The similarity between the Rademacher functions $\{r_k\}_{k=1}^\infty$ on $[0,1]$ and the lacunary frequencies $\{e^{i2^k\theta}\}_{k=0}^\infty$ $[0,2\pi]$ is easy to see.
For the Rademacher functions, we have the following theorem.
For each $1 \le p < \infty$ there is a bound $A_p$ so that $$\|F\|_{L^p} \le A_p \|F\|_{L^2},$$ for all $F \in L^2([0, 1])$ of the form $F(t) = \sum_{n=1}^\infty a_n r_n(t)$.
I want to prove that we have a similar result for $F(t) = \sum_{n=0}^\infty a_n e^{i2^nt}$. Where should I start? Unlike the Rademacher functions, the lacunary frequencies are clearly not probability independent.
EDIT: I found a more general result here. But I still wonder if there is a much simpler proof for this special case.
You may start at chapter XV of "Trigonometric Series" by Zygmund, where Theorem $2.1$ contains the result you are interested in. The proof is highly nontrivial, and I doubt that there exist any simpler proofs.