6
$\begingroup$

I ran into the following problem when I was doing my homework, and I have no thoughts on where I should start with:

(1) If $f\in L^{2}$, show that $\displaystyle \lim_{p \rightarrow 1^{+}}\int_{[0,1]}|f|^{p}=\int_{[0,1]}|f|$

(2) If $0, show that $\displaystyle \lim_{q\rightarrow p^{-}}||f||_{q}=||f||_{p}$

My first thought was Generalized LDCT, but it didn't seem to work. I also made some other attempts but none of them were successful... Can anybody give me some hints on how I should look at this question?

Also, I know if $p\rightarrow\infty$ then $||f||_{p}\rightarrow||f||_{\infty}$ on $[0,1]$, but does similar continuity in p holds for other $L^{p}[0,1]$ norms in general?

Thank you!

Edit:

Sorry if I did not make it clear enough in the question. All $L^{p}$ refers to $L^p[0,1]$.

The first question is found here (thanks to t.b.), but the second question remains, mainly because $f$ is not guaranteed to be in any $L^{p}$.

  • 0
    @Vokram: Oops, sorry, inequality $f$ail. Comment deleted.2012-04-22

2 Answers 2

2

For (2), you've addressed all cases except $f \notin L^p$ with p < \infty. As $q \to p^-$, we have $|f|^q \to |f|^p$ pointwise, so by Fatou's lemma $\liminf_{q \to p^-} \int |f|^q \ge \int |f|^p = \infty.$ This means $\int |f|^q \to \infty$ as $q \to p^-$. Putting the powers of $1/q$ back in is left as an exercise :)

0

I was thinking about this: if q and $f \in L^p(0,1)$, then $f \in L^q(0,1)$. Just Hölder inequality. In case (2), you have to consider two cases: (i) $f \in L^p$ and $\|f\|_p=+\infty$.

  • 0
    Hmm... I'm particularly interested in how you deal with the case $||f||_{p}=\infty$ in (2)2012-04-22