Interchange of $\ell^r$ and $L^p$-norm

Question

Let $(f_i)_{i\in\mathbb{N}}$ be a sequence of $L^p$-functions. What is the relation between $\Vert \Vert (f_i)_{i\in\mathbb{N}}\Vert_{\ell^r}\Vert_{L^p}$ and $\Vert \left(\Vert f_i\Vert_{L^p}\right)_{i\in\mathbb{N}}\Vert_{\ell^r}$, where $\Vert (f_i)_{i\in\mathbb{N}}\Vert_{\ell^r}$ means the pointwise $\ell^r$-norm.

Answer 1

What you are looking for is Minkowski's inequality. If $p \leq r \leq \infty$ you have

$$ \| (\|f_i\|_{L^p}) \|_{\ell^r} \leq \| \|(f_i)\|_{\ell^r} \|_{L^{p}} $$

if on the other hand $r \leq p \leq \infty$ you have

$$ \| \|(f_i)\|_{\ell^r} \|_{L^{p}} \leq \| (\|f_i\|_{L^p}) \|_{\ell^r} $$

The Wikipedia discussion linked to above is brief. Better to consult Inequalities by Hardy, Littlewood, and Polya.