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?
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?
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 $(p_2/p_1')'$...