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Re: a multiplier operator on the n-torus

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