Some classifications for Gauss map of Tubular hypersurfaces in $\mathbb{E}^{4}_{1}$ concerning linearized operators $\mathcal{L}_{k}$

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, $k\in\{1,2\}$.