Partial generalizations of some Conjectures in locally symmetric Lorentz spaces

In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with $P+aH=b$ in a locally symmetric Lorentz space $L_1^{n+1}$. Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces $L_1^{n+1}$ satisfying some curvature conditions. By modifying Cheng-Yau’s operator $\mathrm{\square }$ given in {\cite{ChengYau77}}, we introduce a modified operator $L$ and give new estimates of $L(nH)$ and $\mathrm{\square }(nH)$ of such spacelike hypersurfaces. Finally, we give partial generalizations of some conjectures in locally symmetric Lorentz spaces $L_1^{n+1}$.