The program consists of the following three lectures

13:30 – 14:15

“Classes of Singular Integral Operators and Applications to Boundary Value Problems”

Dorina Mitrea (Baylor University, Texas, USA)

Abstract

A.P. Calderón and A. Zygmund have been at the forefront of a program aimed at developing a theory for singular integral operators as a means for treating problems in partial differential equations. Initially formulated in ${\mathbb{R}}^n$, the theory has evolved to the point of now accommodating the most general geometric setting in which singular integral operators are bounded on Lebesgue spaces namely, uniformly rectifiable sets.

In this talk, I will survey some of the most recent advances, with special emphasis on categorizing subclasses of singular integral operators which are well behaved on various scales of spaces of interest. These include (boundary) Sobolev spaces, Hölder spaces, spaces of functions with bounded mean oscillations, spaces of functions with vanishing mean oscillations, Orlicz spaces, Muckenhoupt weighted Lebesgue spaces, and Morrey spaces.

14:30 – 15:15

“Geometric Estimates for Parabolic Singular Integral Operators”

Marius Mitrea (Baylor University, Texas, USA)

Abstract

While the issue of boundedness of singular integral operators (SIO) is reasonably understood, the specific manner in which the geometry impacts the size of the SIO remains a source of fascinating questions. After the initial breakthrough by S. Hofmann, M. Mitrea, and M. Taylor in 2010, significant progress has been made by D. Mitrea, I. Mitrea, and the present speaker in their series Geometric Harmonic Analysis, vols. I-V, 2022-2023, in the realm of boundary layer potentials associated with elliptic PDE’s. Here, I will report on recent progress aimed at taking the next steps in the direction of accommodating boundary layer potentials associated with parabolic PDE’s. The main tools and techniques originate in Harmonic Analysis and Geometric Measure Theory.

15:30 – 16:15

“Geometric Hardy inequalities on the Heisenberg group”

Gerassimos Barbatis (National and Kapodistrian University of Athens, Greece)

Abstract

We present some new $L^p$ Hardy inequalities on the Heisenberg group which involve the distance to the boundary. We adopt an abstract approach which allows us to obtain inequalities with various distances.

The constants obtained are sharp.

Joint work with M. Chatzakou and A. Tertikas.