# Holder Inequality question...

Dear all, I am trying to prove the following question :

If f is in L^p(0,1) for some p, 1 <= p < infinity, that

lim y^[(1-p)/p] * integral_[0,y] f(x)dx = 0.
y--->0+

I'm not sure if my solution is valid, and I was wondering if someone could
comment : Well,

y^[(1-p)/p] * integral_[0,y] f(x)dx <= y^[(1-p)/p] * integral_[0,y] |f(x)|dx
= y^[(1-p)/p] * integral_[0,y] |f(x)*1|dx <= y^[(1-p)/p] * ||f||_p ||1||_q =
y^[(1-p)/p] * ||f||_p (y)^(1/q) = y^[(1-p)/p + 1/q] ||f||_p = y^(1-1/p)
||f||_p

The last term goes to 0 as y ---> 0+ since f is in L^p.

One question I have is the following : When I write ||f||_p, I must mean
(integral_[0,y] |f|^p)^(1/p). That is, the integral goes from 0 to y, not 0
to 1. The reason is that I want to apply Holder to integral_[0,y]
|f(x)*1|dx, and that integral goes from 0 to y. Am I right? This means
that f must be in L^p(0,y) for all y < 1. Is it true in general that if f
is in L^p(0,1), then f is in L^p(0,y) for all 0 < y < 1?

James

.

## Relevant Pages

• Re: OT: Katrina -- Whom To Blame
... James doesn't wanna poor billions of tax dollars into ... He's a baaaad, baaaad man." ... assistance from Alaska is limited, ... if any hurricane victims are ...
(alt.smokers.cigars)
• Columns
... paragraph.The second columun starts with a new paragraph. ... Thanks for any assistance. ... I am at a dead end with this one. ... Prev by Date: ...
(microsoft.public.word.general)
• USER FILE
... Can anyone help me rectivate the file and tansfer the ... ANy time I as the adminstrator try to open the file I get this ... I am really hoping I can get some assistance. ... Prev by Date: ...
(microsoft.public.windowsxp.general)
• Re: Printing part of merged envelope file
... > I've created a merged file of about 750 envelopes. ... each envelope is a separate section with each page ... > Thanks for any assistance! ... Prev by Date: ...
(microsoft.public.word.mailmerge.fields)
• logging users against an update
... I don’t want to upsize to sql. ... I can then see how many records a user has updated Dailey? ... James ... Prev by Date: ...
(microsoft.public.access.security)