Proof of Minkowski's Inequality in Lebesgue Spaces 📂Lebesgue Spaces

Proof of Minkowski's Inequality in Lebesgue Spaces

Theorem¹

Let $\Omega \subset \mathbb{R}^{n}$ be an open set. If $1 \le p < \infty$ and $u, v \in L^{p}(\Omega)$, then

$$ \left\| u + v \right\|_{p} \le \left\| u \right\|_{p}+\left\| v \right\|_{p} $$

This is called the Minkowski inequality.

Explanation

It means that $\left\| \cdot \right\|_{p}$ satisfies the triangle inequality and thus becomes a norm, and that the $L^{p}$ space becomes a normed space. Note that it holds for $p \ge 1$.

Proof

For the case $p=1$, it holds trivially by the properties of the integral.

$$ \int_{\Omega} \left| u(x) + v(x) \right| dx \le \int_{\Omega} \left| u(x) \right| dx + \int_{\Omega} \left| v(x) \right| dx $$

Let $1 \lt p \lt \infty$. Let $w$ be a function with $w \ge 0$ and $\left\| w \right\|_{p^{\prime}} \le 1$.

Hölder’s inequality
For $1 \le p, p^{\prime} < \infty$ with $\dfrac{1}{p} + \dfrac{1}{p^{\prime}} = 1$, if $u \in L^p(\Omega)$ and $v\in L^{p^{\prime}}(\Omega)$, then
$$ \int_{\Omega} |u(x)v(x)| dx \le \| u \|_{p} \| v \|_{p^{\prime}} $$

Then the following holds by Hölder’s inequality.

$$ \begin{align*} \int_{\Omega} \left| u(x) + v(x) \right| w(x) dx \le & \int_{\Omega} \left| u(x) \right| w(x) dx + \int_{\Omega} \left| v(x) \right| w(x) dx \\ \le & \left\| u \right\|_{p} \left\| w \right\|_{p^{\prime}} + \left\| v \right\|_{p} \left\| w \right\|_{p^{\prime}} \\ \le & \left\| u \right\|_{p} + \left\| v \right\|_{p} \end{align*} $$

Therefore, the following holds.

$$ \sup \left\{ \int_{\Omega} \left| u(x) + v(x) \right| w(x) dx : v(x) \ge 0 \text{ on } \Omega, \left\| w \right\|_{p^{\prime}} \le 1 \right\} \lt \left\| u \right\|_{p} + \left\| v \right\|_{p} $$

Converse of Hölder’s inequality: a sufficient condition for being an $L^{p}$ function
If $\sup \left\{ \int_{\Omega} \left| u(x) \right| v(x) dx : v(x) \ge 0 \text{ on } \Omega, \left\| v \right\|_{p^{\prime}} \le 1 \right\} \lt \infty$ holds, then $$ u\in L^{p}(\Omega) \quad \text{and} \quad \left\| u \right\|_{p} = \sup \left\{ \int_{\Omega} \left| u(x) \right| v(x) dx : v(x) \ge 0 \text{ on } \Omega, \left\| v \right\|_{p^{\prime}} \le 1 \right\} $$

By the above theorem,

$$ \left\| u + v \right\|_{p} \lt \left\| u \right\|_{p} + \left\| v \right\|_{p} $$

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