Comparison of the Lebesgue Seminorms on the Torus

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ and the classes $\mathcal{L}^{t}(\mathbb{T}^{n})$, for a real number $t$ with $1\le t$, are the ones fixed there. Let $\lVert\,\cdot\,\rVert_{t}$ denote the $L^{t}$ seminorm of $(Q,\mathcal{B}_{Q},\lambda_{Q})$.

1. (Comparison of the seminorms)¶ Let $r$ and $s$ be real numbers with $1\le r$ and $r\le s$, and let $v\in\mathcal{L}^{s}(\mathbb{T}^{n})$. Then $v\in\mathcal{L}^{r}(\mathbb{T}^{n})$ and