Proof of The Sobolev Embedding of the First Sobolev Space of the Torus into the Sixth Lebesgue Space in Dimensions at Most Three

Each result cited is universally quantified over the data in its own statement, and is applied here to the data named below. We use silently that the order of $\mathbb{R}$ is reflexive, transitive and antisymmetric, and that equal real numbers satisfy $\le$ in both directions. Powers of nonnegative real numbers and their properties are those of Measure Spaces and the Lebesgue Integral: Standing Notation §powers; for a measurable $f$ on $(Q,\mathcal{B}_{Q},\lambda_{Q})$ and a real number $r$ with $0<r$, $|f|^{r}$ denotes the map $y\mapsto(|f(y)|)^{r}$ of Power-Integrable Functions and the p-Seminorm §measurable-power.

Proof of claim 1. Let $u\in C^{1}_{\mathrm{per}}$. By Elementary Properties of the Weak Partial Derivative on the Torus §classical, $u\in C_{\mathrm{per}}$ and $\partial_{j}u\in C_{\mathrm{per}}$ for every $j\in[n]$, the restrictions $u|_{Q}$ and $(\partial_{j}u)|_{Q}$ lie in $\mathcal{L}^{t}(\mathbb{T}^{n})$ for every real $t$ with $1\le t$, and $[(\partial_{j}u)|_{Q}]$ is the $j$-th weak partial derivative of $[u|_{Q}]$; so $[u|_{Q}]\in H^{1}(\mathbb{T}^{n})$ by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus §space. Write

both nonnegative real numbers, the first by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus §inner-product and the second by Power-Integrable Functions and the p-Seminorm §seminorm.

(R1) The two elementary bounds. By The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space §embedding, applied to $U=[u|_{Q}]$, and by The Lebesgue Space of Power-Integrable Functions §norm, which identifies the norm of a class with the seminorm of a representative,

the norm $\lVert\,\cdot\,\rVert_{L^{2}}$ of that theorem being the norm of $L^{2}(\mathbb{T}^{n})$ fixed in The Flat Torus: Standing Notation §lebesgue and the $j$-th weak partial derivative of $[u|_{Q}]$ being $[(\partial_{j}u)|_{Q}]$.

(R2) Powers of $u$. For $k\in\{2,3,4\}$ let $u^{k}$ denote the pointwise product of $k$ copies of $u$, formed as $u^{2}=uu$, $u^{3}=u^{2}u$ and $u^{4}=u^{3}u$. By claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and Elementary Properties of Lattice-Periodic Functions §algebra, each $u^{k}$ lies in $C^{1}_{\mathrm{per}}$, and by the pointwise product rule of claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set,

each obtained from the previous one, since for instance $\partial_{j}(u^{3}u)=\bigl(3u^{2}\partial_{j}u\bigr)u+u^{3}\partial_{j}u=4u^{3}\partial_{j}u$. Moreover $|u^{k}(y)|=(|u(y)|)^{k}$ for every $y$, by claim 4 of Properties of the Absolute Value in an Ordered Field together with Properties of Real Powers of Nonnegative Real Numbers §exponents and Properties of Real Powers of Nonnegative Real Numbers §agreement, which give $(|u(y)|)^{k}$ as the product of $k$ copies of $|u(y)|$.

Consequently, for $k\in\{2,3,4\}$ and any real $t$ with $1\le t$, the restriction $(u^{k})|_{Q}$ and the map $|u|_{Q}|^{k}$ have the same absolute value at every point of $Q$, so the seminorms $\lVert(u^{k})|_{Q}\rVert_{t}$ and $\bigl\lVert|u|_{Q}|^{k}\bigr\rVert_{t}$ are equal, both being $\bigl(\int_{\mathbb{T}^{n}}(|u(y)|)^{kt}\,dx\bigr)^{1/t}$ by Power-Integrable Functions and the p-Seminorm §seminorm and Properties of Real Powers of Nonnegative Real Numbers §exponents. By Elementary Properties of the p-Seminorm §rescaling, applied to $h=u|_{Q}$ with the exponents $r=k$ and $s=t$,

