Assume $1 I want to prove that
$$\left|\int_X f_1 f_2\cdots f_N\; d\mu \right| \le \lVert f_1\rVert_{p_1} \lVert f_2\rVert_{p_2} \cdots \lVert f_N\rVert_{p_N}.$$ How can I directly adjust Hölder's inequality for it?
Generalization of Hölder's inequality
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analysis
measure-theory
inequality
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3Do you need this? http://en.wikipedia.org/wiki/H%C3%B6lder's_inequality#Generalization_of_H.C3.B6lder.27s_inequality – 2012-06-13
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0Yes. That document is very helpful for this question. – 2012-06-13
1 Answers
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Hint
Start with Hölder on the function $f_1$ and $g_1=f_2f_3\dots f_n$ using $p_1$ and $p_1'$. That is $$\left|\int f_1 g_1 dx\right|\leq\left(\int |f_1|^{p_1}dx\right)^{1/p_1}\left(\int |g_1|^{p_1'}dx\right)^{1/p_1'}$$ where $p_1'=p_1/(p_1-1)$.
Apply Hölder on $|f_2|^{p_1'}$ and $g_2 = |f_3f_4\dots f_n|^{p_1}$ using $p_2/p_1'$ and $(p_2/p_1')'$...
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0Oh then I can do it by induction. – 2012-06-13
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0@japee Sounds like a good idea! – 2012-06-13