On the equivalence between $p$--Poincar\'e inequalities and L$^r$--L$^q$ regularization and decay estimates of certain nonlinear evolutions

Consider a p-homogeneous functional E(p) (p > 2) and suppose that a weighted Poincaré inequality involving it holds. Then all solutions u(t) to the evolution equation driven by the associated weighted p-Laplacian belong to L^r for any time provided the initial datum is in L^q, whenever q<r<+\infty, with a quantitative bound on the L^r norm of the solution. Such bound is in fact equivalent to the Poincaré inequality. There are examples in which the Poincaré inequality holds but there exist solutions whiich are not essentially bounded but correspond to data in L^q. Moreover, if a p-logarithmic Sobolev inequality holds then the Poincaré inequality is shown to hold too, therefore the previous regularization result is valid.