Sobolev embedding on the torus: $H^{\sigma }\subseteq C^j$ for $\sigma >j+3\mathrm{/}2$, with continuous representatives

By [F1] with ${\sigma }^{\mathrm{\prime }}=\sigma$, ${\sum }_k\mathrm{\mid }\widehat{u}(k){\mathrm{\mid }}_2\le W_{\sigma }\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{\sigma }}<+\mathrm{\infty }$. For each $M$ the partial sum $S_M={\sum }_{\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}\le M}\widehat{u}(k)e_k$ is continuous with $\mathrm{\mid }{S}_M(x){\mathrm{\mid }}_2\le {\sum }_{\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}\le M}\mathrm{\mid }\widehat{u}(k){\mathrm{\mid }}_2\le W_{\sigma }\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{\sigma }}$. Applying [F2] componentwise to the series with dominating constants $\mathrm{\mid }\widehat{u}(k){\mathrm{\mid }}_2$ gives: $S_M$ converges uniformly on ${\mathbb{T}}^3$ to a continuous field $\tilde{u}$, and $\mathrm{\parallel }\tilde{u}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\le W_{\sigma }\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{\sigma }}$.

Nonempty open sets have positive measure. For nonempty open $U\subseteq {\mathbb{T}}^3$, its inverse image under the surjective quotient map $Q:{\mathbb{R}}^3\to {\mathbb{T}}^3$ is nonempty open by [F6]. Choose a point $y$ in it with no integer coordinate: such points meet every open box, because each coordinate interval contains a noninteger real. Translate $y$ by an integer vector to $z\in (0,1{)}^3$; periodicity leaves $z\in Q^{-1}[U]$. Openness gives a sufficiently small nonempty box $I$ about $z$ contained in $Q^{-1}[U]\cap (0,1{)}^3$. Its volume is positive, so $dx(U)={\lambda }_3(Q^{-1}[U]\cap [0,1{)}^3)\ge {\lambda }_3(I)>0$ by [F5] and [F6].

For each $m\in {\mathbb{Z}}^3$ and every $M\ge \mathrm{\mid }m{\mathrm{\mid }}_{\mathrm{\infty }}$, the integrals ${\int }_{{\mathbb{T}}^3}{S}_M{e}_{-m}dx$ equal $\widehat{u}(m)$ by orthonormality of the characters; the sequence $S_M$ is uniformly bounded by $W_{\sigma }\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{\sigma }}$ and converges pointwise (indeed uniformly) to $\tilde{u}$, so bounded convergence [F4] gives $\widehat{\tilde{u}}(m)=\int \tilde{u}{e}_{-m}dx={\lim }_M\int S_M{e}_{-m}dx=\widehat{u}(m)$ for every $m$. Hence $u$ and $\tilde{u}$ have the same coefficient family; by [F3] they agree in $L^2$, so $\tilde{u}=u$ almost everywhere.

A continuous field vanishing almost everywhere is zero. Let $f$ be a continuous scalar or vector field with $f=0$ almost everywhere. Then $\mathrm{\mid }f{\mathrm{\mid }}^2=0$ almost everywhere, so $\int \mathrm{\mid }f{\mathrm{\mid }}^2=0$ by [F5]. If $f(x_0)\mathrm{\ne }0$ for some $x_0$, continuity (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement) gives a nonempty open neighborhood $U$ of $x_0$ on which $\mathrm{\mid }f{\mathrm{\mid }}_2>\mathrm{\mid }f(x_0){\mathrm{\mid }}_2\mathrm{/}2>0$; that neighborhood is an open set of positive measure by step 1.2, whence $\int \mathrm{\mid }f{\mathrm{\mid }}^2\ge (\mathrm{\mid }f(x_0){\mathrm{\mid }}_2\mathrm{/}2{)}^{2}dx(U)>0$ by monotonicity [F5], a contradiction. Hence $f=0$. Applying this to the difference of two continuous fields proves that fields equal almost everywhere are equal, proving claim 2.

