Re: Function integrable for exactly one L^p norm
- From: Kira Yamato <kirakun@xxxxxxxxxxxxxxxxxx>
- Date: Wed, 13 Sep 2006 18:39:15 GMT
Ahh... Ok. Thanks for the hint. I will try to complete the problem.
-kira
David C. Ullrich wrote:
On Tue, 12 Sep 2006 11:52:05 GMT, Kira Yamato.
<kirakun@xxxxxxxxxxxxxxxxxx> wrote:
I need some help in showing rather or not:
given a fixed 1<=p<=infty, is there a function such that
\int |f|^r < infty
if and only if r=p?
Whether this is true depends on the measure space
(if M is the set of all mu(A) such that 0 < mu(A)
< infinity then the result is true if and only
if inf(M) = 0 and sup(M) = infinity.)
It's true on the line, for example. A hint to get
you started: Say the A_n (n = 1, 2, ...) are
disjoint sets with m(A_n) = 1/2^n. Let
a_n = 2^n/n^2, and set f = sum a_nb chi_{A_n}.
If p = ??? then you can check that the integral
of f^r is finite if and only if r <= p.
A similar construction gives g such that the
integral of g^r is finite if and only if r >= p;
if you take f and g to have disjoint support
then you're done for this value of p, and taking
powers gives the example you want for other
values of p.
Thanks
-kira
************************
David C. Ullrich
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- From: Kira Yamato
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- From: David C . Ullrich
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