  At the core of this article is an improved, sharp dispersive estimate for the
anisotropic linear semigroup $e^{R_1 t}$ arising in both the study of the
dispersive SQG equation and the inviscid Boussinesq system. We combine the
decay estimate with a blow-up criterion to show how dispersion leads to
long-time existence of solutions to the dispersive SQG equation, improving the
results obtained using hyperbolic methods. In the setting of the inviscid
Boussinesq system it turns out that linearization around a specific stationary
solution leads to the same linear semigroup, so that we can make use of
analogous techniques to obtain stability of the stationary solution for an
increased timespan.
