Motivated by the study of the maximal operator for the Schr\"{ o}dinger equation on the one-dimensional torus $ \mathbb{ T}^1 $, it is conjectured that for any complex sequence $ \{ b_n\}_{ n=1}^N $, $$ \left\| \sup_{ t\in [0,N^2]} \left|\sum_{ n=1}^N b_n e \left(x\frac{ n}{ N} + t\frac{ n^2}{ N^2} \right) \right| \right\|_{ L^4([0,N])} \leq C_\epsilon N^{ \epsilon} N^{ \frac{ 1}{ 2}} \|b_n\|_{ \ell^2} $$ In this note, we show that if we replace the sequence $ \{ \frac{ n^2}{ N^2}\}_{ n=1}^N $ by an arbitrary sequence $ \{ a_n\}_{ n=1}^N $ with only some convex properties, then $$ \left\| \sup_{ t\in [0,N^2]} \left|\sum_{ n=1}^N b_n e \left(x\frac{ n}{ N} + ta_n \right) \right| \right\|_{ L^4([0,N])} \leq C_\epsilon N^\epsilon N^{ \frac{ 7}{ 12}} \|b_n\|_{ \ell^2}. $$ We further show that this bound is sharp up to a $C_\epsilon N^\epsilon$ factor.
Categories: math.CA math.AP