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• ## Re: a multiplier operator on the n-torus

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• Thanks a lot.
May 15, 2009 1 of 4
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Thanks a lot.

--- In harmonicanalysis@yahoogroups.com, "Philip Gressman" <ptgressman@...> wrote:
>
> 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.
> >
>
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