Minkowski’s Inequality

Minkowski’s Inequality

For $1\le p\le \infty$, $f,g\in L^p(X,\mathcal{M},\mu )$, then ${\left({\int }_X|f(x)+g(x){|}^{p}d\mu (x)\right)}^{\frac{1}{p}}\le {\left({\int }_X|f(x){|}^{p}d\mu (x)\right)}^{\frac{1}{p}}+{\left({\int }_X|g(x){|}^{q}d\mu (x)\right)}^{\frac{1}{p}}$

In ${\mathbb{R}}^n$, for all $1\le p\le \infty$, if $x,y\in {\mathbb{R}}^n$ then $\sqrt[p]{(x_1+y_1{)}^p+\cdots +(x_n+y_n{)}^p}\le \sqrt[p]{x_1^p+\cdots +x_n^p}+\sqrt[p]{y_1^p+\cdots +y_n^p}$

Proof For $\frac{1}{p}+\frac{1}{q}=1$, $p=(p-1)q$, then $|f+g{|}^{p-1}\in L^q$ when $f,g\in L^p$. From triangle inequality we have $|f+g{|}^p=|f+g||f+g{|}^{p-1}\le (|f|+|g|)|f+g{|}^{p-1}.$ By Holder inequality, $\int |f||f+g{|}^{p-1}\le \Vert f{\Vert }_{L^p}(\int |f+g{|}^p{)}^{\frac{1}{q}},$ $\int |g||f+g{|}^{p-1}\le \Vert g{\Vert }_{L^p}(\int |f+g{|}^p{)}^{\frac{1}{q}}.$ Combine the two inequalities we get,

$\int |f+g{|}^p\le (\Vert f{\Vert }_{L^p}+\Vert g{\Vert }_{L^p})(\int |f+g{|}^p{)}^{\frac{1}{q}},$ thus $(\int |f+g{|}^p{)}^{\frac{1}{p}}\le \Vert f{\Vert }_{L^p}+\Vert g{\Vert }_{L^p}.$ $\square$

Minkowski's inequality for integrals

Suppose $(X_1,{\mu }_1)$ and $(X_2,{\mu }_2)$ are two measure spaces, and $1\le p\le \infty$. Show that if $f(x_1,x_2)$ is measurable on $X_1\times X_2$ and non-negative, then ${\Vert {\int }_{X_2}f(x_1,x_2)d{\mu }_2\Vert }_{L^p(X_1)}\le {\int }_{X_2}\Vert f(x_1,x_2){\Vert }_{L^p(X_1)}d{\mu }_2$