Re: L-3 space anyone

On 30 joulu, 15:49, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Mon, 29 Dec 2008 20:21:27 -0800 (PST), Gc <Gcut...@xxxxxxxxxxx>
wrote:

On 30 joulu, 03:25, Robert Israel
<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
Its quite well known that fourier series converges to their functions
almost everywhere if that function is periodic and L-2 integrable.

but what about L-3 spaces ?

On a space of finite measure, L_3 is contained in L_2.

Or is it the other way?

No, it's the way he stated it.

I don`t still understand, because in my Rudin 1966, I have this
question 5 on page 70.
" ´Suppose f is a complex measurable function on X, m is a positive
measure on X, and m(X) = 1.
5,a) prove that norm(f)_r =< norm(F)_s, if 0 < r < s < oo."
Doensn`t this mean that L_r is contained in L_s?
Also I would think that when the exponent s is more bigger than 1 than
r , more bigger the integral is when the values of f are bigger than
1.

An easy way to keep this straight: On a space of
finite measure it's clear that a bounded (measurable)
function is integrable. So L^infinity is contained
in L^1.

Actually, Carleson's theorem is true in L_p for any p > 1.

It wasn`t trivial to extend the Carleson theorem for greater p:s, it
was done by Hunt.

If "greater ps" means values of p greater than 2 then yes, this
was trivial, in fact it's not an extension at all. The not quite so
trivial part was 1 < p < 2.

The general version is known as Carleson-Hunt
theorem. There is also extension to non-integer p:s, and there is a
"race" how close to p=1 you can go keeping the theorem valid.

??? Maybe you mean there _was_ a race?

I need to learn more about this.

no book references plz.

It might do you good to read a book once in a while.
--
Robert Israel              isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics        http://www.math.ubc.ca/~israel
University of British Columbia            Vancouver, BC, Canada

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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