Sharp Variational estimates for Schr" o dinger equation in R squared
The paper establishes a sharp strong r-variation estimate for solutions to the free Schrödinger equation in $\mathbb{R}^2$. It determines the minimum Sobolev regularity ($s(p, r)$) required to control time variations. The result is achieved by deriving local estimates using Littlewood–Paley decomposition and parabolic scaling, showing that this critical index governs the global boundedness of these solutions.