Suppose $f\in L^{p}(\mathbb{R}^{n}) \cap L^{q}(\mathbb{R}^{n})$. How can I prove that for any $p \lt r \lt q$, $ \lVert f \rVert_{r} \leq (\lVert f \rVert_{p})^{(1/r-1/q)/(1/p-1/q)} (\lVert f \rVert_{q})^{(1/r-1/q)/(1/p-1/q)}\:? $ I tried using Hölder to show that $f$ is in $L^{r}$ but I'm completely lost on how to go about it...
Sharp interpolation inequality for Lebesgue spaces
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real-analysis
lp-spaces
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0See also: http://math.stackexchange.com/q/163042/ and http://math.stackexchange.com/q/31683 – 2012-06-25
1 Answers
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If $p
$|f|^r=|f|^{r\theta}\ |f|^{r(1-\theta)} =|f|^{p\ \frac{r\theta}{p}}\ |f|^{q\ \frac{r(1-\theta)}{q}}$,
and you can apply Hölder's inequality, because $|f|^{p \frac{r\theta}{p}} \in L^{\frac{p}{r\theta}}$ and $|f|^{q\ \frac{r(1-\theta)}{q}} \in L^{\frac{q}{r(1-\theta)}}$ and the exponents $\frac{p}{r\theta}, \frac{q}{r(1-\theta)}$ are conjugate.
Finally, to get your claim it suffices to compute explicitly the value of $\theta$.
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0@squ1d: I'm glad you succeeded... Finding the right exponents to apply Hölder's inequality can be quite messy sometimes. XD – 2011-03-23