Periodic Sobolev embedding: H^s on the three-torus controls C^m when s minus m exceeds three halves

(Absolute convergence bound.) For $\mathrm{\mid }\alpha \mathrm{\mid }\le m$, finite Cauchy--Schwarz gives ${\sum }_{k\in F}\mathrm{\mid }{k}^{\alpha }\mathrm{\mid }\mathrm{\mid }\widehat{u}(k)\mathrm{\mid }\le ({\sum }_{k\in F}\mathrm{\mid }{k}^{\alpha }{\mathrm{\mid }}^2(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{-s}{)}^{1\mathrm{/}2}{N}_s(u)\le ({\sum }_k(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{-(s-m)}{)}^{1\mathrm{/}2}{N}_s(u)$. Finite-subset suprema preserve this inequality.

(The weight sum is finite.) For $\theta >3\mathrm{/}2$, ${\sum }_{k\in {\mathbb{Z}}^3}(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{-\theta }<+\mathrm{\infty }$: the ball $\mathrm{\mid }k\mathrm{\mid }\le 1$ contributes at most $27$ terms, and the dyadic shell $2^j\le \mathrm{\mid }k\mathrm{\mid }<2^{j+1}$ ($j\ge 0$) contains at most $C_3{2}^{3j}$ lattice points each weighted by at most $2^{-2j\theta }$, so the shell sum is at most $C_3{2}^{-j(2\theta -3)}$ and ${\sum }_j{2}^{-j(2\theta -3)}<\mathrm{\infty }$ by the geometric series. Hence with $\theta :=s-m$ step 1.1 gives ${\sum }_k\mathrm{\mid }{k}^{\alpha }\mathrm{\mid }\mathrm{\mid }\widehat{u}(k)\mathrm{\mid }\le C_{s,m}{N}_s(u)\le C_{s,m}^{\mathrm{\prime }}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}$ by [L1].

(The limit series gives a $C^m$ representative.) For finite $F$ put $S_F:={\sum }_{k\in F}\widehat{u}(k){\chi }_k$, a $C^{\mathrm{\infty }}$ function. If $F,G$ are finite then $\mathrm{\parallel }{S}_F-S_G{\mathrm{\parallel }}_{C^m}\le {\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m}{\sum }_{k\in F\mathrm{△}G}\mathrm{\mid }{k}^{\alpha }\mathrm{\mid }\mathrm{\mid }\widehat{u}(k)\mathrm{\mid }$, and the tail sums tend to zero along the finite-subset net by [L3] and step 2.1; hence the derivative series for every $\mathrm{\mid }\alpha \mathrm{\mid }\le m$ satisfies the uniform Cauchy property on ${\mathbb{T}}_{2\pi }^3$. Applying the interval differentiation theorem [L5] successively along each coordinate direction and for each order gives a $C^m$ function $u^{\ast }$ with $D^{\alpha }{u}^{\ast }={\sum }_k(ik{)}^{\alpha }\widehat{u}(k){\chi }_k$ for every $\mathrm{\mid }\alpha \mathrm{\mid }\le m$, the series converging absolutely and uniformly by step 2.1; and $\mathrm{\parallel }{u}^{\ast }{\mathrm{\parallel }}_{C^m}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}$ by step 2.1. Since $S_F\to u$ in $L^2$ by [L2] and $S_F\to u^{\ast }$ uniformly, $u^{\ast }$ represents the class $u$; two continuous representatives of one $L^2$ class agree everywhere, giving uniqueness. This proves (a).

(The Hölder endpoint.) Put $d=d(x,y)$ and $q=s-m$, an integer at least $2$. Cauchy--Schwarz and [L6] bound the square of the derivative difference by $N_s(u{)}^{2}S(d)$, where $S(d)={\sum }_k(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{-q}\min \{4,\mathrm{\mid }k{\mathrm{\mid }}^2{d}^2\}$. For $0<d\le 1$ split at $\mathrm{\mid }k\mathrm{\mid }=d^{-1}$. A dyadic shell of radius $L\ge 1$ has at most $C{L}^3$ points. If $q=2$, its low-frequency contribution is at most $C{d}^{2}L$; summing shells up to $d^{-1}$ gives at most $Cd$, while high shells contribute at most $C{L}^{-1}$ and sum to at most $Cd$. If $q\ge 3$, the entire series ${\sum }_k\mathrm{\mid }k{\mathrm{\mid }}^2(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{-q}$ converges by the same shell estimate, so $S(d)\le C{d}^2$. Thus the endpoint exponent is $1\mathrm{/}2$ for $q=2$ and $1$ for $q\ge 3$, exactly $\min \{1,q-3\mathrm{/}2\}$. Smaller positive exponents follow since $d^{{\theta }_0}\le \max \{1,\mathrm{diam}({\mathbb{T}}^3{)}^{{\theta }_0-\theta }\}{d}^{\theta }$; for $d>1$ use the sup norm from (a), and for $d=0$ the claim is immediate. This proves (b) with all its stated endpoints.

(Linearity and convergence (c).) $D^{\alpha }$ is linear and the series representation is linear in the coefficients, so $u\mapsto u^{\ast }$ is linear; and for $u_j\to u$ in $H^s$, applying (a) to the difference gives $\mathrm{\parallel }{u}_j^{\ast }-u^{\ast }{\mathrm{\parallel }}_{C^m}\le C\mathrm{\parallel }{u}_j-u{\mathrm{\parallel }}_{H^s}\to 0$, which is uniform convergence of all derivatives of order at most $m$.