  In this note we show how to obtain regularity estimates for the fractional
Laplacian on the multidimensional torus $\mathbb{T}^n$ from the fractional
Laplacian on $\mathbb{R}^n$. Though at first glance this may seem quite
natural, it must be carefully precised. A reason for that is the simple fact
that $L^2$ functions on the torus can not be identified with $L^2$ functions on
$\mathbb{R}^n$. The transference is achieved through a formula that holds in
the distributional sense. Such an identity allows us to transfer Harnack
inequalities, to relate the extension problems, and to obtain pointwise
formulas and H\"older regularity estimates.
