Rodrigo Bañuelos

Math 545.  Spring 2017, MWF 1:30-2:20, UNIV 103

Instructor: Rodrigo Bañuelos
Office Hours: F 9:30-10:30 

Course Description

This course will cover some of the basic tools of analysis that are extremely useful in many areas of mathematics, including PDE's, stochastic analysis, harmonic analysis and complex analysis. Specific topics covered in the course include: Geometric lemmas (Vitali, Wiener, etc.) and geometric decomposition theorems (Whitney, etc.) and their applications to differentiation theory and to the Hardy-Littlewood maximal function; convolutions; approximations to the identity and their applications to boundary value problems in Rd with Lp-data; the Fourier transform and its basic properties on L1 and L2 (including Plancherel's theorem); interpolation theorems for linear operators (Marcinkiewicz, Riesz-Thorin); the basic Calderón-Zygmund singular integral theory and some of its applications; the Hardy-Littlewood-Sobolev inequalities for fractional integration and powers of the Laplacian and other elliptic operators; the inequalities of Nash and Sobolev viewed from the point of the heat semigroup.  The basics properties of Lusin and Littlewood-Paley square functions and applications to Hormander's multiplier theorem.

Textbooks:   Not RequiredRecommended and reserved in the Mathematics  Library              

E. M. Stein (ES)
Singular integrals
L. GrafaKos (LG) Classical and Modern Fourier Analysis
R. Bañuelos (RB)
Lecture notes will be provided for some topics covered in last few weeks.


Attend and participate in class, do a few homework assignments and a final.

Homework Assignments

Due Date Homework  Comments
Assignment 1
Math 544 Review Problems
Assignment 2

Assignment 3
Assignment 4