96772 - REAL AND HARMONIC ANALYSIS

Conoscenze e abilità da conseguire

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

Contenuti

Conoscenze preliminari:

Spazi di Lebesgue L^p; Teorema di Hahn-Banach; Trasformata di Fourier in L^2; distribuzioni temperate.

Contenuti del corso:

Spazi di Lebesgue deboli e loro proprieta'. Teoremi di interpolazione per gli spazi di Lebesgue deboli (Marcinckiewicz) e forti (Riesz-Thorin). Lemma di ricoprimento di Besicovich. Funzioni massimali di Hardy-Littlewood. Teorema di differenziazione di Lebesgue. Trasformata di Hilbert. Trasformate di Riesz. Integrali singolari omogenei. Decomposizione di Calderon-Zygmund.

Tempo permettendo, integrali singolari a valori vettoriali. 

Testi/Bibliografia

We will use the following textbook:

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

Metodi didattici

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.

Modalità di verifica e valutazione dell'apprendimento

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.

Strumenti a supporto della didattica

Prerequisites for this course includes the following topics, which were covered in the ``Analisi Superiore'' component of the ``ANALISI SUPERIORE E GEOMETRIA DIFFERENZIALE'' course taught by Prof. Lanzani in Fall 2021, and can be reviewed 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).

Orario di ricevimento

Consulta il sito web di Loredana Lanzani