Dispersive Estimates for Linearized Water Wave-Type Equations in R^d

We derive a Lx1(Rd)-Lx∞(Rd) decay estimate of order O(t^-d/2) for the linear propagators exp(±it|D|(1+β|D|2)tanh|D|),β∈{0,1}.D=-i∇, with a loss of 3d/4 or d/4–derivatives in the case β= 0 or β= 1 , respectively. These linear propagators are known to be associated with the linearized water wave equations, where the parameter β measures surface tension effects. As an application, we prove low regularity well-posedness for a Whitham–Boussinesq-type system in R^d , d≥ 2 . This generalizes a recent result by Dinvay, Selberg and the third author where they proved low regularity well-posedness in R and R² .