Estimates for eigenvalues of the operator $L_r$

In this paper, we consider an eigenvalue problem of the elliptic operator $$L_r={\rm div}(T^r\nabla\cdot )$$ on compact submanifolds in arbitrary codimension of space forms $\mathbb{R}^N(c)$ with $c\geq0$. Our estimates on eigenvalues are sharp.