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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 $0<p$, 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 $L^{p}$.

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Vokram
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    See http://math.stackexchange.com/q/133773/5363 – t.b. Apr 22 '12 at 12:26
  • Thanks @t.b. ! But I am still confused about the second question. How should I deal with it when $f$ is not guaranteed to be in any $L^{p}$? DCT seems to require convergence of the integral. – Vokram Apr 22 '12 at 12:35
  • @NateEldredge On $[0,1]$ if $q<p$ then $L^{p}\subset{}L^{q}$ but not other other way round, so $f\notin{L^{p}}\nRightarrow{}f\notin{L^{q}}$. – Vokram Apr 22 '12 at 17:54
  • @Vokram: Oops, sorry, inequality fail. Comment deleted. – Nate Eldredge Apr 22 '12 at 18:45

2 Answers2

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For (2), you've addressed all cases except $f \notin L^p$ with $p &lt \infty$. 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 :)

Nate Eldredge
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I was thinking about this: if $q&ltp$ 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 $\|f\|_p=+\infty$.

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