## Course Information

Lecturer:

Dr. Pierre Portal, The Australian National University

Synopsis:

Harmonic analysis is a branch of analysis inspired by the decomposition of square integrable functions on the circle into Fourier series.

It includes a range of methods to decompose functions defined on certain metric measure spaces (e.g. euclidean space, Lie groups, Riemannian manifolds) into pieces in such a way that various properties of these functions (e.g. smoothness, integrability, oscillations) can be easily uncovered. This is helpful for a range of problems, from PDE, to geometry, to number theory.

The Harmonic Analysis course at AMSI Summer School 2017 is sponsored by AustMS

In this course, we study the foundations of harmonic analysis on Rn, and aim to reach, at the very least, its first milestone result: Mihlin-Hormander’s Fourier multiplier theorem. This theorem, which has wide reaching applications, is concerned with linear operators defined by multiplying the Fourier transform of a function by a function m, then taking the inverse Fourier transform. Mihlin-Hormander’s theorem gives useful sufficient conditions on m to ensure the continuity of the corresponding operator on Lp. Perhaps surprisingly, finding necessary and sufficient conditions for such a theorem is still very much an open problem.

Course Overview:

• Lp spaces
• Distributions
• Fourier transform
• Maximal functions
• Interpolation
• Calderon-Zygmund decomposition
• Fourier multipliers

Contact Hours:

28 hours.

Timetable:

Please see the Timetable for scheduled class times during the AMSI Summer School 2017.
Please note: Harmonic Analysis is held concurrently with Mathematics and Statistics of Big Data and students cannot attend both courses in 2017.

Prerequisites:

The required background for this course is some knowledge of Lebesgue measure and integration theory. Knowledge of functional analysis (Hilbert and Lp spaces) is not required, but would be an advantage.

Assessment:

• Mid-School assignment: 40%
• Final examination: 60%

Resources:

• “Functional Analysis” by E. Stein and R. Shakarchi.
• “Classical Fourier Analysis” by L. Grafakos.

## Lecturer Biography

Dr. Pierre Portal, The Australian National University

Pierre is an Australian Research Council Future Fellow and Senior Lecturer at the ANU. Prior to that, he was a lecturer at the Universite Lille 1 in France.

Pierre’s research focuses on harmonic analysis in rough contexts: situations where either the geometry or the background noise renders the relevant functions non-differentiable. This applies, for instance, to Partial Differential Equations on domains with non-smooth boundaries, and to stochastic PDE.