The Lp-to-Lq boundedness of commutators with applications to the Jacobian operator

Abstract

Supplying the missing necessary conditions, we complete the characterisation of the Lp→Lq boundedness of commutators [b,T] of pointwise multiplication and Calderón–Zygmund operators, for arbitrary pairs of 1<p,q<∞ and under minimal non-degeneracy hypotheses on T.For p≤q (and especially p=q), this extends a long line of results under more restrictive assumptions on T. In particular, we answer a recent question of Lerner, Ombrosi, and Rivera-Ríos by showing that b∈BMO is necessary for the Lp-boundedness of [b,T] for any non-zero homogeneous singular integral T. We also deal with iterated commutators and weighted spaces.For p>q, our results are new even for special classical operators with smooth kernels. As an application, we show that every f∈Lp(Rd) can be represented as a convergent series of normalised Jacobians Ju=det⁡∇u of u∈W˙1,dp(Rd)d. This extends, from p=1 to p>1, a result of Coifman, Lions, Meyer and Semmes about J:W˙1,d(Rd)d→H1(Rd), and supports a conjecture of Iwaniec about the solvability of the equation Ju=f∈Lp(Rd).