Nonstandard estimates for a class of 1D dispersive equations and   applications to linearized water waves
  In this work, we obtain decay bounds for a class of ID dispersive equations that includes the linearized water wave. These decay bounds display a surprising growth factor, which we show is sharp, The proofs rely on careful analysis of certain oscillatory integral operators. In addition, these results have applications to the linearized water wave operator for low regularity data. 
Jennifer Beichman
math.AP