Which rate of growth of the Sobolev norms guarantees analyticity? Let $u\in C^\infty(\mathbb T^k)$, where $\mathbb T^k$ is the $k$-dimensional torus. (Equivalently, $u\in\mathbb R^k$ and $u$ is $2\pi$-periodic with respect of each argument.) 
We define the semi-norm 
$$
\|u\|_s=\|(-\Delta)^{s/2}u\|_{L^2(\mathbb T^k)}=\Big(\sum_{\ell_1,\ldots,\ell_k\in\mathbb Z}(\ell_1^2+\cdots+\ell^2_k)^{s/2}\big|\hat{u}_{\ell_1,\ldots,\ell_k}\big|^2\Big)^{1/2},
$$ 
where $s>0$ and $u(x_1,\ldots,x_k)=\sum_{\ell_1,\ldots,\ell_k\in\mathbb Z}\mathrm{e}^{2\pi i(\ell_1x_1+\cdots\ell_kx_k)}\hat{u}_{\ell_1,\ldots,\ell_k}$.
My question is the following: Which rate of growth of $\|u\|_s$, as $s\to\infty$, implies that $u$ extends holomorphically to an open neighborhood of $\mathbb R^k$ in $\mathbb C^k$? 
More specifically: Which rate of growth of $\|u\|_s$, as $s\to\infty$, guarantees that $u$ extends holomorphically to 
$$
\Omega_\alpha= \{(x_1+iy_1,\ldots,x_k+iy_k): x_1,y_1,\ldots,x_k,y_k\in\mathbb R\,\&\,|y_1|,\ldots,|y_k|<\alpha\} \subset \mathbb C^k,
$$
for a given $\alpha>0$.
 A: The rate of growth must be $(cs)^s$ for some $c>0$.
In my sketch of the proof I assume for simplicity
that $k=1$ and Fourier coefficients $a_n$ are zero for $n<0$.
The function has an analytic extension in a neighborhood of the
unit circle if $|a_n|$ decrease faster than a geometric
progression, that is $\log |a_n|^2\leq-\delta n$ for some $\delta>0$.
Then your norm squared is
$$\Phi(s)=\sum_{1}^\infty n^s|a_n|^2\leq\sum e^{s\log n-\delta n}.$$
We consider this as a Dirichlet series of one complex variable $s$.
Let $m(s)$ be the maximal term of the sum in the RHS.
It is known from the theory of Dirichlet series that for every $\epsilon$
$$\Phi(s)\leq A(\epsilon)m(s+1+\epsilon).$$
The maximim term on the right hand side is found by calculus, by maximizing 
with respect to $n$, and we get
$$\Phi(s)\leq (cs)^s$$ for some $c>0.$
In the opposite direction, suppose that
 $\Phi(s)\leq (cs)^s$. Then
$$|a_n|^2n^s\leq (cs)^s.$$
This gives an estimate for $|a_n|$ with parameter $s$.
Minimizing with respect to $s$ 
we obtain the desired $\log|a_n|\leq -\delta n$, for some $\delta>0$.
For the inequality from the theory of Dirichlet series
(this is also no more than calculus) I refer to
the proof of Theorem III.2.1 of S. Mandelbrojt, Series de Dirichlet. Principes
et methodes, Paris Gauthier-Villars, 1969. The only references
that I know are in French, Russian and Ukrainian, sorry.
