Pseudodifferential operators associated with a semigroup of operators

Related to a semigroup of operators on a metric measure space, we define and study pseudodifferential operators (including the setting of Riemannian manifold, fractals, graphs …). Boundedness on $L^p$ for pseudodifferential operators of order 0 are proved. Mainly, we focus on symbols belonging to the class $S_{1,\delta }^0$ for $\delta \in [0,1)$. For the limit class $S_{1,1}^0$, we describe some results by restricting our attention to the case of a sub-Laplacian operator on a Riemannian manifold.