Beijing Institute of Mathematical Sciences and Applications Beijing Institute of Mathematical Sciences and Applications

  • About
    • President
    • Governance
    • Partner Institutions
    • Visit
  • People
    • Management
    • Faculty
    • Postdocs
    • Visiting Scholars
    • Staff
  • Research
    • Research Groups
    • Courses
    • Seminars
  • Join Us
    • Faculty
    • Postdocs
    • Students
  • Events
    • Conferences
    • Workshops
    • Forum
  • Life @ BIMSA
    • Accommodation
    • Transportation
    • Facilities
    • Tour
  • News
    • News
    • Announcement
    • Downloads
About
President
Governance
Partner Institutions
Visit
People
Management
Faculty
Postdocs
Visiting Scholars
Staff
Research
Research Groups
Courses
Seminars
Join Us
Faculty
Postdocs
Students
Events
Conferences
Workshops
Forum
Life @ BIMSA
Accommodation
Transportation
Facilities
Tour
News
News
Announcement
Downloads
Qiuzhen College, Tsinghua University
Yau Mathematical Sciences Center, Tsinghua University (YMSC)
Tsinghua Sanya International  Mathematics Forum (TSIMF)
Shanghai Institute for Mathematics and  Interdisciplinary Sciences (SIMIS)
BIMSA > Introduction to Harmonic Analysis
Introduction to Harmonic Analysis

In this course, we begin by exploring foundational concepts such as the Hardy-Littlewood maximal function, the Lebesgue differentiation theorem, and the Marcinkiewicz interpolation theorem. We then delve into the Calderón-Zygmund decomposition, which effectively separates functions and sets into "good" and "bad" parts, allowing for distinct treatment through techniques in real variable theory and harmonic analysis.

Next, we examine the space of functions of bounded mean oscillation (BMO), which consists of functions whose mean oscillation over cubes is uniformly bounded. BMO is significant in the regularity theory of nonlinear partial differential equations. We rigorously analyze the proofs of key theorems, including the John-Nirenberg inequality, the Fefferman-Stein theorem, and the Stampacchia interpolation theorem.

Additionally, we study Muckenhoupt’s \(A_p\) weights and their associated weighted norm inequalities, which provide characterizations of BMO functions. A primary objective of the course is to demonstrate that the Hardy-Littlewood maximal operator is of weighted strong type \((p, p)\) for \(1 < p < \infty\) if and only if the weight satisfies Muckenhoupt’s \(A_p\) condition.

Finally, we examine the statement and proof of the celebrated Gehring Lemma, demonstrating how this lemma establishes that every $A_p$ weight satisfies a reverse Hölder inequality.
Lecturer
Mahdi Hormozi
Date
20th September ~ 20th December, 2024
Location
Weekday Time Venue Online ID Password
Friday 20:55 - 19:20 A7-201 ZOOM 06 537 192 5549 BIMSA
Reference
[1] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,
Princeton University Press, 1970.
[2]E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality,
and Oscillatory Integrals, Princeton University Press, 1993.
[3] E. M. Stein and G. Weiss, Fourier Analysis on Euclidean Spaces, Princeton
University Press, 1970.
[4] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press,
1988.
[5] J. Duoandikoetxea, Fourier Analysis, American Mathematical Society,
2001.
[6] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities
and Related Topics, North-Holland, 1985.
[7] J. Kinnunen, Harmonic Analysis, 2017
[8] S. Lu, Y. Ding and D. Yan, Singular Integrals and Related Topics, World
Scientific, 2007.
[9] B. Simon, Harmonic Analysis, A Comprehenesive Course in Analysis,
Part 3, American Mathematical Society, 2015.
[10] A. Torchinsky, Real Variable Methods in Harmonic Analysis, Academic
Press, 1986.
[11] L. Grafakos, Classical Fourier Analysis, Springer, 2008.
[12] L. Grafakos, Modern Fourier Analysis, Springer, 2008.
Audience
Advanced Undergraduate , Graduate , Postdoc , Researcher
Video Public
No
Notes Public
No
Language
English
Beijing Institute of Mathematical Sciences and Applications
CONTACT

No. 544, Hefangkou Village Huaibei Town, Huairou District Beijing 101408

北京市怀柔区 河防口村544号
北京雁栖湖应用数学研究院 101408

Tel. 010-60661855
Email. administration@bimsa.cn

Copyright © Beijing Institute of Mathematical Sciences and Applications

京ICP备2022029550号-1

京公网安备11011602001060 京公网安备11011602001060