$L^p$ maximal estimates for quadratic Weyl sums

Let $S(x,t)$ denote the Weyl sum with associated polynomial $xn+t{n}^2$. Suppose that $\mathrm{\mid }S(x,t)\mathrm{\mid }$ attains its maximum for given $x$ at $t=t(x)$. We give upper and lower bounds of the same order of magnitude for the $L^p$ norm of $S(x,t(x))$.