To show $\|\hat{f}\|_q \lesssim \|f\|_p$ does not hold for $p > 2$ where $1/p + 1/q = 1$, I was given a hint to consider a Schwartz function $\phi(\xi)$ whose support is on $B(0,1)$, and let $$ \hat{f_N}(\xi) = \sum_{n=1}^N e^{2\pi in\xi_1}\phi(\xi-10ne_1). $$ It is straightforward to see that $\|\hat{f_N}\|_q = N^{1/q}\|\phi\|_q$ since the support of $\phi(\xi-10ne_1)$ for different $n$ are disjoint. Now to show $p \le 2$, it suffices to show $\|f_N\|_p \lesssim N^{1/p}$, which after some simplification is equivalent to $$ \bigg\|\sum_{n=1}^N e^{20\pi i nx_1}\check{\phi}(x+ne_1) \bigg\|_p^p \lesssim N. $$ Can anyone offer some help on estimating the left hand side?
BTW I know there is another proof which is similar to this construction, and uses Khinchin's Inequality to estimate(Using Khinchin's inequality). But here we do not want to introduce the Rademacher variable.
Intuitively, each summand of $f_N(x)$ looks like a Gaussian bump of width $\sim 1$ and height $\sim 1$, centered at points that are $\sim 1$-separated, so $f_N(x)$ should look not much different than $\sum_{n=1}^N\chi_{B(0,1)}(x+ne_1)$, where $B(x,r)$ is notation for the ball of radius $r$ centered at $x$. Of course $\|\sum_{n=1}^N\chi_{B(0,1)}(x+ne_1)\| \sim N^{1/p}$.
Here is a sketch. Estimate, \begin{align*} \int|\sum_n\check\phi(x+ne_1)e^{20\pi inx_1}|^p\,dx\le \int(\sum_n|\check\phi(x+ne_1)|)^p\,dx. \end{align*} The key point is that for any $x\in\bigcup_{n}B(-ne_1,10)$, $\sum_n|\check\phi(x+ne_1)|$ is dominated by the contribution of at most $O(1)$ terms of the sum, and for $x\notin\bigcup_{n}B(-ne_1,10)$, you can use the smallness of the Schwartz tails. (The choice of $10$ is somewhat arbitrary, any constant $c>1$ would probably suffice, with a constant depending on $c$ hiding in the $\sim$ notation.)