  In this study, we deal with the Gauss map of tubular hypersurfaces in
4-dimensional Lorentz-Minkowski space concerning the linearized operators
$\mathcal{L}_{1}$ (Cheng-Yau) and $\mathcal{L}_{2}$. We obtain the
$\mathcal{L}_{1}$ (Cheng-Yau) operator of the Gauss map of tubular
hypersurfaces that are formed as the envelope of a family of pseudo
hyperspheres {or pseudo hyperbolic hyperspheres} whose centers lie on timelike
or spacelike curves with non-null Frenet vectors in $\mathbb{E}^{4}_{1}$ and
give some classifications for these hypersurfaces which have generalized
$\mathcal{L}_{k}$ 1-type Gauss map, first kind $\mathcal{L}_{k}$-pointwise
1-type Gauss map, second kind $\mathcal{L}_{k}$-pointwise 1-type Gauss map and
$\mathcal{L}_{k}$-harmonic Gauss map, $k\in\{1,2\}$.
