2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/228053In this paper we introduce a local approach for the study of maximal surfaces immersed into a Lorentzian product space of the form $M^2\times R_1$, where $M^2$ is a connected Riemannian surface and $M^2\times R_1$ is endowed with the product Lorentzian metric. Specifically, we establish a local integral inequality for the squared norm of the second fundamental form of the surface, which allows us to derive an alternative proof of our Calabi-Bernstein theorem given in \cite{AA}.To appear in Matematica Contemporanea. Dedicated to Professor Manfredo P. do Carmo on the occasion of his 80th birthdayDifferential Geometry53C42; 53C50text