Master's and Bachelor's Projects, Carlos Mudarra

I offer master or bachelor projects within harmonic and/or functional analysis. Your specific project can vary depending on your interests and background. If you are interested, please contact me. Your project will be co-supervised by Karl-Mikael Perfekt.

(MSc1) Harmonic analysis and Muckenhoupt \( A_p \) weights (suitable for Master)

Harmonic analysis is the study of real-variable function and their Fourier transforms, integral operators, oscillation, harmonic functions, etc. Topics of study may include:

The Poisson kernel, the Hilbert transform, and Calderón–Zygmund operators.
Maximal functions, \( A_p \) weights and the Muckenhoupt theorem (the weighted maximal function is bounded in weighted \( L^p \) iff the weight is in \( A_p \)). The Reverse Hölder Inequalities, with emphasis on obtaining quasi-optimal bounds.
Functions of bounded mean oscillation (BMO), the John-Nirenberg theorem, with emphasis on obtaining quasi-optimal bounds. The connection with \( A_p \) theory.

Recommended background: measure and integration theory. Elementary functional analysis for \( L^p \) spaces.

(BSc1) Sets of Divergence for Fourier Series (suitable for Bachelor)

We know that, for example, a Lipschitz function has Fourier Series uniformly convergent to the function. The situation becomes much more delicate when the functions is merely continuous or only integrable (i.e., \( L^1 \)). In those cases, we can construct functions whose Fourier series diverges essentially everywhere. The goal of this project is to understand those constructions, at the same time you learn how to apply elementary complex, functional and harmonic analysis to the theory of Fourier Series. The project may include:

Convergence of Fourier Series for Hölder functions.
The Du Bois-Reymond theorem: There are continuous functions whose Fourier Series diverges at every rational point.
The Kahane-Katznelson theorem: For every set E of measure zero, there are Fourier Series of continuous functions diverging at every point of E.
The Kolmogorov theorem: There are integrable functions whose Fourier Series diverges at every point.

Recommended background: it may be helpful to have taken (or take at the same time) a Fourier Analysis course.

(MSc2) Extensions of Lipschitz Mappings between Banach spaces (suitable for Master; the most challenging of the projects)

Extending Lipschitz real-valued mappings from subsets of metric spaces, preserving the Lipschitz constant, is not a difficult task if one uses the sup/inf convolution formulas. The problem becomes much more challenging if we consider vector-valued mappings or if we are interested in extending via a linear operator. One of the most famous results in this direction is the theorem of Kirszbraun's-Valentine (1934).

The project would cover the following topics.

The real-valued case.
A Linear Extension Operator for Banach-valued maps in \( \mathbb R^n \) with dimensional distortion.
The Kirszbraun's Theorem in the Hilbert case. Validity in more general settings.
Continuous extension operators in the Hilbert case.
Analysis of the problem via convex analysis and \( C^{1,1} \) functions.

Recommended background: basic functional analysis.

(BSc2) Smooth and Analytic Approximation (suitable for Bachelor)

This project is focused on the study of approximation techniques for continuous functions in \( \mathbb R^n \) by smooth or even analytic functions, with the possibility of preserving their geometric properties (such as the convexity or the Lipschitzness) or their order of smoothness (if any). We will mainly cover:

Smooth approximation via integral convolution with mollifiers.
Analytic approximation via convolution with the Heat Kernel.
The Whitney Approximation Theorem.
Perhaps the Lasry-Lions \( C^{1,1} \)-regularization in a Hilbert space.

Recommended background: only basic elements from measure and integration.

(MSc3) Smooth extension in \( \mathbb R^n \), with applications (suitable for Master)

Given a set E of \( \mathbb R^n \) and a real function f on E, how can we decide whether f can be extended to a \( C^m \) function in all of \( \mathbb R^n \)? The Whitney Extension Theorem (1934) was one of the first steps towards the solution to this (although simple to formulate) very difficult question, and has provided a large number of applications in the last hundred years. The project would cover the following topics.

The Whitney Extension Theorem.
Analytic version of the Whitney's Theorem via analytic approximation.
Application 1: The construction of non-constant \( C^1 \) functions with null gradients along continuous paths.
Application 2: The Lusin-type theorem for smooth functions: Every \( C^m \) function with Lipschitz m-th derivative coincides with a \( C^{m+1} \) function up to a set of arbitrarily small measure.

Recommended background: only very basic elements from measure theory.

(BSc3) Differentiability of Lipschitz and Convex Functions (suitable for Bachelor)

This project is focused on two classical (and beautiful) theorems in analysis: the Rademacher Theorem and the Alexandrov Theorem. More precisely,

Rademacher's Theorem. Every Lipschitz function in \( \mathbb R^n \) is differentiable at almost every point.
Alexandrov's Theorem. Every Convex function in \( \mathbb R^n \) is twice differentiable at almost every point, understanding that f admits a Taylor approximation of order 2 at almost every point.
Perhaps, we can additionally review the existence/non-existence of infinite-dimensional versions of these results.

Recommended background: very basic measure theory, and calculus in \( \mathbb R^n \).