Poincaré inequalities for the maximal function

We study generalized Poincare´ inequalities. We prove that if a function satisfies an inequality of Poincare´ type, then the Hardy-Littlewood maximal function also obeys an estimate of similar form. As a by-product, we get a unified approach to proving that the maximal operator is bounded on Sobolev, Lipschitz and BMO spaces.