Department Mathematik
print


Navigationspfad


Inhaltsbereich

Harmonic analysis and applications (Summer 2023)

Phan Thành Nam, Larry Read, Uni2work, Notes, Homework

General Information

Goal: This course gives an introduction to Harmonic analysis, focusing on applications in number theory, functional analysis, partial differential equations and mathematical physics. Among other things, we will discuss Fourier transform, maximal functions, interpolation, Calderón-Zygmund and Littlewood-Paley theories.

The course is suitable for master students and motivated bachelor students. Prerequisites: Lebesgue integration and L^2 theory of the Fourier transform.

Audience: Bachelor and Master students of Mathematics, TMP Master. Other participants will get "Schein" if pass the course.

Time and place:
  • Lectures: Wednesday 10:15-12:00 (B 046), Friday 10:15-12:00 (B 252).
  • Exercises: Wednesday 14:15-16:00 (B 133).
  • Tutorial: Friday 14:15-16:00 (A 027).

References:
  • Elias M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1971).
  • Tristan Rivière. Fourier Analysis in Function Space Theory. Lecture notes.
  • Loukas Grafakos. Classical Fourier Analysis. Book.
  • Terence Tao. Introduction to harmonic analysis and Fourier analysis methods. Lecture notes.

Grade: Final grade is determined by your total performance of the exams:
  • You can get up to 100 points in the final exam.
  • You can get up to 25 points in the midterm exam.
You need 50 points to pass the course and 85 points to get the final grade 1.0.

Exams: We will have exams on June 14 (midterm) and July 21 (final). You can use your notes (e.g. lecture notes, homework sheets) in the exams.



Contents of the lectures

26.4. Introduction.

28.4. Chapter 1: L^p spaces. Measure, integration, L^p spaces, weak L^p spaces.

3.5. Chapter 2: Hardy-Littewood maximal inequalities. Motivation: Lebesgue differentiation theorem. Maximal function. Vitali's covering lemma. Weak L^1 maximal inequality.

5.5. Remarks on the use of Vitali's covering lemma. Strong L^p maximal inequality. Fefferman - de la Llave decomposition. Hardy-Littewood Sobolev inequality.

10.5. Density functional theory in quantum mechanics. Lieb-Oxford inequality.

12.5. Chapter 3: Interpolation theory. Motivation. Complex interpolation method: Hadamard three-lines theorem. A complex-interpolation proof of the Hölder L^p interpolation inequality.

17.5. Riesz–Thorin interpolation theorem. Real interpolation method: Marcinkiewicz interpolation theorem. Maligranda's notes about Marcinkiewicz.

19.5. Chapter 4: Lorentz spaces. Quasi-normed vector spaces. Metrizability and the Aoki-Rolewicz theorem.

24.5. Decreasing rearrangement. Equivalent definition of Lorentz norm in terms of rearrangement. Normability of Lorentz spaces.

26.5. The inclusion monotonicity of Lorentz spaces. Hölder's inequality. The dual of Lorentz spaces.

31.5. From the weak Young to the standard Sobolev inequality. Young's inequality for Lorentz spaces. Improved Sobolev inequality.

2.6. Dyadic decomposition. An alternative proof of the improved Sobolev inequality.

6.6. Chapter 5: Fourier transform. Fourier transform on L^1, L^2. The Hausdorff-Young inequality. Schwartz space. Fréchet spaces. Fourier transform on Schwartz space.

9.6. Tempered distributions. Examples: L^p functions, slowly increasing functions, Dirac delta, finite measures. The relationship of Fourier transform with derivatives and multiplication by polynomials. Liouville's theorem.

14.6. Midterm exam.

16.6. Remarks on general distributions (dual of C_c^infty). Hardy's uncertainty principle. Sobolev's uncertainty principle and compact embedding.

21.6. Further remarks on tempered distributions.

28.6. Chapter 6: Applications of Fourier transform. Green function in PDE. Regularity of Poisson equation.

30.6. Dispersive estimates for heat equation and Schrödinger equation. Ill-posedness and regularization of backward heat equation.

5.7. Oscillatory integrals. Van der Corput theorem. Applications to Bessel functions.

7.7. Some applications of Fourier transform in number theory. Gauss circle problem. Equidistributed sequence. First digit of powers of 2. Riemann zeta function and Basel problem.

12.7. Chapter 7: Decomposition methods. Calderón–Zygmund decomposition. Application: a reverse maximal inequality. L^1 log L^1 space and integrability of maximal function.

14.7. A variant of Calderón–Zygmund decomposition. Application: A proof of the Lieb-Thirring inequality. A review of Lieb-Thirring inequalities (we discussed the Lundholm-Solovej method in Section 3).

19.7. Littlewood–Paley L^p theory. Sabin's proof of the LT inequality via Littlewood–Paley decomposition.

21.7. Final exam.