  In this article, an $L^p$-approach to the primitive equations is developed.
In particular, it is shown that the three dimensional primitive equations admit
a unique, global strong solution for all initial data $a \in
[X_p,D(A_p)]_{1/p}$ provided $p \in [6/5,\infty)$. To this end, the hydrostatic
Stokes operator $A_p$ defined on $X_p$, the subspace of $L^p$ associated with
the hydrostatic Helmholtz projection, is introduced and investigated. Choosing
$p$ large, one obtains global well-posedness of the primitive equations for
strong solutions for initial data $a$ having less differentiability properties
than $H^1$, hereby generalizing in particular a result by Cao and Titi (Ann.
Math. 166 (2007), pp. 245-267) to the case of non-smooth initial data.
