Motivated by the study of the maximal operator for the Schr\"{o}dinger
equation on the one-dimensional torus T1, it is conjectured that
for any complex sequence {bnβ}n=1Nβ, βtβ[0,N2]supββn=1βNβbnβe(xNnβ+tN2n2β)ββL4([0,N])ββ€CΟ΅βNΟ΅N21ββ₯bnββ₯β2β In this note, we show that if we replace the sequence {N2n2β}n=1Nβ by an arbitrary sequence {anβ}n=1Nβ with
only some convex properties, then βtβ[0,N2]supββn=1βNβbnβe(xNnβ+tanβ)ββL4([0,N])ββ€CΟ΅βNΟ΅N127ββ₯bnββ₯β2β. We further show that this bound is sharp up to a
CΟ΅βNΟ΅ factor.Comment: 13 page