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 $\alpha>0$.*



Which rate of growth of the Sobolev norms guarantees analyticity?
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.


