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
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.
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