96772 - Real and Harmonic Analysis

Learning outcomes

At the end of the course, the student has a knowledge of some basic features and methods of real and harmonic analysis.

Course contents

Prerequisites:

Lebesgue spaces L^p; Hahn-Banach theorem; Fourier transform on L^2; tempered distributions.

Course contents:

Weak Lebesgue spaces and their properties proprieta'. Interpolation theorems for weak, resp. strong Lebesgue spaces (Marcinckiewicz; resp. Riesz-Thorin). Besicovich covering Lemma. Hardy-Littlewood maximal functions. Lebesgue differentiation theorem. Hilbert transform. Riesz transforms. Homogeneous singular integral operators. Calderon-Zygmund decomposition.

Time permitting, vector-valued singular integral operators.

 

 

 

Topics covered include: Interpolation of Lebesgue spaces; Fourier coefficients; Hilbert and Riesz transforms; homogeneous singular integrals; Calderon-Zygmund decompositions. Time permitting, vector-valued singular integrals.

Readings/Bibliography

We will use the following textbook:

Loukas Grafakos, ``Classical Fourier Analysis'', 3d Edition (2014), Springer Graduate Texts in Mathematics v. 249.

Teaching methods

In-class lectures on the theory along with exercises, examples and applications, also aimed to students in the applied curriculum.

This course is taught in English; however students enrolled in the regular master program (not the international master) may choose to take the oral exam in Italian. During class, students who wish to ask a question may choose to do so in Italian.

Assessment methods

The evaluation is based on an oral examination that starts with the discussion of a topic chosen by the student among the topics covered during the course. The student will then answer questions pertaining to the proof of theorems that were demonstrated during lecture; the solution of exercises shown in class by the professor, or assigned by the professor as practice problems; the discussion of examples shown in class by the professor, or assigned by the professor as supplemental reading.

Students enrolled in the General Curriculum may choose to take the oral exam in Italian.

Students with learning disorders and\or temporary or permanent disabilities: please, contact the office responsible (https://site.unibo.it/studenti-con-disabilita-e-dsa/en/for-students) as soon as possible so that they can propose acceptable adjustments. The request for adaptation must be submitted in advance (15 days before the exam date) to the lecturer, who will assess the appropriateness of the adjustments, taking into account the teaching objectives.

Teaching tools

Prerequisites for this course includes the following topics, which were covered  e.g., in the ``Analisi Superiore'' component of the ``ANALISI SUPERIORE E GEOMETRIA DIFFERENZIALE'' course taught by Prof. Lanzani in Fall 2021. This material is available for study or review in the following chapters in the textbook: Section 2.2 (Schwartz class & Fourier Transform); Section 2.3 (Tempered Distributions); Section 2.4.1 (Distributions supported at a point).

Office hours

See the website of Loredana Lanzani