We shall present a summary of a forthcoming book with Daniel Hauer (University of Sydney). The aim of this monograph is to introduce natural and simple functional analytic methods to deduce $L^1-L^\infty$ regularization estimates on nonlinear semigroups from natural Gagliardo-Nirenberg inequalities satisfied by their infinitesimal generator. This enables one to treat in an optimal and unified way a wealth of examples, including the p-Laplace operator, the porous medium operator, as well as variations and combinations of them.