I'm trying to go through a proof of Khintchine inequality, available here: sadly only a polish version available, but perhaps language will not be a barrier here. I understand everything perfectly up to the third line on page 5, where we make a jump from
$||S||_{L^p}^{1-s}||S||^s_{L^3}\\$
to
$3^{6(1/p - 1/2)}||S||_{L^p}$ ,
where $S:= \sum_{k=1}^n x_k r_k$
and at this point I am completely lost. Where on earth is this bound coming from? Is it only true because we deal with Radamacher variables defines as functions $r_i$ of the form of $t \mapsto sgn(sin(2i\pi t))$ or does it hold in general for functions in $L^p$ spaces? I thought it was just some sort of a trick but after a considerable time of playing with exponents etc. I still do not have any clue as to why this might be true.
PS. If you struggle with the language, I'm quite happy to translate the details, but I suppose it is understandable (I might a bit biased on that one though since I actually know the language :) ). Any help will be greatly appreciated. Thanks!
I think the argument is the following. Thanks to the "$p=3$" case, you have $$\Vert S\Vert_3\leq \sqrt{C_3}\, \Vert S\Vert_2\,.$$
This gives you $$\Vert S\Vert_2\leq \Vert S\Vert_p^{1-s}\times C_3^{s/2}\Vert S\Vert_2^s\, ;$$ in other words $$\Vert S\Vert_2^{1-s}\leq C_3^{s/2}\, \Vert S\Vert_p^{1-s}.$$ So you obtain $$\Vert S\Vert_2\leq C_3^{s/2(1-s)}\, \Vert S\Vert_p= C_3^{3(\frac1p-\frac12)} \,\Vert S\Vert_p\,,$$ because, as it turns out (and if I didn't make any mistake) $\frac{s}{1-s}=6\bigl(\frac1p-\frac12\bigr)$.
Now, apparently $C_3\leq 2\times 3-1=5$; so what you get in the end seems to be $$\Vert S\Vert_p\leq 5^{3\bigl(\frac1p-\frac12\bigr)}\,\Vert S\Vert_2\, .$$
This is not exactly what is given in the notes. Quite possibly, I made some mistake somewhere; but anyway, the method (i.e. to use the "$p=3$" case) is the correct one.