Lp-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,) is non smooth with respect to x and whose derivatives with respect to have a decay of order rho with 0 < rho <= 1. The algebra property for some classes of weighted Sobolev spaces is proved and an application to multi - quasi - elliptic semilinear equations is given.