Internet Analysis Seminar

Dear Colleagues and Participants,

The first lecture for the Fall 2011 -- Spring 2012 has been posted to the website. This semester we will compare and contrast some results from harmonic analysis, in particular multi-parameter harmonic analysis. The goal of the lectures is to give a flavor of the similarities and differences between one and the multi-parameter situation.

The first lecture deals with the usual maximal function in harmonic analysis. The maximal function is defined by
$$
Mf(x)=\sup_{r>0}\frac{1}{r^{n}}\int_{B_r(x)}|f(y)|dy.
$$
This is a very nice operator that appears naturally in the study of harmonic analysis. For example, most people encounter this operator when studying the Lebesgue Differentiation Theorem.

In this lecture we show that the maximal function is bounded on all $L^p$ spaces. The method of proof will introduce several standard tools in real and harmonic analysis: covering lemmas and interpolation theorems.

Regards,
Brett