The H^1 to C^0 bound fails at s minus m equal to one: dyadic shells have an unbounded C^0/H^1 ratio

Given: the torus ${\mathbb{T}}_{2\pi }^3$, its characters ${\chi }_k$, the norms $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{H^1}$ and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{C^0}$, and the shell family $(u_N)$.

Fourier norm formula: for $u={\sum }_k{a}_k{\chi }_k$ with finitely many nonzero coefficients, $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^1}^2={\sum }_k(1+\mathrm{\mid }k{\mathrm{\mid }}^2)\mathrm{\mid }{a}_k{\mathrm{\mid }}^2$ and $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^0}={\sup }_x\mathrm{\mid }{\sum }_k{a}_k{\chi }_k(x)\mathrm{\mid }$; in particular $u_N(0)=\mathrm{\#}\{k:2^N\le \mathrm{\mid }k\mathrm{\mid }<2^{N+1}\}$ for the shell family, whose coefficients are $1$ on the shell (Periodic Fourier differentiation identity and equivalence of the derivative and Fourier-weight Sobolev norms, The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s).

Lattice-point counts: there are constants $0<c\le C$ with $c{2}^{3N}\le \mathrm{\#}\{k\in {\mathbb{Z}}^3:2^N\le \mathrm{\mid }k\mathrm{\mid }<2^{N+1}\}\le C{2}^{3N}$ for all $N\ge 0$ (the shell is a spherical annulus of radius comparable to $2^N$; each unit cube contains at most one lattice point and volume comparison gives both bounds) (The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s). [L2]

In the embedding lemma the hypothesis $s-m>3\mathrm{/}2$ is used exactly at the finiteness of ${\sum }_k(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{-(s-m)}$; at $s-m=1$ this sum diverges (Periodic Sobolev embedding: H^s on the three-torus controls C^m when s minus m exceeds three halves). [L3]