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 associated eigenfunctions. We introduce a new notion of "quasi"-uniform maximum principle, named k-uniform maximum principle, which holds for A belonging to certain neighborhoods of 0 depending on k is an element of R+. Our approach actually also covers the case of a "quasi"-uniform antimaximum principle, and is based on an L-infinity - L-2 estimate. As an application, we prove some generalization of known results for elliptic Neumann problems and new results for parabolic problems with time-periodic boundary conditions.