### ${L}^{p}$-boundedness for pseudodifferential operators with non-smooth symbols and applications

Starting from a general formulation of the characterization by dyadic crowns of Sobolev spaces, the authors give a result of ${L}^{p}$ continuity for pseudodifferential operators whose symbol a(x,ξ) is non smooth with respect to x and whose derivatives with respect to ξ have a decay of order ρ with $0<\rho \le 1$. The algebra property for some classes of weighted Sobolev spaces is proved and an application to multi - quasi - elliptic semilinear equations is given.