In particular $\lVert(u^{3})|_{Q}\rVert_{2}=S^{3}$, $\lVert(u^{4})|_{Q}\rVert_{3/2}=S^{4}$, $\bigl\lVert|u|_{Q}|^{3}\bigr\rVert_{2}=S^{3}$ and $\bigl\lVert|u|_{Q}|^{2}\bigr\rVert_{2}=\bigl(\lVert u|_{Q}\rVert_{4}\bigr)^{2}$.

(R3) Four integral estimates. The numbers $2$ and $2$ are conjugate exponents, since $1<2$ and $\tfrac12+\tfrac12=1$. Hence Hoelder's Inequality, for Two and for Finitely Many Factors §holder, applied to the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ with $p=q=2$, gives $\int_{\mathbb{T}^{n}}|fg|\,dx\le\lVert f\rVert_{2}\lVert g\rVert_{2}$ for all $f,g\in\mathcal{L}^{2}(\mathbb{T}^{n})$. The maps $|u|_{Q}|^{2}$ and $|u|_{Q}|^{3}$ belong to $\mathcal{L}^{2}(\mathbb{T}^{n})$: by Elementary Properties of the p-Seminorm §rescaling, applied to $h=u|_{Q}$ with $s=2$ and with $r=2$ and $r=3$ in turn, the map $|h|^{r}$ lies in $\mathcal{L}^{2}(\mathbb{T}^{n})$ if and only if $h$ lies in $\mathcal{L}^{2r}(\mathbb{T}^{n})$, which it does, $u|_{Q}$ belonging to $\mathcal{L}^{t}(\mathbb{T}^{n})$ for every real $t$ with $1\le t$. Taking in turn $f=|u|_{Q}|^{3}$ with $g=(\partial_{j}u)|_{Q}$, $f=|u|_{Q}|^{3}$ with $g=u|_{Q}$, $f=|u|_{Q}|^{2}$ with $g=(\partial_{j}u)|_{Q}$ and $f=|u|_{Q}|^{2}$ with $g=u|_{Q}$, and using (R1), (R2) and the fact that the values of $|u|_{Q}|^{k}$ are nonnegative by Properties of Real Powers of Nonnegative Real Numbers §values, we obtain for every $j\in[n]$

the second and fourth because the pointwise products $|u|_{Q}|^{3}\,u|_{Q}$ and $|u|_{Q}|^{2}\,u|_{Q}$ have absolute values $|u|_{Q}|^{4}$ and $|u|_{Q}|^{3}$, by claim 4 of Properties of the Absolute Value in an Ordered Field and Properties of Real Powers of Nonnegative Real Numbers §exponents. By Comparison of the Lebesgue Seminorms on the Torus §comparison, applied with $r=4$ and $s=6$, we have $\lVert u|_{Q}\rVert_{4}\le S$, so Properties of Real Powers of Nonnegative Real Numbers §monotone turns the last two bounds into $S^{2}N$.

(R4) Cancelling a power. Let $\sigma$ and $\tau$ be nonnegative real numbers and let $a$ and $b$ be positive real numbers with $\sigma^{a+b}\le\tau^{a}\,\sigma^{b}$. Then $\sigma\le\tau$.

Indeed, if $\sigma=0$ then $\sigma\le\tau$ since $\tau$ is nonnegative. Otherwise $0<\sigma$, so $0<\sigma^{b}$ by Properties of Real Powers of Nonnegative Real Numbers §values, and the multiplicative inverse $c$ of $\sigma^{b}$ is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field. Since $\sigma^{a+b}=\sigma^{a}\sigma^{b}$ by Properties of Real Powers of Nonnegative Real Numbers §exponents, multiplying the hypothesis by $c$, which is legitimate by claim 5 of Elementary Arithmetic in an Ordered Field, gives $\sigma^{a}\le\tau^{a}$. Now Properties of Real Powers of Nonnegative Real Numbers §inverse, in the form $s^{a}\le t\iff s\le t^{1/a}$ with $t=\tau^{a}$, gives $\sigma\le(\tau^{a})^{1/a}=\tau$, the last equality by that same clause.

