This reply assumes that you left out a square root: (1+|m|^2)^{(n/2-1)/2}.

This operator may be expressed as convolution with a kernel k(x) which has a
singularity like |x|^{-n/2-1} at the origin. The kernel just fails to belong to
L^{2n/(n-2)} (it's in the corresponding weak space). If it actually did belong
to this particular L^p space, the boundedness question you asked would be an
immediate consequence of Young's inequality for convolutions. To take care of
the details, you should look at the proof of the Hardy-Littlewood-Sobolev
inequality appearing in Stein's _Harmonic Analysis_ (the proof there is for R^n,
but it goes through on the torus without any significant changes).

-Philip

--- In harmonicanalysis@yahoogroups.com, "lakhmau" <lakhmau@...> wrote:
>
> Dear all,
>
> I would like to know why the (Bessel-like ?) operator which maps
>
> exp(2i.pi.m.x) |-> exp(2i.pi.m.x) / (1 + |m|^2)^{n/2 - 1}
>
> for any multi-index m \in \Z^n maps the space
>
> L^{n/(n-1)}((0,1)^n)
>
> into L^2((0,1)^n) ?
>
> Sorry, the post is not really readable...
>
> Thanks,
> L.
>