Dispersive estimates for linearized water wave type equations in $\mathbb R^d$

We derive a $L^1_x (\mathbb R^d)-L^{\infty}_x ( \mathbb R^d)$ decay estimate of order $\mathcal O \left( t^{-d/2}\right)$ for the linear propagators $$\exp \left( {\pm it \sqrt{ |D|\left(1+ \beta |D|^2\right) \tanh |D | } }\right), \qquad \beta \in \{0, 1\}. \quad D = -i\nabla,$$ with a loss of $3d/4$ or $d/4$-derivatives in the case $\beta=0$ or $\beta=1$, respectively. These linear propagators are known to be associated with the linearized water wave equations, where the parameter $\beta$ measures surface tension effects. As an application we prove low regularity well-posedness for a Whitham-Boussinesq type system in $\mathbb R^d$, $d\ge 2$. This generalizes a recent result by Dinvay, Selberg and the third author where they proved low regularity well-posedness in $\mathbb R$ and $\mathbb R^2$.