This is the webpage of the Curso Avanzado de Análisis (Advanced Analysis Course) of the Máster en Matemáticas y Aplicaciones corresponding to the academic year 2022-2023. The official syllabus of the course can be found here (English here). The language of the course is English.
Schedule and class location: Tuesday and Thursday, 10:00h to 11:30h, room 320, módulo 17 (facultad de Ciencias).
Office hours: by appointment.
Announcements
(2/6/23) Slides of the talks have been uploaded.
(1/6/23) Last version of the notes has been uploaded.
(20/4/23) Notes of week 10 have been uploaded.
(14/4/23) Notes of weeks 8 and 9 have been uploaded. The last new exercises are available.
(10/4/23) Only exercises up to (and including) block 2 will be available for typing.
(25/3/23) Notes of week 7 have been uploaded. Some new exercises are available.
(21/3/23) Notes of week 6 have been uploaded. Some new exercises are available.
(10/3/23) A revision of the notes, including weeks 4 and 5, has been uploaded. Final essay rubric uploaded as well.
(8/3/23) Notes of weeks 4 and 5 have been uploaded. Some new exercises are available.
(3/3/23) A revision of the notes of week 3 has been uploaded.
(27/2/23) Notes of week 3 have been uploaded. Some new exercises are available.
(17/2/23) Notes of week 2 have been uploaded. Some new exercises are available.
(8/2/23) Notes of block 0 updated after corrections.
(8/2/23) List of exercises to write has been created. Let me know if you want me to save a particular one for you.
(3/2/23) Note taking list created. If you are enrolled you have to choose a week to take notes and another one revise someone else's note taking.
(8/9/22) Web created. Classes start on Tuesday, January 31 2023.
Materials
Presentations of the final essays.
Collaborative notes.
Template for final assignment. It is the usual amsart template, set to 11pt letter size. The final assignment should not be longer than 11 pages, plus the title page with the abstract and the references.
Rubric for the final assignment. I will use it to grade, so you can use it to inform your writing.
Summary of topics covered in class
Tuesday, January 31: Presentation. Measurable functions. Lp spaces: definition, Hölder's inequality, duality, layer cake formula. Weak Lp spaces: Chebyshev's inequality, motivation, definition, quasi-norm.
Thursday, February 2: Intersection and sum of Banach spaces. Linear, sublinear, quasi-linear and bounded operators. Interpolation: Marcinkiewicz theorem, Riesz-Thorin theorem.
Tuesday, February 7: Conditional expectations: definition, properties and Lp boundedness.
Thursday , February 9: Martingales, Lp bounded martingales, characterization for p in strictly convex range.
Tuesday, February 14: Characterization of L1 and essentially bounded martingales. Krickeberg decomposition. BMO norm. Doob's maximal function. Boundedness on Lp of Doob's maximal function.
Thursday, February 16: Almost everywhere convergence of martingales in Lp. Statement of Calderón-Zygmund decomposition and first part of the proof.
Tuesday, February 21: Second part of the proof of the Calderón-Zygmund decomposition.
Thursday, February 23: Square functions of martingale transforms: weak L1 boundedness.
Tuesday, February 28: Square functions of martingale transforms: Lp, BMO boundedness. Burkholder-Gundy inequalities. Khintchine inequality. The dyadic martingale.
Thursday, March 2: Dyadic BMO on the Euclidean space. Regular martingales and doubling measures. The Hardy-Littlewood and other maximal functions.
Tuesday, March 7: Construction of several separated dyadic systems.
Thursday, March 9: Maximal functions are Lp bounded. Lebesgue's differentiation theorem. Schwartz class.
Tuesday, March 14: Tempered distributions. The Fourier transform.
Thursday, March 16: The Hilbert transform: definition, L2 boundedness.
Tuesday, March 21: The Hilbert transform: weak L1 and Lp estimates.
Thursday, March 23: The Hilbert transform: convergence of truncations in principal value sense.
Tuesday, March 28: Calderón-Zygmund operators: definition, Lp and BMO boundedness.
Thursday, March 30: BMO and H1 estimates for Calderón-Zygmund operators. Examples.
Tuesday, April 11: Convergence of Fourier integrals. Fourier multipliers: definition and examples.
Thursday, April 13: Fourier multipliers: general properties.
Tuesday, April 18: Vector valued operators and square functions. Smooth dyadic annuli in dimension 1.
Thursday, April 20: Littlewood-Paley theory in dimension 1.
Tuesday, April 25: General relation between martingale transforms and Caalderón-Zygmund operators. Littlewood-Paley theory in general dimension: coronas.
Thursday, April 27: Littlewood-Paley theory in general dimension: rectangles.
Thursday, May 4: Sufficiency conditions: Marcinkiewicz theorem.
Tuesday, May 9: Sufficiency conditions: Hörmander-Mikhlin theorem. Smooth approximations of identity.
Thursday, May 11: The multiplier of the ball.
Tuesday, May 16: Review and essay preparation.
Thursday, May 18: Review and essay preparation.
Note taking list
Week of Jan 31. Notes: Jose. Revision: David Muñoz.
