A note on maximal operators for the Schrödinger equation on $\mathbb{T}^1.$ (2307.12870v1)
Abstract: 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 $, $$ \left| \sup{t\in [0,N2]} \left|\sum_{n=1}N b_n e \left(x\frac{n}{N} + t\frac{n2}{N2} \right) \right| \right|{L4([0,N])} \leq C\epsilon N{\epsilon} N{\frac{1}{2}} |b_n|{\ell2} $$ In this note, we show that if we replace the sequence $ {\frac{n2}{N2}}{n=1}N $ by an arbitrary sequence $ {a_n}{n=1}N $ with only some convex properties, then $$ \left| \sup{t\in [0,N2]} \left|\sum_{n=1}N b_n e \left(x\frac{n}{N} + ta_n \right) \right| \right|{L4([0,N])} \leq C\epsilon N\epsilon N{\frac{7}{12}} |b_n|{\ell2}. $$ We further show that this bound is sharp up to a $C\epsilon N\epsilon$ factor.