After recalling some basic notions about traces of Sobolev functions, I
will focus on a few recent developments of the theory of trace
embeddings of Sobolev type. In particular:
- Criteria for the validity of trace inequalities involving general rearrangement-invariant norms will be presented, with specific applications to Orlicz and Lorentz norms.
- Sharp constants in certain trace inequalities for functions of bounded variation will be exhibited, in connection with certain unconventional isoperimetric inequalities.
- Sobolev type inequalities on arbitrary domains will be described, where domain regularity is replaced with suitable trace integrabilty information on trial functions.
In these lectures I will discuss the theory of Muckenhoupt weights
and the related theories of factorization and extrapolation. The topics
of my lectures will include:
(1) The maximal operator and Muckenhoupt Ap weights; (2) The fine structure of Ap weights and the reverse Hölder inequality RHs; (3) The Rubio de Francia iteration algorithm; (4) The Jones factorization theorem of weights in Ap ∩ RHs; (5) Rubio de Francia extrapolation from the perspective of families of extrapolation pairs; (6) Off-diagonal, limited range, A∞ and bilinear extrapolation; (7) Generalizations of extrapolation and factorization to other scales of function spaces, especially variable Lebesgue spaces and Musielak-Orlicz spaces.
Abstract: The lectures will provide a self-contained and elementary introduction
to the geometry of the Heisenberg groups. In particular I will discuss:
(1) The Carnot-Carathéodory metric and the structure of geodesics,
(2) The Hausdorff dimension of the group,
(3) The Heisenberg group as a unit ball in Cn,
(4) The Heisenberg group as a canonical contact structure,
(5) The Pansu-Rademacher theorem and non existence of the bi-Lipschitz embedding into the Euclidean space,
(6) Non-rectifiability of the Heisenberg group,
(7) Poincaré and Sobolev inequalities.