Sharp Strichartz estimates for some variable coefficient Schrödinger operators on \(\mathbb{R}\times\mathbb{T}^2 \) (Q2167610)

This research article is focused on the two-dimensional analysis of Schrödinger operators on toroidal manifolds. Mainly, on the investigation of Strichartz estimates and the well-posedness of IVPs in the periodic setting. In the proof of two of the main results, the authors have shown that the \(L^4-\)Strichartz estimate on \(\mathbb{T}^2\) -- formely obtained by \textit{J. Bourgain} [Geom. Funct. Anal. 3, No. 3, 209--262 (1993; Zbl 0787.35098)] -- can be wisely exploited for two dimensional \textit{constant} coefficient Schrödinger operators \(\mathbb{T}^2\), with the aid of an appropriated gauge transform (see Theorem 4.1 \& Theorem 1.3). Noteworthy, by proving that \(H^2\) regularity is sufficient to ensure the well-posedness of IVPs carrying non-constant coefficients \(a_1\) and \(a_2\), they have concluded in turn that the lack of a \(C^\infty\) condition for the \(a_j'\)s is not an obstacle for the strategy of proof adopted throughout the manuscript. Thus, the sharp local well-podsedness results hold true for the aforementioned class of IVPs, even if the constant coefficients have Sobolev regularity.