Week of Feb 7. Notes: Víctor Mialdea. Revision: Francisco Pina.
Week of Feb 14. Notes: Fernando Ballesta. Revision: David Muñoz.
Week of Feb 21. Notes: Francisco Pina. Revision: Victoria Pelayo.
Week of Feb 28. Notes: Pablo García. Revision: Víctor Mialdea.
Week of Mar 7. Notes: Victoria Pelayo. Revision: Pablo García.
Week of Mar 14. Notes: David Muñoz. Revision: Patrick Schurk.
Week of Mar 21. Notes: Louise Ethier. Revision: Elena Petrunina.
Week of Mar 28. Notes: Patrick Schurk. Revision: Anne Grünbeck.
Week of Apr 11. Notes: Anne Grünbeck. Revision: Jose.
Week of Apr 18. Notes: Javier Minguillón. Revision: Louise Ethier.
Week of Apr 25. Notes: Elena Petrunina. Revision: Javier Minguillón.
Week of May 2. Notes: Diego González. Revision: José Antonio Lucas.
Week of May 9. Notes: José Antonio Lucas. Revision: Diego González.
Exercises to be written (with author). Only exercises from chapters 0, 1 and 2 can be picked up.
Block 0
Exercise 0.2. Anne Grünbeck (pub).
Exercise 0.4. Francisco Pina (pub).
Exercise 0.6. David Muñoz (pub).
Exercise 0.8. David Nieto (pub).
Exercise 0.9. David Muñoz (pub).
Exercise 0.11. Victoria Pelayo (pub).
Exercise 0.12. Elena Petrunina (pub).
Exercise 0.14. Elena Petrunina.
Exercise 0.15. Pablo García (pub).
Exercise 0.16. David Muñoz (pub).
Exercise 0.17. Elena Petrunina (pub).
Exercise 0.18. David Muñoz (pub).
Exercise 0.19. Elena Petrunina (pub).
Exercise 0.21. David Nieto (pub).
Exercise 0.22. David Nieto (pub).
Block 1
Exercise 1.2. Diego González (pub).
Exercise 1.3. Victoria Pelayo (pub).
Exercise 1.4. Francisco Pina (pub).
Exercise 1.8. José Antonio Lucas (pub).
Exercise 1.13. Francisco Pina (pub).
Exercise 1.19. Louise Ethier.
Exercise 1.24. David Muñoz (pub).
Exercise 1.25. Víctor Mialdea (pub).
Exercise 1.26. Patrick Schurk.
Exercise 1.29. Víctor Mialdea (pub).
Exercise 1.33. Louise Ethier.
Exercise 1.35. David Nieto (pub).
Exercise 1.36. David Nieto.
Exercise 1.38. David Muñoz (pub).
Exercise 1.42.
Exercise 1.44.
Exercise 1.45. David Nieto.
Exercise 1.51.
Exercise 1.52.
Exercise 1.53.
Exercise 1.55. José Antonio Lucas.
Exercise 1.57. José Antonio Lucas (pub).
Exercise 1.64.
Exercise 1.65.
Exercise 1.66. Victoria Pelayo (pub).
Exercise 1.67. David Nieto.
Exercise 1.68. Victoria Pelayo (pub).
Exercise 1.70. Francisco Pina (pub).
Exercise 1.71. David Nieto.
Exercise 1.72. Victoria Pelayo (pub).
Exercise 1.74. Pablo García.
Exercise 1.76. Víctor Mialdea.
Exercise 1.77.
Exercise 1.78.
Exercise 1.82. David Muñoz (pub).
Exercise 1.83. David Muñoz (pub).
Exercise 1.84. David Muñoz (pub).
Block 2
Exercise 2.2. Victoria Pelayo (pub).
Exercise 2.3. Víctor Mialdea.
Exercise 2.4.
Exercise 2.5.
Exercise 2.6. Anne Grünbeck.
Exercise 2.10. Diego González (pub).
Exercise 2.15.
Exercise 2.17. Diego González.
Exercise 2.18.
Exercise 2.19.
Exercise 2.21. Patrick Schurk (pub).
Exercise 2.23. Víctor Mialdea.
Exercise 2.24. Francisco Pina (pub).
Exercise 2.27.
Exercise 2.28. David Nieto (pub).
Exercise 2.30. David Nieto.
Exercise 2.32.
Exercise 2.34.
Exercise 2.37. Anne Grünbeck (pub).
Exercise 2.38.
Exercise 2.39.
Exercise 2.42.
Exercise 2.43.
Exercise 2.44.
Exercise 2.46.
Exercise 2.49.
Exercise 2.52.
Exercise 2.55. Diego González (pub).
Exercise 2.58. Diego González (pub).
Exercise 2.60.
Exercise 2.63.
Exercise 2.64.
Exercise 2.65.
Exercise 2.68.
Exercise 2.72. Patrick Schurk.
Exercise 2.75. Anne Grünbeck (pub).
Exercise 2.76. Patrick Schurk.
Exercise 2.80.
Exercise 2.85.
Exercise 2.86.
Exercise 2.89.
Exercise 2.90.
Exercise 2.91.
Exercise 2.92.