📂Lebesgue Spaces

Minkowski's Inequality

Theorem¹

Let’s denote $\Omega \subset \mathbb{R}^{n}$ as an open set, and $0 \lt p \lt 1$. If $u, v \in L^p(\Omega)$ then $u+v \in L^p(\Omega)$.

Explanation

This is called the reverse Minkowski’s inequality. It’s not the converse of the Minkowski’s inequality proposition, but the direction of the inequality is reversed.

The Minkowski inequality shows that when $1 \le p \lt \infty$, the defined $\left\| \cdot \right\|_{p}$ satisfies the triangle inequality and becomes the norm in the $L^{p}$ space.

However, in the case of the reverse Minkowski inequality, when $0 \lt p \lt 1$, $\left\| \cdot \right\|_{p}$ does not satisfy the definition of a norm, indicating that $L^{p}$ is not a normed space.

Proof

If $u = v = 0$, the proof is trivial, so let’s assume at least one of $u, v$ is not $0$. To compute $\left\| |u| + |v| \right\|_{p}^{p}$, rearrange the equation as follows:

Rearranging the equation gives us

Moreover, since we assumed that at least one of $u, v$ is not $0$,

At this point, $p^{\prime}$ is defined as the conjugate exponent. Therefore, if $(p-1)p^{\prime} = p$ and $|u|, |v| \in L^{p}$, then ${|u| + |v| \in L^{p}}$,

From the above two inequalities, we obtain:

This is a sufficient condition for the reverse Hölder’s inequality to hold.

Reverse Hölder’s inequality

Let $0 < p < 1$ and $p^{\prime} = \dfrac{p}{p-1} < 0$. If $f \in L^{p}(\Omega)$, ${fg\in L^{1}(\Omega)}$, and

Then, the following inequality holds:

Let $f = u$ and denote $g = \left( |u| + |v| \right)^{p-1}$,

Similarly, let $f = v$ and denote $g = \left( |u| + |v| \right)^{p-1}$,

Substituting the above two inequalities for $(1)$,

Multiplying both sides by $\left\| |u| + |v| \right\|_{p}^{-p/p^{\prime}}$ gives $p - \dfrac{p}{p^{\prime}} = 1$,

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