  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 $\mathbb R^2$.
