A k-uniform maximum principle when 0 is an eigenvalue

In this paper we consider linear operators of the form $L+\lambda I$ between suitable functions spaces, when 0 is an eigenvalue of L with constant eigenfunctions. We introduce a new notion of \quasi" uniform maximum principle, named k-uniform maximum principle, which holds for $\lambda$ belonging to certain neighborhoods of 0 depending on $k\in R^+$. Our approach is based on a $L^\infty-L^2$ estimate, which lets us prove some generalization of known results for elliptic and parabolic problems with Neumann or periodic boundary conditions.