  This master thesis is based on the paper "Sharp commutator estimates via
harmonic extensions" by Lenzmann and Schikorra, in which they proposed a method
to prove estimates for commutators involving Riesz transforms, fractional
Laplacians and Riesz potentials, see arXiv:1609.08547. These proofs only
involve harmonic extensions to the upper half-space and integration by parts
next to some elementary transfromations, since the deeper theory is
concentrated in a variety of trace characterization results which can be used
as a blackbox.
  In the first half of this thesis, after collecting some elementary results
for the s-harmonic extension by Caffarelli and Silvestre, we use this method to
prove a variety of commutator estimates, closely following Lenzmann and
Schikorra except for shortening some proofs.
  In the second half, we prove generalized versions of the blackbox estimates
listed by Lenzmann and Schikorra and discuss the different building blocks
which make up these blackbox estimates, including Triebel-Lizorkin and
Besov-Lipschitz space characterizations as well as square function estimates.
