Differentiability properties of functions, fall 2013

Lecturer

Pedro Caro
Miren Zubeldia

Scope

5 sp.

Type

Advanced studies

Prerequisites

Basic and intermediate studies in analysis.

Real Analysis in Rⁿ (L^p spaces). Properties of the Fourier transform would be useful but not necessary.

Description

We will present certain differentiability properties of functions in terms of Sobolev and Besov spaces. We will mainly focus on a frequency space analysis but we will also present some other techniques. In order to accomplish this goal, we will start by reviewing quickly the Fourier transform and its properties, specially the uncertainty principle which will motivate a principle to prove a wide range of estimates. We will proceed studying the so-called Bernstein's inequalities and the Littlewood-Paley theory. These tools will allow us to obtain fundamental estimates in analysis as Sobolev's embeddings, Hardy's inequalities and fractional Leibniz's rule, among others.

Lectures

Weeks 44-50, Thursdays 9-11 in room CK108 and Fridays 10-12 in room C130. First lecture on Thursday 31st October.

Evaluation

The course can be passed by doing exercises during the course and giving a presentation of a specific topic related to the course.

Bibliography

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