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 (vi) 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

$$\bigl\lVert\,|v|^{r}\,\bigr\rVert_{q}=\bigl(\lVert v\rVert_{s}\bigr)^{r}.$$

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 (iv) 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

$$\bigl\lVert\,|v|^{r}\,\bigr\rVert_{1}\le\bigl\lVert\,|v|^{r}\,\bigr\rVert_{q}.$$

Step 3. Apply Elementary Properties of the p-Seminorm (vi) 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

$$\bigl\lVert\,|v|^{r}\,\bigr\rVert_{1}=\bigl(\lVert v\rVert_{r}\bigr)^{r}.$$

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,

$$\bigl(\lVert v\rVert_{r}\bigr)^{r}=\bigl\lVert\,|v|^{r}\,\bigr\rVert_{1}\le\bigl\lVert\,|v|^{r}\,\bigr\rVert_{q}=\bigl(\lVert v\rVert_{s}\bigr)^{r}.$$

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

$$\lVert v\rVert_{r}\le\Bigl(\bigl(\lVert v\rVert_{s}\bigr)^{r}\Bigr)^{1/r}=\lVert v\rVert_{s},$$

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