“Padova Spring Lectures in Harmonic Analysis”

Abstract

In these lectures we will be concerned with the basic problem of almost everywhere pointwise convergence problems for a variety of families of linear operators acting on functions in Lebesgue spaces defined on some measure space. We will introduce tools (maximal functions, oscillation and variational semi-norms, jump functions) that quantify the rate of convergence and enable us to establish almost everywhere pointwise convergence in very general settings. We will apply these tools to pointwise convergence problems in ergodic theory, and examine results of Bourgain extending Birkhoff’s original result from the 1930s to ergodic averages over sparse arithmetic sets. There is a nice interplay between analytic number theory and harmonic analysis which arises. A main goal of these lectures is to show the usefulness of oscillation semi-norms (it seems to be the only viable tool) when we try to generalise Bourgain’s results to the multi-parameter setting.

• Thursday 7, h. 15:30 – Room 2BC60
• Friday 8, h. 15:30 – Room 2BC60
• Monday 11, h. 15:30 – Room 2BC60
• Tuesday 12, h. 15:30 – Room 2AB40
• Wednesday 13, h. 15:30 – Room 2AB40