Let $j\ge 0$ and $\sigma >j+3\mathrm{/}2$. For each multi-index $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le j$ the family $c_{\alpha }(k):=(2\pi ik{)}^{\alpha }\widehat{u}(k)$ has $H^{\sigma -\mathrm{\mid }\alpha \mathrm{\mid }}$-norm at most $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{\sigma }}$ (because $\mathrm{\mid }(2\pi k{)}^{\alpha }\mathrm{\mid }\le w_k^{\mathrm{\mid }\alpha \mathrm{\mid }}$), and $\sigma -\mathrm{\mid }\alpha \mathrm{\mid }>3\mathrm{/}2$, so by [F1] the series $f_{\alpha }:={\sum }_k(2\pi ik{)}^{\alpha }\widehat{u}(k)e_k$ converges absolutely and uniformly with $\mathrm{\parallel }{f}_{\alpha }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\le W_{\sigma -\mathrm{\mid }\alpha \mathrm{\mid }}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{\sigma }}$ by the argument of step 1.1.

Termwise differentiation. We prove ${\mathrm{\partial }}^{\alpha }\tilde{u}=f_{\alpha }$ for every $\mathrm{\mid }\alpha \mathrm{\mid }\le j$ by induction on $\mathrm{\mid }\alpha \mathrm{\mid }$. Fix an index $i$ and a multi-index $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }+1\le j$, and assume ${\mathrm{\partial }}^{\alpha }\tilde{u}=f_{\alpha }$. Put $v_M:={\mathrm{\partial }}^{\alpha }{S}_M$; then $v_M\to f_{\alpha }$ uniformly by step 2.3, and ${\mathrm{\partial }}_i{v}_M={\sum }_{\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}\le M}(2\pi ik{)}^{\alpha +e_i}\widehat{u}(k)e_k\to f_{\alpha +e_i}$ uniformly. For fixed $x$ and small $h$ with $x+h{e}_i\in {\mathbb{T}}^3$, the componentwise fundamental theorem (If $f:[a,b]\to {\mathbb{R}}^m$ is differentiable with integrable $f^{\mathrm{\prime }}$ then ${\int }_a^b{f}^{\mathrm{\prime }}=f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz, claim 1), applied to the smooth real curve $s\mapsto v_M(x+sh{e}_i)$ on $[0,1]$, gives $v_M(x+h{e}_i)-v_M(x)=h{\int }_0^1{\mathrm{\partial }}_i{v}_M(x+sh{e}_i)ds$ for each $M$; for continuous coordinate functions its integral agrees with the tagged-sum integral (upper/lower sums squeeze tagged sums by uniform continuity). The norm bound in Riemann integration of continuous curves in a complete normed space, with the fundamental theorem of calculus and coefficient functionals, claim 1, therefore allows passage to the uniform limit, giving $f_{\alpha }(x+h{e}_i)-f_{\alpha }(x)=h{\int }_0^1{f}_{\alpha +e_i}(x+sh{e}_i)ds$, and continuity of $f_{\alpha +e_i}$ gives $(f_{\alpha }(x+h{e}_i)-f_{\alpha }(x))\mathrm{/}h\to f_{\alpha +e_i}(x)$ as $h\to 0$. Hence ${\mathrm{\partial }}_i{f}_{\alpha }=f_{\alpha +e_i}$, which is the induction step; the base $\mathrm{\mid }\alpha \mathrm{\mid }=0$ is the definition of $\tilde{u}$, so all claimed identities hold and $\tilde{u}\in C^j$ with the displayed sup bounds. This proves claim 3.

Claim 4: for $u\in H^{\sigma +1}$ with ${\mathrm{\partial }}_{j}u$ continuous, ${\mathrm{\partial }}_{j}u\in H^{\sigma }$ and the continuous field $w:={\sum }_k\widehat{{\mathrm{\partial }}_{j}u}(k)e_k$ produced by steps 2.1-2.2 satisfies $w={\mathrm{\partial }}_{j}u$ almost everywhere, hence $w={\mathrm{\partial }}_{j}u$ everywhere by step 2.2; therefore $\sup \mathrm{\mid }{\mathrm{\partial }}_{j}u{\mathrm{\mid }}_2=\mathrm{\parallel }w{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\le W_{\sigma }\mathrm{\parallel }{\mathrm{\partial }}_{j}u{\mathrm{\parallel }}_{H^{\sigma }}\le W_{\sigma }\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{\sigma +1}}$, because $\mathrm{\parallel }{\mathrm{\partial }}_{j}u{\mathrm{\parallel }}_{H^{\sigma }}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H^{\sigma +1}}$ by Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection.