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We consider the prescribed mean curvature equation for entire spacelikehypersurfaces in the Lorentz-Minkowski space, namely \begin{equation*}-\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nablau|^2}}\right)= \rho \quad \hbox{in }\mathbb{R}^N, \end{equation*} where $N\geq3$. We first prove a new gradient estimate for classical solutions with smoothdata $\rho$. As a consequence we obtain that the unique weak solution of theequation satisfying a homogeneous boundary condition at infinity is locally ofclass $W^{2,q}$ and strictly spacelike in $\mathbb{R}^N$, provided that$\rho\in L^q(\mathbb{R}^N) \cap L^m(\mathbb{R}^N)$ with $q>N$ and$m\in[1,\frac{2N}{N+2}]$.

A sharp gradient estimate and $W^{2,q}$ regularity for the prescribed mean curvature equation in the Lorentz-Minkowski space