Smoothing of Commutators for a Hörmander Class of Bilinear Pseudodifferential Operators

Commutators of bilinear pseudodifferential operators with symbols in the Hörmander class BS 1,0 and multiplication by Lipschitz functions are shown to be bilinear Calderón-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made. 1. Motivation, preliminaries and statements of main results The work of Calderón and Zygmund on singular integrals and Calderón’s ideas [8, 9] about improving a pseudodifferential calculus, where the smoothness assumptions on the coefficients are minimal, have greatly affected research in quasilinear and nonlinear PDEs. The subsequent investigations about multilinear operators initiated by Coifman and Meyer [13] in the late 70s have added to the success of Calderón’s work on commutators. A classical bilinear estimate, the so-called Kato-Ponce commutator estimate [26], is crucial in the study of the Navier-Stokes equations. Its original formulation is the following: if 1 < r < ∞, s > 0, and f, g are Schwartz functions, then ∥[J, f ](g)∥Lr . ∥∇f∥L∞∥Jg∥Lr + ∥Jf∥Lr∥g∥L∞ (1.1) where J := (I − ∆) denotes the Bessel potential of order s and [J, f ](·) := J(f ·)− f(J·) is the commutator of J with f . The previous estimate (1.1) has been recast later on into a general Leibniz-type rule (still commonly known as Kato-Ponce’s inequality) which takes the form ∥D(fg)∥Lr . ∥Df∥Lp∥g∥Lq + ∥f∥Lp∥Dg∥Lq , (1.2) for 1 < p, q ≤ ∞, 1 < r < ∞, 1/p+1/q = 1/r and α > 0. An extension of the estimate (1.2) to the range 1/2 < r < ∞ can be found in the recent work of Grafakos and S. Oh [17], while the ∞-end point result is explored by Grafakos, Maldonado and Naibo in [16]. More general Leibniz-type rules that apply to bilinear pseudodifferential operators with symbols in the bilinear Hörmander classes BS ρ,δ (see (1.8) below for Date: November 12, 2013. 1991 Mathematics Subject Classification. Primary 35S05, 47G30; Secondary 42B15, 42B20, 42B25, 47B07, 47G99.