MA4J0 Advanced Real Analysis
Lecturer: Cagri Sert
Term(s): Term 2
Status for Mathematics students: List C
Commitment: 30 lectures
Assessment: 3 hour exam (100%)
Formal registration prerequisites: None
Assumed knowledge:
- MA3G7 Functional Analysis I: Banach spaces, Lebesgue spaces, dual spaces, linear operators.
- MA359 Measure Theory: General measures, Lebesgue measure & integral, properties of Lebesgue integral, convergence theorems.
Useful background:
- MA3G8 Functional Analysis II: More familiarity with Banach spaces and their main theorems.
- MA433 Fourier Analysis: Fourier transform & its properties.
Synergies:
- MA4A2 Advanced PDEs: Applications & motivation and graduate courses in Analysis.
Content: The module builds upon modules from the second and third year like MA222 Metric Spaces, MA359 Measure Theory and MA3G7 Functional Analysis I to present the fundamental tools in Harmonic Analysis and some applications, primarily in Partial Differential Equations. Some of the main aims include:
- Setting up a rigorous calculus of rough objects, such as distributions.
- Studying the boundedness of singular integrals and their applications.
- Understanding the scaling properties of inequalities.
- Defining Sobolev spaces using the Fourier Transform and the connections between the decay of the Fourier Transform and the regularity of functions.
Outline:
- Distributions on Euclidean space.
- Tempered distributions and Fourier transforms.
- Singular integral operators and Calderon-Zygmund theory.
- Theory of Fourier multipliers.
- Littlewood-Paley theory.
Books:
- Friedlander, G. and Joshi, M. : Introduction to the Theory of Distributions, 2nd edition, Cambridge University Press, 1998.
- Duoandikoetxea, J. : Fourier Analysis - American Mathematical Society, Graduate Studies in Mathematics, 2001.
- Muscalu C. and Schlag, W. : Classical and Multilinear Harmonic Analysis, Cambridge Studies in advanced Mathematics, 2013.
- Folland, G. Real Analysis: Modern Techniques and their Applications, Wiley 1999.
- Grafakos, L. : Classical Fourier Analysis - Springer 2008.
- Grafakos, L.: Modern Fourier Analysis - Springer 2008.
- Stein, E.M.: Singular Integrals and Differentiability Properties of Functions and Differentiability Properties of Functions - Princeton University Press, 1970.