  We give an alternative proof of several sharp commutator estimates involving
Riesz transforms, Riesz potentials, and fractional Laplacians. Our methods only
involve harmonic extensions to the upper half-space, integration by parts, and
trace space characterizations.
  The commutators we investigate are Jacobians, more generally
Coifman-Rochberg-Weiss commutators, Chanillo's commutator with the Riesz
potential, Coifman-Meyer or Kato-Ponce-Vega type commutators, and the Da
Lio-Rivi\`{e}re three-term commutators. We also give a new limiting
$L^1$-estimate for a double commutator of Coifman-Rochberg-Weiss-type, and
several intermediate estimates.
  The beauty of our method is that all those commutator estimates, which are
originally proven by various specific methods or by general paraproduct
arguments, can be obtained purely from integration by parts and trace theorems.
Another interesting feature is that in all these cases the cancellation effect
responsible for the commutator estimate simply follows from the product rule
for classical derivatives and can be traced in a precise way.
