Let $f_n\in L_{\infty}([0,1])$ and $(r_n)_n$ be the Radermacher sequence. For each $n\in\mathbb{N}$ we define the function $g:[0,1]^2\to\mathbb{R}$ by $g_n(x_1,x_2)=r_n(x_1)f_n(x_2)$. Show that for every $\infty>p\geq 1$ there exists a constant $c(p)$ such that: $\|\sum_{n=1}^kg_n\|_{L_p([0,1]^2)}\geq c(p)\|(\sum_{n=1}^k{f_n}^2)^{\frac{1}{2}}\|_{L_p([0,1])}$.
I can see that this is essentially Khintchine's Inequality with $f_n$'s relacing the constants so I suspect that a proof might mimic the original proof but I am not certain how to proceed. Also is there any way to estimate $c(p)$?
Ok I revisited and it is rather trivial: It is just the regular Kintchine with the $f_i$ as constants by applying the $L_p$ norm to both sides and noticing that the $L_p[0,1]^2$ norm of $g(x,y)$ is just $(\int_0^1\int_0^1|g(x,y)|^pdxdy)^{1/p}$.