Sharp estimates for pseudodifferential operators with symbols of limited smoothness and commutators

We consider here pseudo-differential operators whose symbol $\sigma (x,\xi )$ is not infinitely smooth with respect to $x$. Decomposing such symbols into four -sometimes five- components and using tools of paradifferential calculus, we derive sharp estimates on the action of such pseudo-differential operators on Sobolev spaces and give explicit expressions for their operator norm in terms of the symbol $\sigma (x,\xi )$. We also study commutator estimates involving such operators, and generalize or improve the so-called Kato-Ponce and Calderon-Coifman-Meyer estimates in various ways.