ソボレフ共役
u λ ( x ) := u ( λ x ) {\displaystyle u_{\lambda }(x):=u(\lambda x)} とすると、次が成り立つ。
‖ u λ ‖ L q ( R n ) q = ∫ R n | u ( λ x ) | q d x = 1 λ n ∫ R n | u ( y ) | q d y = λ − n ‖ u ‖ L q ( R n ) q {\displaystyle \|u_{\lambda }\|_{L^{q}(\mathbb {R} ^{n})}^{q}=\int _{\mathbb {R} ^{n}}|u(\lambda x)|^{q}dx={\frac {1}{\lambda ^{n}}}\int _{\mathbb {R} ^{n}}|u(y)|^{q}dy=\lambda ^{-n}\|u\|_{L^{q}(\mathbb {R} ^{n})}^{q}}
‖ D u λ ‖ L p ( R n ) p = ∫ R n | λ D u ( λ x ) | p d x = λ p λ n ∫ R n | D u ( y ) | p d y = λ p − n ‖ D u ‖ L p ( R n ) p {\displaystyle \|Du_{\lambda }\|_{L^{p}(\mathbb {R} ^{n})}^{p}=\int _{\mathbb {R} ^{n}}|\lambda Du(\lambda x)|^{p}dx={\frac {\lambda ^{p}}{\lambda ^{n}}}\int _{\mathbb {R} ^{n}}|Du(y)|^{p}dy=\lambda ^{p-n}\|Du\|_{L^{p}(\mathbb {R} ^{n})}^{p}}