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Using Holder inequality, is it correct to express $$\parallel \int\limits_0^t f(s)g(s)ds \parallel ^p \leq T^{p-1}\int \limits_0^t\parallel f(s)\parallel^pds\int \limits_0^t\parallel g(s)\parallel^pds $$

madhuri
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1 Answers1

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No. Take $f(s)=g(s)=1$, $p=1$ and $t=1/2$.

Fred
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  • Then how to express $$\parallel \int\limits_0^t f(s)g(s)ds \parallel ^p $$ as $$ \int \limits_0^t\parallel f(s)\parallel^pds\int \limits_0^t\parallel g(s)\parallel^p ds $$ – madhuri Oct 05 '17 at 10:29