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Assume $1<p_k< \infty$ for $k=1,\ldots,N$ , and $\displaystyle\sum^N_{k=1}\frac{1}{p_k} =1$.

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?

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Do you need this? en.wikipedia.org/wiki/… –  ziyuang Jun 13 '12 at 5:05
    
Yes. That document is very helpful for this question. –  japee Jun 13 '12 at 5:11

1 Answer 1

up vote 5 down vote accepted

Hint

  1. 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 $p_1'=p_1/(p_1-1)$.

  2. 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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Oh then I can do it by induction. –  japee Jun 13 '12 at 6:07
    
@japee Sounds like a good idea! –  AD. Jun 13 '12 at 11:58

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