The case $n=1$. By The Gagliardo-Nirenberg Inequality on the Torus in Dimensions at Most Three §one, $|u(x)|\le A_{1}$ for every $x\in\mathbb{R}^{n}$, where $A_{1}=\lVert u|_{Q}\rVert_{1}+\lVert(\partial_{1}u)|_{Q}\rVert_{1}$. By Comparison of the Lebesgue Seminorms on the Torus §comparison, applied with $r=1$ and $s=2$, and by (R1), each of the two seminorms is at most $N$, so $A_{1}\le N+N=2N$. Hence $|u(x)|\le2N$ for every $x$, and Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member, applied with $p=6$ and $M=2N$, gives $\lVert u|_{Q}\rVert_{6}\le2N$. Finally $2N\le5N$ by claim 5 of Elementary Arithmetic in an Ordered Field, since $2\le5$ and $0\le N$.

The case $n=2$. Put $v=u^{3}$. By Elementary Properties of the p-Seminorm §power, applied with $p=1$, and Properties of Real Powers of Nonnegative Real Numbers §agreement,

by (R2), (R3) and claim 4 of Properties of the Absolute Value in an Ordered Field, which gives $|3u^{2}\partial_{j}u|=3|u|^{2}|\partial_{j}u|$, together with the linearity in claim 2 of Linearity and Monotonicity of the Lebesgue Integral. So the numbers $A_{j}=\lVert v|_{Q}\rVert_{1}+\lVert(\partial_{j}v)|_{Q}\rVert_{1}$ of The Gagliardo-Nirenberg Inequality on the Torus in Dimensions at Most Three satisfy $A_{j}\le4S^{2}N$ for $j\in[2]$. By The Gagliardo-Nirenberg Inequality on the Torus in Dimensions at Most Three §two, applied to $v\in C^{1}_{\mathrm{per}}$, and by (R2),

using Properties of Real Powers of Nonnegative Real Numbers §exponents for $(S^{3})^{2}=S^{6}$, claim 5 of Elementary Arithmetic in an Ordered Field twice for the product of the two bounds, and Properties of Real Powers of Nonnegative Real Numbers §product for the last equality. Since $16N^{2}=(4N)^{2}$ by Properties of Real Powers of Nonnegative Real Numbers §product, (R4) applied with $\sigma=S$, $\tau=4N$, $a=2$ and $b=4$ gives $S\le4N$, and $4N\le5N$ by claim 5 of Elementary Arithmetic in an Ordered Field.

The case $n=3$. Put $v=u^{4}$. Exactly as in the previous case, with the first and second estimates of (R3) in place of the third and fourth,

so $A_{j}\le5S^{3}N$ for $j\in[3]$. By The Gagliardo-Nirenberg Inequality on the Torus in Dimensions at Most Three §three, applied to $v$, together with (R2), Properties of Real Powers of Nonnegative Real Numbers §monotone, Properties of Real Powers of Nonnegative Real Numbers §exponents and Properties of Real Powers of Nonnegative Real Numbers §product,

the second inequality because $A_{1}A_{2}A_{3}\le(5S^{3}N)^{3}$ by claim 5 of Elementary Arithmetic in an Ordered Field applied twice, all factors being nonnegative. Now (R4), applied with $\sigma=S$, $\tau=5N$, $a=3/2$ and $b=9/2$, gives $S\le5N$.

In each of the three cases $\lVert u|_{Q}\rVert_{6}=S\le5N$, which is claim 1.

Proof of claim 2. Let $U\in H^{1}(\mathbb{T}^{n})$, let $v$ be a representative of $U$ and put $N=\lVert U\rVert_{H^{1}}$, a nonnegative real number.

