Abstracts

Abstract: 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.

Abstract: 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 A_p weights;
(2) The fine structure of A_p weights and the reverse Hölder inequality RH_s;
(3) The Rubio de Francia iteration algorithm;
(4) The Jones factorization theorem of weights in A_p ∩ RH_s;
(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.