Vectorial $L^p$ spaces

Question

Consider the space

$$Ł^p(\Omega):=(L^p(\Omega))^N=L^p(\Omega)×L^p(\Omega)×...×L^p(\Omega), \, N \ge 1,\, \Omega \subset \mathbb{R^n},$$

as the vectorial $L^p$ space associated to the scalar one.

My questions are:

1. What is the norm $||\cdot||_{Ł^p}$ and can we relate it to $||\cdot||_{L^p}$?

2. How to apply the Hölder Inequality on

$$\int_{\Omega}{u(x)\cdot v(x)dx},$$

where

$u \in Ł^q$, $v \in Ł^p$, $\frac{1}{p}+\frac{1}{q}=1$, $u=(u_1,...,u_N)^t$, $v=(v_1,....,v_N)^t$, $u_i \in L^q(\Omega)$, $v_i \in L^p(\Omega)$, and $u(x)\cdot v(x)$ is the scalar product between?

Answer 1

A nice norm is $$ \lVert u \rVert_{L^p(\Omega)^N} := \sqrt[p]{\sum_{i = 1}^N \lVert u_i \Vert_{L^p(\Omega)}^p}. $$ If $\lVert \cdot \rVert_{\mathbb{R}^N}$ is any norm on $\mathbb{R}^N$, then it is easy to see that $\lVert u \rVert_{L^p(\Omega)^N}$ is equivalent to $$ \lVert u \rVert := \lVert (\lVert u_1 \rVert_{L^p(\Omega)}, ..., \lVert u_N \rVert_{L^p(\Omega)})^\top \rVert_{\mathbb{R}^N}. $$ In the case of $\lVert \cdot \rVert_{L^p(\Omega)^N}$ we had set $\lVert \cdot \rVert_{\mathbb{R}^N} := \lVert \cdot \rVert_p$.

The Hölder inequality also holds in the following sense: We first use the Hölder-inequality of $\mathbb{R}^N$ inside the integral and then use the Hölder-inequality for Lebesgue-integrals: $$ \int_\Omega u \cdot v ~\mathrm{d}x \leq \int_\Omega \lVert u \Vert_p \lVert v \Vert_q ~\mathrm{d}x \leq \sqrt[p]{\int_\Omega \lVert u \rVert_p^p ~\mathrm{d}x} \sqrt[q]{\int_\Omega \lVert v \rVert_q^q ~\mathrm{d}x} = $$ $$ \sqrt[p]{\sum_{i = 1}^N \int_\Omega \lvert u_i \rvert^p~\mathrm{d}x}\sqrt[q]{\sum_{i = 1}^N \int_\Omega \lvert v_i \rvert^q~\mathrm{d}x} = \sqrt[p]{\sum_{i = 1}^N \lVert u_i \rVert_{L^p(\Omega)}^p} \sqrt[q]{\sum_{i = 1}^N \lVert v_i \rVert_{L^q(\Omega)}^q} = \lVert u \rVert_{L^p(\Omega)^N} \lVert v \rVert_{L^q(\Omega)^N} $$