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In this work, we obtain decay bounds for a class of ID dispersive equationsthat includes the linearized water wave. These decay bounds display asurprising growth factor, which we show is sharp, The proofs rely on carefulanalysis of certain oscillatory integral operators. In addition, these resultshave applications to the linearized water wave operator for low regularitydata.

Nonstandard estimates for a class of 1D dispersive equations and applications to linearized water waves