  Supplying the missing necessary conditions, we complete the characterisation
of the $L^p\to L^q$ boundedness of commutators $[b,T]$ of pointwise
multiplication and Calder\'on-Zygmund operators, for arbitrary pairs of
$1<p,q<\infty$ and under minimal non-degeneracy hypotheses on $T$.
  For $p\leq q$ (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-R\'ios by showing that $b\in BMO$ is
necessary for the $L^p$-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 $f\in L^p(R^d)$ can be
represented as a convergent series of normalised Jacobians $Ju=\det\nabla u$ of
$u\in \dot W^{1,dp}(R^d)^d$. This extends, from $p=1$ to $p>1$, a result of
Coifman, Lions, Meyer and Semmes about $J:\dot W^{1,d}(R^d)^d\to H^1(R^d)$, and
supports a conjecture of Iwaniec about the solvability of the equation $Ju=f\in
L^p(R^d)$.
