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MA4J0 Advanced Real Analysis

Lecturer: Hendrik Weber

Term(s): Term 2

Status for Mathematics students: List C

Commitment: 30 lectures

Assessment: 3 hour exam (100%).

Prerequisites: MA3G7 Functional Analysis I & MA359 Measure Theory. Desirable: MA3G8 Functional Analysis II, MA433 Fourier Analysis

Content: The module builds upon modules from the second and third year like Metric Spaces, Measure Theory and 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.


  • Distributions on Euclidean space.
  • Tempered distributions and Fourier transforms.
  • Singular integral operators and Calderon-Zygmund theory.
  • Theory of Fourier multipliers.
  • Littlewood-Paley theory.
  • Wavelets.


- 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.

- Daubechies, I. : Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 1992.

- 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 Univesity Press, 1970.

Additional Resources

Year 1 regs and modules
G100 G103 GL11 G1NC

Year 2 regs and modules
G100 G103 GL11 G1NC

Year 3 regs and modules
G100 G103

Year 4 regs and modules

Archived Material
Past Exams
Core module averages