A note on maximal operators for the Schr\"{o}dinger equation on $\mathbb{T}^1.$

TL;DR

This paper investigates maximal operators related to the Schrödinger equation on the one-dimensional torus, establishing bounds for generalized sequences and demonstrating their sharpness, advancing understanding of oscillatory integral estimates.

Contribution

It extends known bounds for maximal operators to more general sequences with convex properties, showing these bounds are nearly optimal.

Findings

Established bounds for generalized sequences in maximal Schrödinger operators.

Proved the bounds are sharp up to a small epsilon factor.

Extended the understanding of oscillatory sums on the torus.

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 $, $$ \left\| \sup_{t\in [0,N^2]} \left|\sum_{n=1}^N b_n e \left(x\frac{n}{N} + t\frac{n^2}{N^2} \right) \right| \right\|_{L^4([0,N])} \leq C_\epsilon N^{\epsilon} N^{\frac{1}{2}} \|b_n\|_{\ell^2} $$ 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.