  In this work, we aim to prove algebra properties for generalized Sobolev
spaces $W^{s,p} \cap L^\infty$ on a Riemannian manifold, 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 don't require any assumption on
the gradient of the semigroup. To do that, we propose two different approaches
(one by a new kind of paraproducts and another one using functionals). We also
give a chain rule and 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 the Euclidean space for
Sobolev spaces associated to second order uniformly elliptic operators in
divergence form.
