A smoothing property of Schrodinger equations in the critical case
  In this paper a global smoothing property of Schrodinger equations is established in the critical case in dimensions two and higher. It is shown that the critical smoothing estimate is attained if the smoothing operator has some structure. This structure is related to the geometric properties of the equations. Results for critical cases for operators of higher orders as well as for hyperbolic equations are also given. 
Michael Ruzhansky and Mitsuru Sugimoto