Re: Fourier transform

On Thu, 06 Aug 2009 07:32:02 EDT, omega
<blue_jaguar@xxxxxxxxx> wrote:

It is known that the Fourier transform takes
L^p(R^m)
to L^{p'}(R^m),
p\in [1,2]
Please, is for the case p\in (1,2) the Fourier
transform a
homeomorphism ?

Well, it suffices to remark that

- the Fourier transform is a linear operator and
- the Banach spaces L^p(R^m) and L^q(R^m) are
linearly homeomorphic iff p = q.

How do you prove that last statement?

Say, using type and cotype (of course, this is not the unique way).

Namely, if the above spaces are isomorphic, then their types and cotypes must coincide, i.e.,
min {2, p} = min {2, q}, and
max {2, p} = max {2, q},
so that ...


So that, the answer is negative.

David C. Ullrich

"Understanding Godel isn't about following his formal
proof.
That would make a mockery of everything Godel was up
to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

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