Algebra properties for Sobolev spaces — applications to semilinear PDEs on manifolds

Abstract

In this work, we aim to prove algebra properties for generalized Sobolev spaces W s,p ∩ L ∞ on a Riemannian manifold (or more general homogeneous type space as graphs), where W s,p is of Bessel-type W s,p := (1+L)−s/m (L p ) with an operator L generating a heat semigroup satisfying off-diagonal decays. We do not require any assumption on the gradient of the semigroup. Instead, we propose two different approaches (one by paraproducts associated to the heat semigroup and another one using functionals). We also study the action of nonlinearities on these spaces and give applications to semi-linear PDEs. These results are new on Riemannian manifolds (with a non-bounded geometry) and even in euclidean space for Sobolev spaces associated to second order uniformly elliptic operators in divergence form.