Singular Integrals
The course covers singular integrals, maximal functions, and applications to differentiation theory and partial differential equations. If time permits, we will also discuss oscillatory integrals, in particular Hörmander’s Fourier integral operators.
Lecturer
Date
23rd September ~ 18th December, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 09:50 - 11:25 | Shuangqing-B626 | Zoom 16 | 468 248 1222 | BIMSA |
Prerequisite
I plan to remind the main notions/facts and try to explain them informally if need be.
• Real Analysis (basic measure theory, Lebesgue integration).
• Basic Functional Analysis (Hilbert spaces).
• Introductory Fourier analysis.
• Real Analysis (basic measure theory, Lebesgue integration).
• Basic Functional Analysis (Hilbert spaces).
• Introductory Fourier analysis.
Syllabus
Week 1 – Introduction & Motivation
• Lecture 1: What are singular integrals? Examples from harmonic analysis and PDEs.
• Lecture 2: Preliminaries: $L^p$ spaces, Hölder/Minkowski inequalities, Fourier transform basics.
Week 2 – Classical Examples
• Lecture 3: The Hilbert transform on $\mathbb{R}$. Principal value definition.
• Lecture 4: Riesz transforms on $\mathbb{R}^n$.
Week 3 – Calderón–Zygmund Kernels
• Lecture 5: Kernel conditions (size & smoothness).
• Lecture 6: Singular integral operators (formal definition).
Week 4 – $L^2$ Theory
• Lecture 7: Boundedness of singular integrals on $L^2$.
• Lecture 8: Weak $(1,1)$ bounds (sketch proof using Calderón–Zygmund decomposition).
Week 5 – Interpolation and $L^p$ Theory
• Lecture 9: Marcinkiewicz interpolation theorem.
• Lecture 10: Boundedness of Calderón–Zygmund operators on all $L^p$, $ 1< p<\infty$.
Week 6 – Maximal Functions
• Lecture 11: Hardy–Littlewood maximal operator, weak $(1,1)$.
• Lecture 12: Differentiation of integrals, Vitali covering lemma.
Week 7 – Applications I
• Lecture 13: Fourier series, Carleson’s theorem (motivation).
• Lecture 14: Harmonic functions and Poisson kernel.
Week 8 – Applications II
• Lecture 15: Calderón’s problem in elliptic PDEs (basic sketch).
• Lecture 16: Singular integrals in boundary value problems.
Week 9 – Weighted Inequalities
• Lecture 17: $A_p$ weights (definition, examples).
• Lecture 18: Weighted norm inequalities for singular integrals.
Week 10 - Application III
• Lecture 19: Weighted Hardy spaces.
• Lecture 20: Application in signal analysis
Week 11 – Maximal Singular Integrals
• Lecture 19: Maximal truncations and their boundedness.
• Lecture 20: Comparison with Hardy–Littlewood maximal function.
Week 12 – Oscillatory Integrals (Introductory)
• Lecture 21: Examples of oscillatory integrals (Fourier integral operators).
• Lecture 22: Hörmander–Mikhlin multiplier theorem (statement, applications).
If time permits:
• Survey of the course, say something about further developments
• Lecture 1: What are singular integrals? Examples from harmonic analysis and PDEs.
• Lecture 2: Preliminaries: $L^p$ spaces, Hölder/Minkowski inequalities, Fourier transform basics.
Week 2 – Classical Examples
• Lecture 3: The Hilbert transform on $\mathbb{R}$. Principal value definition.
• Lecture 4: Riesz transforms on $\mathbb{R}^n$.
Week 3 – Calderón–Zygmund Kernels
• Lecture 5: Kernel conditions (size & smoothness).
• Lecture 6: Singular integral operators (formal definition).
Week 4 – $L^2$ Theory
• Lecture 7: Boundedness of singular integrals on $L^2$.
• Lecture 8: Weak $(1,1)$ bounds (sketch proof using Calderón–Zygmund decomposition).
Week 5 – Interpolation and $L^p$ Theory
• Lecture 9: Marcinkiewicz interpolation theorem.
• Lecture 10: Boundedness of Calderón–Zygmund operators on all $L^p$, $ 1< p<\infty$.
Week 6 – Maximal Functions
• Lecture 11: Hardy–Littlewood maximal operator, weak $(1,1)$.
• Lecture 12: Differentiation of integrals, Vitali covering lemma.
Week 7 – Applications I
• Lecture 13: Fourier series, Carleson’s theorem (motivation).
• Lecture 14: Harmonic functions and Poisson kernel.
Week 8 – Applications II
• Lecture 15: Calderón’s problem in elliptic PDEs (basic sketch).
• Lecture 16: Singular integrals in boundary value problems.
Week 9 – Weighted Inequalities
• Lecture 17: $A_p$ weights (definition, examples).
• Lecture 18: Weighted norm inequalities for singular integrals.
Week 10 - Application III
• Lecture 19: Weighted Hardy spaces.
• Lecture 20: Application in signal analysis
Week 11 – Maximal Singular Integrals
• Lecture 19: Maximal truncations and their boundedness.
• Lecture 20: Comparison with Hardy–Littlewood maximal function.
Week 12 – Oscillatory Integrals (Introductory)
• Lecture 21: Examples of oscillatory integrals (Fourier integral operators).
• Lecture 22: Hörmander–Mikhlin multiplier theorem (statement, applications).
If time permits:
• Survey of the course, say something about further developments
Reference
• E.M. Stein, Singular Integrals and Differentiability Properties of Functions, (Princeton, 1970)
• E.M. Stein & R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, (Princeton Lectures in Analysis III, 2005)
• J. Duoandikoetxea, Fourier Analysis (AMS Graduate Studies in Mathematics, Vol. 29, 2001)
• L. Grafakos, Classical Fourier Analysis (3rd ed.) (Springer, 2014)
• E.M. Stein & R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, (Princeton Lectures in Analysis III, 2005)
• J. Duoandikoetxea, Fourier Analysis (AMS Graduate Studies in Mathematics, Vol. 29, 2001)
• L. Grafakos, Classical Fourier Analysis (3rd ed.) (Springer, 2014)
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Yurii Lyubarskii received PhD degree in mathematics in 1974 in the Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences and Doctor degree in St. Petersburg branch of Russian mathematical institute in 1990. He taught at the Norwegian University of Science and Technology and also at St. Petersburg State University. Scientific interests of Yu. Lyubarskii include complex and harmonic analysis and applications to the signal analysis.