# Khintchine's Inequality variant

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 $$c(p)$$?

## 1 Answer

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 $$(\int_0^1\int_0^1|g(x,y)|^pdxdy)^{1/p}$$.