Step 1: mollification. Let $\rho$ be a mollifier kernel of radius $1$ on $\mathbb{R}^{n}$, which exists by Existence of Mollifier Kernels of Every Radius, and for a natural number $k$ let $\rho_{1/k}$ be its rescaling with parameter $1/k$. By The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space §mollify, applied with $\delta=1$, with $U$ and with the representative $v$, the maps $u_{k}=\rho_{1/k}\star v$ lie in $C^{\infty}_{\mathrm{per}}$ and the classes $U_{k}=[u_{k}|_{Q}]$ lie in $H^{1}(\mathbb{T}^{n})$; moreover $\lVert U_{k}\rVert_{H^{1}}\le N$ by The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space §mollify-bound, and the sequence $(U_{k})_{k\in\mathbb{N}}$ converges to $U$ in $H^{1}(\mathbb{T}^{n})$ by The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space §mollify-converge.

A smooth map on $\mathbb{R}^{n}$ is of class $C^{1}$ by Smooth Map on a Euclidean Open Set, so each $u_{k}$, being $\mathbb{Z}^{n}$-periodic and smooth, lies in $C^{1}_{\mathrm{per}}$ by Lattice-Periodic Functions and the Periodic Function Classes §classes. Claim 1, applied to $u_{k}$, therefore gives

the second inequality by claim 5 of Elementary Arithmetic in an Ordered Field.

Step 2: an almost everywhere convergent subsequence. By The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space §embedding the distance in $L^{2}(\mathbb{T}^{n})$ between two members of $H^{1}(\mathbb{T}^{n})$ is at most their distance in $H^{1}(\mathbb{T}^{n})$, so any positive real witnessing the convergence of $(U_{k})$ to $U$ in $H^{1}(\mathbb{T}^{n})$ witnesses it in $L^{2}(\mathbb{T}^{n})$ as well; thus $(U_{k})$ converges to $U$ in $L^{2}(\mathbb{T}^{n})$ by Convergent Sequence in a Metric Space. By The Riesz-Fischer Theorem: the Lebesgue Space is a Real Banach Space §subsequence, applied with $p=2$ to this sequence, to the representatives $u_{k}|_{Q}$ of $U_{k}$ and to the representative $v$ of $U$, there are natural numbers $m_{1}<m_{2}<\dots$ and a null set $E_{0}$ such that for every $y\in Q\setminus E_{0}$ the sequence $(u_{m_{j}}(y))_{j\in\mathbb{N}}$ converges to $v(y)$. That clause also provides a dominating function, which is not needed here. A null set is not required to lie in $\mathcal{B}_{Q}$, so we enlarge it: by Null Set of a Measure there is $E\in\mathcal{B}_{Q}$ with $E_{0}\subseteq E$ and $\lambda_{Q}(E)=0$, and this $E$ is itself a null set. Since $Q\setminus E\subseteq Q\setminus E_{0}$, the sequence $(u_{m_{j}}(y))_{j\in\mathbb{N}}$ converges to $v(y)$ for every $y\in Q\setminus E$.

Step 3: two maps into the extended half-line. For $j\in\mathbb{N}$ let $g_{j}:Q\to[0,\infty]$ be the map taking the value $(|u_{m_{j}}(y)|)^{6}$ at $y\in Q\setminus E$ and the value $0$ at $y\in E$, and let $g:Q\to[0,\infty]$ take the value $(|v(y)|)^{6}$ at $y\in Q\setminus E$ and the value $0$ at $y\in E$. The maps $|u_{m_{j}}|_{Q}|^{6}$ and $|v|^{6}$ are measurable with respect to $\mathcal{B}_{Q}$ by Power-Integrable Functions and the p-Seminorm §measurable-power, the set $Q\setminus E$ lies in $\mathcal{B}_{Q}$, that being a $\sigma$-algebra on $Q$ of which $E$ is a member, and its indicator is measurable by claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and $g_{j}$ and $g$ are the pointwise products of these, hence measurable by claim 3 of that lemma; their values are nonnegative by Properties of Real Powers of Nonnegative Real Numbers §values, so they are measurable as maps into $[0,\infty]$ by Measure Spaces and the Lebesgue Integral: Standing Notation §measurable.

