I am trying to prove Problem 6.10 in "Classical and Multilinear Harmonic Analysis" by by Camil Muscalu and Wilhelm Schlag.
Here are some attempts,
\begin{align*}
\mathbb{E}\Big\|\sum_{n=-N}^N\epsilon_n\hat f(n)e(n\theta)\Big\|_p&=\mathbb{E}\Big(\int_0^1|\sum_{n=-N}^N\epsilon_n\hat f(n)e(n\theta)|^pd\theta\Big)^{1/p}\\
&\leq\Big(\mathbb{E}\int_0^1|\sum_{n=-N}^N\epsilon_n\hat f(n)e(n\theta)|^pd\theta\Big)^{1/p}\quad(\text{since $g(x)=x^{1/p}$ is concave for $p\geq1$})\\
&\leq C(\sum_{n=-N}^N|\hat f(n)|^2)^{1/2}\quad(\text{Khinchine's inequality})\\
&\leq C\|f\|_2
\end{align*}
But
$$
\mathbb{E}\Big\|\sum_{n=-N}^N\epsilon_n\hat f(n)e(n\theta)\Big\|_p\leq\sup_{\epsilon_n}\Big\|\sum_{n=-N}^N\epsilon_n\hat f(n)e(n\theta)\Big\|_p
$$
the inequality seems in the wrong direction. Any suggestion on how to do next? Thanks in advance!