- Thread starter
- Moderator
- #1
- Jan 26, 2012
- 995
Here's this week's problem.
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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.
Remember to read the POTW submission guidelines to find out how to submit your answers!
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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!