Abstract of the first lecture

During the first lecture, we are going to give  the basic properties
of the Heisenberg group concerning left invariant distance, measure and
derivations.
Then we are going to define the equivalent of Sobolev spaces and the
Laplace operator in this context
and prove some basic result  which are analogous of thoses for the
analysis on the euclidian space
(Poincar\'e inequality, Hardy inequality, Sobolev inequality, Dirichlet
problem). The theory of inhomogeneous Dirichlet
problem will be the motivation of the following lectures through the
problem of trace of the Sobolev space $H^1$.

Abstract of the second lecture

We want to study the problem of trace of function in $H^1$ for the
Heisenberg group. The key point is the sub-ellipticity  of the
laplacian.  This allows the use of  Weyl-H\"ormander  calculus. After
recalling some basic definition and basic properties of Weyl-H\"ormander
calculus, we shall explain  how (locally) Sobolev spaces on the
Heisenberg  group can be seen as Sobolev spaces in the framework of
Weyl-H\"ormander calculus.