The course will center in the study of basic and widely used operators on L^p spaces that are treated by means of harmonic analysis.
The operators to be considered include multipliers and singular operators (for instance, Hilbert transform), and the techniques involved include interpolation and stopping time argument.
Prerequisites: (introduction to) analysis and linear algebra are required; either Math 563 (measure theory) or Math 472/572 (Fourier analysis and wavelets) is desirable. Concepts and results of measure theory and Fourier analysis will be introduced as necessary.
Textbook: L. Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc, Upper Saddle River, NJ, 2004, ISBN 0-13-035399-X
We will cover selected material of chapters 1 - 4, 8.
Supplementary reading: 1. Linear operators, functional analysis: K. Yosida, Functional analysis, (reprint of ) the 6th edition, 1980.
2. L^p spaces, distributions, applications to differential equations: E. M. Stein, R. S. Shakarchi, Functional analysis. Introduction to further topics in analysis, Princeton Lectures in Analysis, 4. Princeton University Press, Princeton, NJ, 2011
3. Real analysis, L^1 & L^2 spaces: E. M. Stein, R. S. Shakarchi, Real analysis. Measure theory, integration, and Hilbert spaces, Princeton Lectures in Analysis, 3. Princeton University Press, Princeton, NJ, 2005
4. Fourier series, transforms, and their applications:
E. M. Stein, R. S. Shakarchi, Fourier analysis. An introduction, Princeton Lectures in Analysis, 1. Princeton University Press, Princeton, NJ, 2003
M. C. Pereyra, L. A. Ward, Harmonic analysis: from Fourier to wavelets, Student Mathematical Library Series, Volume 63, American Mathematical Society, 2012.
5. Exposition on evolution of the subject and its applications: S. Krantz, A panorama of harmonic analysis, Carus Mathematical Monographs, 27, Mathematical Association of America, Washington, DC, 1999
Tue & Thur 2 - 3:15 p.m., in SMLC 352
Tue and Thur 3:30 - 4:45 p.m., and also by appointment, in SMLC 212
will be determined by homework and in-class presentations
Jan 15, 17 1.1.a:L^p spaces, distribution function, weak L^p spaces. Jan 22, 24 1.1.a,c:Weak L^p spaces and some interpolation results. Linear operators. Jan 29, 31 1.3.a,b:Real and complex interpolation methods. Feb 5, 7 1.2.d, 2.2.a,b:Three lines lemma. Approximate identities. Fourier transform. Feb 12, 14 2.2.d,b, 2.3.c: Fourier and inverse Fourier transforms. Tempered distributions. Feb 19, 21 4.1.a: Hilbert transform on S(R) and L^2(R). Feb 26, 28 4.1.b, 2.1.a: Convergence of truncated Hilbert transforms on L^2(R). Maximal function. Mar 5, 7 2.1.c, 4.1.c: Control of maximal operators. Hilbert transform on L^p(R). Mar 19, 21 2.1.b,c, 4.3.a, 4.3.c: Maximal operators. Calderon-Zygmund decomposition. H is of weak type (1,1). Mar 26, 28 4.3.c, 4.4.a: Singular integrals of convolution type. Apr 2, 4 overview of 8.1 - 8.3, 7.1.a: Singular integrals of nonconvolution type. BMO