(New page: 8a) Given such an <math>r<s</math> for any <math>p \in (r,s)</math> we have <math>\int_X |f|^p = \int_{\{x:|f|\leq 1 \}} |f|^p+\int_{\{x:|f|> 1 \}} |f|^p \leq \int_{\{x:|f|\leq 1 \}} |f|...) |
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<math>\int_X |f|^p = \int_{\{x:|f|\leq 1 \}} |f|^p+\int_{\{x:|f|> 1 \}} |f|^p \leq \int_{\{x:|f|\leq 1 \}} |f|^r+\int_{\{x:|f|> 1 \}} |f|^s <\infty</math> | <math>\int_X |f|^p = \int_{\{x:|f|\leq 1 \}} |f|^p+\int_{\{x:|f|> 1 \}} |f|^p \leq \int_{\{x:|f|\leq 1 \}} |f|^r+\int_{\{x:|f|> 1 \}} |f|^s <\infty</math> | ||
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| + | b) Recall that a function <math>\sigma (x):(r,s)\rightarrow \mathbb{R}</math> is convex iff <math>\sigma(tp_1+(1-t)p_2) \leq t\sigma (p_1)+(1-t)\sigma( p_2)</math> | ||
| + | for any <math>t\in [0,1]</math> and any <math>r<p_1<p_2<s</math>. | ||
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| + | Now, by Holder, | ||
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| + | <math>\int|f|^{tp_1}|f|^{(1-t)p_2} \leq \|f^{tp_1}\|_{\frac{1}{t}}\|f^{tp_2}\|_{\frac{1}{1-t}}</math> since f is in <math>L^{p_1}</math> and <math>L^{p_2}</math> and since <math>t+(1-t)=1</math>. | ||
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| + | Since log is increasing we may take log of both sides while preserving the inequality: | ||
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| + | <math>log[\int|f|^{tp_1}|f|^{(1-t)p_2}] \leq log[\|f^{tp_1}\|_{\frac{1}{t}}\|f^{tp_2}\|_{\frac{1}{1-t}}] | ||
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| + | =log[(\int|f|^{p_1})^t(\int|f|^{p_2})^{1-t}]= log[(\int|f|^{p_1})^t]+log[(\int|f|^{p_2})^{1-t}] | ||
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| + | </math> <math> = tlog[\int|f|^{p_1}]+(1-t)log[\int|f|^{p_2}]= tlog(\phi (p_1))+(1-t)log(\phi (p_2))</math> | ||
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| + | So <math>tlog(\phi (p_1))+(1-t)log(\phi (p_2)) \geq log(\int|f|^{tp_1}|f|^{(1-t)p_2}) = log(\int|f|^{tp_1+(1-t)p_2})=log(\phi (tp_1+(1-t)p_2)).</math> | ||
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| + | Thus <math>log(\phi)</math> is convex on <math>(r,s).</math> | ||
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| + | --[[User:Wardbc|Wardbc]] 19:57, 14 July 2008 (EDT) | ||
Revision as of 19:57, 14 July 2008
8a)
Given such an $ r<s $ for any $ p \in (r,s) $ we have
$ \int_X |f|^p = \int_{\{x:|f|\leq 1 \}} |f|^p+\int_{\{x:|f|> 1 \}} |f|^p \leq \int_{\{x:|f|\leq 1 \}} |f|^r+\int_{\{x:|f|> 1 \}} |f|^s <\infty $
b) Recall that a function $ \sigma (x):(r,s)\rightarrow \mathbb{R} $ is convex iff $ \sigma(tp_1+(1-t)p_2) \leq t\sigma (p_1)+(1-t)\sigma( p_2) $ for any $ t\in [0,1] $ and any $ r<p_1<p_2<s $.
Now, by Holder,
$ \int|f|^{tp_1}|f|^{(1-t)p_2} \leq \|f^{tp_1}\|_{\frac{1}{t}}\|f^{tp_2}\|_{\frac{1}{1-t}} $ since f is in $ L^{p_1} $ and $ L^{p_2} $ and since $ t+(1-t)=1 $.
Since log is increasing we may take log of both sides while preserving the inequality:
$ log[\int|f|^{tp_1}|f|^{(1-t)p_2}] \leq log[\|f^{tp_1}\|_{\frac{1}{t}}\|f^{tp_2}\|_{\frac{1}{1-t}}] =log[(\int|f|^{p_1})^t(\int|f|^{p_2})^{1-t}]= log[(\int|f|^{p_1})^t]+log[(\int|f|^{p_2})^{1-t}] $ $ = tlog[\int|f|^{p_1}]+(1-t)log[\int|f|^{p_2}]= tlog(\phi (p_1))+(1-t)log(\phi (p_2)) $
So $ tlog(\phi (p_1))+(1-t)log(\phi (p_2)) \geq log(\int|f|^{tp_1}|f|^{(1-t)p_2}) = log(\int|f|^{tp_1+(1-t)p_2})=log(\phi (tp_1+(1-t)p_2)). $
Thus $ log(\phi) $ is convex on $ (r,s). $
--Wardbc 19:57, 14 July 2008 (EDT)
