Partial generalizations of some Conjectures in locally symmetric Lorentz spaces

In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces $M^{n}$with $R=aH+b_{1}$ in a locally symmetric Lorentz space $L_{1}^{n+1}$,where $R$ and $H$ are the normalized scalar curvature and the mean curvature of $M^{n}$, respectively.Furthermore, we study complete or compact linear Weingarten spacelike hypersurfacesin locally symmetric Lorentz spaces $L_{1}^{n+1}$ satisfying some curvature conditions.By modifying Cheng-Yau's operator $\square$ given in {\cite{ChengYau77}},we introduce a modified operator $L$ and give new estimates of $L(nH)$ and $\square(nH)$ ofsuch spacelike hypersurfaces.Finally, we give partial generalizations of some Conjectures in locally symmetric Lorentz spaces $L_{1}^{n+1}$.