Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities

We establish boundedness estimates for solutions of general-ized porous medium equations of the form partial differential tu + (-2)[um] = 0 in RN x (0, T), where m >= 1 and -2 is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Levy operators. Our quantitative esti-mates take the form of precise L1-L infinity-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of -2 and I - 2. In the linear case m = 1, it is well-known that the L1-L infinity- smoothing effect, or ultracontractivity, is equivalent to Nash inequalities. This is also equivalent to heat kernel estimates, which imply the Green function estimates that represent a key ingredient in our techniques. We establish a similar scenario in the nonlinear setting m > 1. First, we can show that operators for which ultracontractivity holds, also provide L1-L infinity-smoothing effects in the nonlin-ear case. The converse implication is not true in general. A counterexample is given by 0-order Levy operators like