left({\int }_{{\mathrm{\Omega }}_2}|w(x){|}^{p_2}d{x}_2\right)}^{1/p_2}d{x}_1\\ =& \frac{1}{|{\mathrm{\Omega }}_1^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}\frac{1}{\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_2^{\prime }/p_2}}{\left({\int }_{{\mathrm{\Omega }}_2}|v(x){|}^{p_2(p_2^{\prime }-1)}d{x}_2\right)}^{1/p_2}d{x}_1\\ =& \frac{1}{|{\mathrm{\Omega }}_1^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}\frac{1}{\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_2^{\prime }/p_2}}{\left({\int }_{{\mathrm{\Omega }}_2}|v(x){|}^{p_2^{\prime }}d{x}_2\right)}^{1/p_2}d{x}_1\\ =& \frac{1}{|{\mathrm{\Omega }}_1^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}d{x}_1=1\end{array}$$

だから

$$\begin{array}{rl}& \underset{\Vert u{\Vert }_{p_1,p_2}=1}{sup}\left|{\int }_{{\mathrm{\Omega }}_1}\left({\int }_{{\mathrm{\Omega }}_2}u(x)v(x)d{x}_2\right)d{x}_1\right|\\ \ge & \left|{\int }_{{\mathrm{\Omega }}_1}\left({\int }_{{\mathrm{\Omega }}_2}w(x)v(x)d{x}_2\right)d{x}_1\right|\\ =& \left|\frac{1}{|{\mathrm{\Omega }}_1^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}\left(\frac{1}{\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_2^{\prime }/p_2}}{\int }_{{\mathrm{\Omega }}_2}|v(x){|}^{p_2^{\prime }-1}{e}^{-i{\theta }_x}v(x)d{x}_2\right)d{x}_1\right|\\ =& \frac{1}{|{\mathrm{\Omega }}_1^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}\left(\frac{1}{\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_2^{\prime }/p_2}}{\int }_{{\mathrm{\Omega }}_2}|v(x){|}^{p_2^{\prime }}d{x}_2\right)d{x}_1\\ =& \frac{1}{|{\mathrm{\Omega }}_1^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_2^{\prime }-\frac{p_2^{\prime }}{p_2}}d{x}_1=\frac{1}{|{\mathrm{\Omega }}_1^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}d{x}_1\\ >& \frac{1}{|{\mathrm{\Omega }}_1^{\prime }|}{\int }_{{\mathrm{\Omega }}_1^{\prime }}(\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}-ϵ)d{x}_1=\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}-ϵ\end{array}$$

となって，$ϵ$の任意性から$\underset{\Vert u{\Vert }_{p_1,p_2}=1}{sup}\left|{\displaystyle {\int }_{{\mathrm{\Omega }}_1}\left({\int }_{{\mathrm{\Omega }}_2}u(x)v(x)d{x}_2\right)d{x}_1}\right|\ge \Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}$が従う．

[３]　$p_1,p_2\in (1,\mathrm{\infty })$のとき，$p_1^{\prime },p_2^{\prime }\in (1,\mathrm{\infty })$である．

${\theta }_x:=\arg v(x)$とし，$w:{\mathrm{\Omega }}_1\times {\mathrm{\Omega }}_2\to \mathbb{C}$を

$$\begin{array}{r}w(x):=\frac{|v(x){|}^{p_2^{\prime }-1}}{\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }/p_1}\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{1-p_1^{\prime }+p_2^{\prime }/p_2}}{e}^{-i{\theta }_x}\end{array}$$

で定めると

$$\begin{array}{rl}& {\left({\int }_{{\mathrm{\Omega }}_1}{\left|{\int }_{{\mathrm{\Omega }}_2}|w(x){|}^{p_2}d{x}_2\right|}^{p_1/p_2}d{x}_1\right)}^{1/p_2}\\ =& \frac{1}{\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }/p_1}}{\left({\int }_{{\mathrm{\Omega }}_1^{\prime }}\frac{1}{\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_1(1-p_1^{\prime }+p_2^{\prime }/p_2)}}{\left({\int }_{{\mathrm{\Omega }}_2}|v(x){|}^{p_2(p_2^{\prime }-1)}d{x}_2\right)}^{p_1/p_2}d{x}_1\right)}^{1/p_1}\\ =& \frac{1}{\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }/p_1}}{\left({\int }_{{\mathrm{\Omega }}_1^{\prime }}\frac{1}{\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_1(1-p_1^{\prime }+p_2^{\prime }/p_2)}}{\left({\int }_{{\mathrm{\Omega }}_2}|v(x){|}^{p_2^{\prime }}d{x}_2\right)}^{p_1/p_2}d{x}_1\right)}^{1/p_1}\\ =& \frac{1}{\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }/p_1}}{\left({\int }_{{\mathrm{\Omega }}_1^{\prime }}\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_1(p_1^{\prime }-1)}d{x}_1\right)}^{1/p_1}\\ =& \frac{1}{\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }/p_1}}{\left({\int }_{{\mathrm{\Omega }}_1^{\prime }}\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_1^{\prime }}d{x}_1\right)}^{1/p_1}=1\end{array}$$

である．すなわち$\Vert w{\Vert }_{p_1,p_2}=1$だから

$$\begin{array}{rl}& \underset{\Vert u{\Vert }_{p_1,p_2}=1}{sup}\left|{\int }_{{\mathrm{\Omega }}_1}\left({\int }_{{\mathrm{\Omega }}_2}u(x)v(x)d{x}_2\right)d{x}_1\right|\\ \ge & \left|{\int }_{{\mathrm{\Omega }}_1}\left({\int }_{{\mathrm{\Omega }}_2}w(x)v(x)d{x}_2\right)d{x}_1\right|\\ =& \left|\frac{1}{\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }/p_1}}{\int }_{{\mathrm{\Omega }}_1}\left(\frac{1}{\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{1-p_1^{\prime }+p_2^{\prime }/p_2}}{\int }_{{\mathrm{\Omega }}_2}|v(x){|}^{p_2^{\prime }-1}{e}^{-i{\theta }_x}v(x)d{x}_2\right)d{x}_1\right|\\ =& \frac{1}{\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }/p_1}}{\int }_{{\mathrm{\Omega }}_1}\left(\frac{1}{\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{1-p_1^{\prime }+p_2^{\prime }/p_2}}{\int }_{{\mathrm{\Omega }}_2}|v(x){|}^{p_2^{\prime }}d{x}_2\right)d{x}_1\\ =& \frac{1}{\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }/p_1}}{\int }_{{\mathrm{\Omega }}_1}\left(\frac{1}{\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{1-p_1^{\prime }+p_2^{\prime }/p_2}}\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_2^{\prime }}\right)d{x}_1\\ =& \frac{1}{\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }/p_1}}{\int }_{{\mathrm{\Omega }}_1}\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_1^{\prime }-1+p_2^{\prime }(1-1/p_2)}d{x}_1\\ =& \frac{1}{\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }/p_1}}{\int }_{{\mathrm{\Omega }}_1}\Vert v(x_1,\cdot ){\Vert }_{L^{p_2^{\prime }}({\mathrm{\Omega }}_2)}^{p_1^{\prime }}d{x}_1\\ =& \frac{1}{\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }/p_1}}\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }}=\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}^{p_1^{\prime }(1-1/p_1)}=\Vert v{\Vert }_{p_1^{\prime },p_2^{\prime }}\end{array}$$

が従う．