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- Jan 26, 2012

- 995

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**Problem**: Suppose that $(X,\mu)$ is a measure space. For $0<r<p<s<\infty$, assume that $f\in L_{\mu}^r(X)\cap L_{\mu}^s(X)$. Show that $f\in L_{\mu}^p(X)$ and that

\[\|f\|_{L_{\mu}^p(X)} \leq \|f\|_{L_{\mu}^r(X)}^{\theta}\|f\|_{L_{\mu}^s(X)}^{1-\theta}\qquad\text{for}\qquad \frac{1}{p}=\frac{\theta}{r}+\frac{1-\theta}{s}.\]

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Here's a hint for this week's question.

Use the generalized Hölder inequality for the second half of the problem.

Remember to read the POTW submission guidelines to find out how to submit your answers!