Proof of Comparison of the Lebesgue Seminorms on the Torus

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named in the statement above.

We have $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field and $1\le r$ by hypothesis, so $0<r$ by claim 2 of that lemma. Also $1\le s$, by transitivity from $1\le r$ and $r\le s$. Put $q=s\,r^{-1}$, where $r^{-1}$ is the multiplicative inverse of $r$, positive by claim 7 of Elementary Order Arithmetic in an Ordered Field. From $r\le s$ we get $1\le q$ on multiplying by $r^{-1}$, using claim 5 of Elementary Arithmetic in an Ordered Field, and $r\,q=s$.

Write $|v|^{r}$ for the map on $Q$ sending $x$ to the power $(|v(x)|)^{r}$.

Step 1. Apply Elementary Properties of the p-Seminorm §rescaling to the measurable map $v$, with its $r$ taken to be $r$ and its $s$ taken to be $q$; the hypotheses hold because $0<r$, $1\le q$ and $1\le r\,q=s$. It gives that $|v|^{r}$ is measurable and that $v\in\mathcal{L}^{s}(\mathbb{T}^{n})$ if and only if $|v|^{r}\in\mathcal{L}^{q}(\mathbb{T}^{n})$, and that in that case

Since $v\in\mathcal{L}^{s}(\mathbb{T}^{n})$ by hypothesis, $|v|^{r}\in\mathcal{L}^{q}(\mathbb{T}^{n})$ and this identity holds.

Step 2. Apply The Periodic Extension of a Function on the Unit Cell §finite-measure to $|v|^{r}\in\mathcal{L}^{q}(\mathbb{T}^{n})$, which is legitimate since $1\le q$. It gives $|v|^{r}\in\mathcal{L}^{1}(\mathbb{T}^{n})$ and

Step 3. Apply Elementary Properties of the p-Seminorm §rescaling again to $v$, this time with its $r$ taken to be $r$ and its $s$ taken to be $1$; the hypotheses hold because $0<r$, $1\le 1$ and $1\le r\cdot 1=r$. It gives that $v\in\mathcal{L}^{r}(\mathbb{T}^{n})$ if and only if $|v|^{r}\in\mathcal{L}^{1}(\mathbb{T}^{n})$, and that in that case

By Step 2 the right-hand condition holds, so $v\in\mathcal{L}^{r}(\mathbb{T}^{n})$, which is the first assertion of the claim, and this identity holds.

Step 4. Combining the three steps,

The seminorms $\lVert v\rVert_{r}$ and $\lVert v\rVert_{s}$ are nonnegative real numbers by Power-Integrable Functions and the p-Seminorm §seminorm. Applying Properties of Real Powers of Nonnegative Real Numbers §inverse with exponent $r$, the inequality $\bigl(\lVert v\rVert_{r}\bigr)^{r}\le\bigl(\lVert v\rVert_{s}\bigr)^{r}$ gives

the last equality by the identity $(t^{r})^{1/r}=t$ of that same clause. This is the second assertion of the claim.