We claim that $g(y)\le\liminf_{j}g_{j}(y)$ for every $y\in Q$, the limit inferior being the one defined in Fatou's Lemma. For $y\in E$ this holds because $g(y)=0$ and every value $g_{j}(y)$ is nonnegative, so every greatest lower bound occurring in the definition is nonnegative and hence so is their least upper bound. Let then $y\in Q\setminus E$ and write $\alpha_{j}=g_{j}(y)$ and $\alpha=g(y)$. By claim 7 of Properties of the Absolute Value in an Ordered Field the sequence $(|u_{m_{j}}(y)|)_{j}$ converges to $|v(y)|$, and then Properties of Real Powers of Nonnegative Real Numbers §continuity, applied with the exponent $6$, shows that $(\alpha_{j})_{j}$ converges to $\alpha$. Let $\varepsilon$ be a real number with $0<\varepsilon$. There is $k\in\mathbb{N}$ with $|\alpha_{j}-\alpha|<\varepsilon$ for every $j\in\mathbb{N}$ with $k\le j$, so $\alpha-\varepsilon\le\alpha_{j}$ for all such $j$ by claim 3 of Properties of the Absolute Value in an Ordered Field; hence $\alpha-\varepsilon$ is a lower bound of the values $\alpha_{j}$ with $k\le j$, so $\alpha-\varepsilon\le\inf_{j\ge k}\alpha_{j}\le\liminf_{j}\alpha_{j}$, the last step because that greatest lower bound is one of the numbers whose least upper bound is the limit inferior. Write $L=\liminf_{j}\alpha_{j}$. If $L=\infty$ then $\alpha\le L$; otherwise $L$ is a nonnegative real number, and if $L<\alpha$ we may take $\varepsilon=(\alpha-L)/2$, which is positive by claim 3 of Elementary Arithmetic in an Ordered Field and claim 8 of Elementary Order Arithmetic in an Ordered Field, and then $\alpha-\varepsilon\le L$ together with $\varepsilon<\alpha-L$, again by claim 8, gives $L<\alpha-\varepsilon\le L$, a contradiction. So $\alpha\le L$ in every case.

Step 4: Fatou's lemma. For every $j$ and every $y\in Q$ we have $g_{j}(y)\le|u_{m_{j}}|_{Q}|^{6}(y)$, the two being equal off $E$ and the right-hand side being nonnegative on $E$. So The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, Elementary Properties of the p-Seminorm §power applied with $p=6$, Step 1 and Properties of Real Powers of Nonnegative Real Numbers §monotone give

Consequently $(5N)^{6}$ is an upper bound of every greatest lower bound occurring in the definition of $\liminf_{j}\int_{\mathbb{T}^{n}}g_{j}\,dx$, so that limit inferior is at most $(5N)^{6}$. By Fatou's Lemma, applied to the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ and the sequence $(g_{j})$, and by Step 3 with The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison,

Step 5: conclusion. The maps $g$ and $|v|^{6}$ agree at every point of $Q\setminus E$, and $E$ is null, so they agree almost everywhere and The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison gives

which is finite. The map $v$ is measurable with respect to $\mathcal{B}_{Q}$, being a member of $\mathcal{L}^{2}(\mathbb{T}^{n})$, so $v\in\mathcal{L}^{6}(\mathbb{T}^{n})$ by Power-Integrable Functions and the p-Seminorm §space. Finally, by Power-Integrable Functions and the p-Seminorm §seminorm, Properties of Real Powers of Nonnegative Real Numbers §monotone and Properties of Real Powers of Nonnegative Real Numbers §inverse,

which is claim 2.