Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators

We consider here pseudo-differential operators whose symbol σ(x,ξ) 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 σ(x,ξ). 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.