(New page: #4. Use Ho(e)lder's Inequality to obtain <math>0\leq \int_0^xf\leq [ \int_0^x(f^2) ] ^\frac{1}{2} [\int_0^x 1^2]^\frac{1}{2}=||f\chi_{[0,x]}||_2 x^\frac{1}{2}</math>) |
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#4. Use Ho(e)lder's Inequality to obtain | #4. Use Ho(e)lder's Inequality to obtain | ||
| − | <math>0\leq \int_0^xf\leq [ \int_0^x(f^2) ] ^\frac{1}{2} [\int_0^x 1^2]^\frac{1}{2}=||f\chi_{[0,x]}||_2 x^\frac{1}{2}</math> | + | <math>0\leq \int_0^xf\leq \big[ \int_0^x(f^2) ] ^\frac{1}{2} [ \int_0^x 1^2]^\frac{1}{2}= ||f\chi_{[0,x]}||_2 x^\frac{1}{2}</math> |
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| + | Assuming <math>x>0 </math> we have | ||
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| + | <math>\dfrac{F(x)}{x} \leq ||f\chi_{[0,x]}||_2 \to 0 </math> as <math>x\to 0_+ </math> | ||
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| + | since <math>f^2 \in L^1 </math> | ||
Revision as of 10:37, 18 July 2008
- 4. Use Ho(e)lder's Inequality to obtain
$ 0\leq \int_0^xf\leq \big[ \int_0^x(f^2) ] ^\frac{1}{2} [ \int_0^x 1^2]^\frac{1}{2}= ||f\chi_{[0,x]}||_2 x^\frac{1}{2} $
Assuming $ x>0 $ we have
$ \dfrac{F(x)}{x} \leq ||f\chi_{[0,x]}||_2 \to 0 $ as $ x\to 0_+ $
since $ f^2 \in L^1 $
