Some of my lecture notes for this course later appeared on my blog "Zeros and Ones".

  1. Convergence of Fourier Series in Lp.

  2. Harmonic functions, Brownian motions, and Stopping times.

  3. Subharmonic functions.

  4. Fourier multipliers: examples on the torus .

  5. Singular integrals.

Course Syllabus

This course starts from Fourier series.

The question is to understand "how can we decompose a function as a sum of simple functions"?

Which functions do we consider (on which sets are they given, what values do they output)? What do we mean by "simple functions"? What do we mean to "decompose"? What do we mean by sum (if the number of simple functions is infinity and countable, there are many ways to sum them up, in which sense the sum would converge?).

As we see there are already many questions which are not well understood. However, there are certain instances when these questions are "kind of" well understood:

Example: any vector in R^n we can decompose as a sum of basis vectors with certain coordinates, and this problem in linear algebra is well understood: in this example by "simple functions" we mean any basis in R^n. And when the family of vectors is basis in R^n? Well, when there are n of them and they are linearly independent. Looks like simple description, but let us do not forget that verification of linear independence takes some time.

So in "finite dimensions" the decomposition is more or less well understood, we do not have issues with summability as all sums are finite.

Now in infinity dimensions , like functions on [0,1], the question is more delicate, there are many (a lot) candidates for "simple functions".

One classical candidate is {exp(2 \pi i k x)}, where k ranges among all integers, and in this case we know something (but not everything). In my course we will be working with this basis "{exp(2 \pi i k x)}", we will see that questions about convergence reduce to understand whether certain "operators" are bounded, say in Lp. Many of these operators have certain singularity, and in harmonic analysis we have developed certain techniques (Calderon--Zygmund theory, Mikhlin multiplier theorems) which give "good" sufficient conditions under which these operators are bounded, but not "if and only if" type conditions. Several subjects come into play (probability, combinatorics, analysis, complex analysis) to better understand how to better bound these operators. My goal is to get to these operators and to describe certain techniques how to bound them (and sometimes find the best possible bounds).

There is no book which contains the way I want to present this subject to you . You can find useful the following lecture notes and books

[1] A. Zygmund, Trigonometric Series, vols. I and II, 2nd ed., Cambridge Univ. Press, London, 1959

[2] T. Tao, Lecture notes "Harmonic Analysis", Part I, Part II

[3] Th. Wolff, Lectures on Harmonic Analysis

[4] J. Duoandikoetxea, Fourier Analysis, AMS Graduate Studies in Mathematics, Vol. 29 (2001)