BCAM Severo Ochoa Course - Introduction to Harmonic Analysis
BCAM Severo Ochoa Course - Introduction to Harmonic Analysis
The course will start on December 2nd, and we will have a total of 8 classes of 2h. I'll also schedule some sessions for questions along the way.
The idea will be to present mostly of what is in the first 3 chapters of
L. Grafakos - Fundamentals of Fourier Analysis. Graduate Texts in Mathematics, 302, Springer, New York, NY 2024.
And then some different results from one of the many great references one can follow, such as:
J. Duoandikoetxea - Fourier Analysis. Graduate Studies in Mathematics, 29, AMS, Providence, RI 2001.
L. Grafakos - Classical Fourier Analysis. Graduate Texts in Mathematics, 249, Springer, New York, NY 2014.
C. Muscalu, and W. Schlag - Classical and Multilinear Harmonic Analysis. Cambridge University Press, New York, NY 2013.
E. Stein - Harmonic Analysis. Princeton University Press, Princeton, NJ 1993.
E. Stein - Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ 1970.
E. Stein, and G. Weiss - Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, NJ 1971.
T. Wolff - Lectures on Harmonic Analysis. American Mathematical Society, Providence, RI 2003.
Differentiation
Fundamental Theorem of Calculus.
The centred and uncentered Hardy-Littlewood (HL) maximal operator.
Lack of boundedness in L1.
Vitali’s covering Lemma.
Weak type estimate for the HL maximal operator.
Lebesgue differentiation theorem.
Convolutions
Definition, basic properties and useful examples.
Young’s inequality.
L^p and pointwise convergence of approximate identities.
Domination by the HL maximal function.
Interpolation
Layer-cake formula.
Weak Lp spaces.
The distribution function and the decreasing rearrangement.
Statement of Marcinkiewicz's interpolation theorem - Diagonal case.
Interpolation
Proof of Marcinkiewicz's interpolation theorem - Diagonal case.
Definition and basic properties of Lorentz spaces
Marcinkiewicz's interpolation theorem - L^{p,q} case.
Hadamard's 3 line lemma.
Riesz-Thorin interpolation theorem.
Fourier transform
Basic properties and Riemann-Lebesgue Lemma.
Fourier inversion.
Plancherel theorem.
Hausdorff-Young inequality.
The Schwartz class and its basic properties.
Fourier transform
Tempered distributions and their basic properties.
Fourier multipliers in L².
The uncertainty principle.
Fourier multipliers in Lp and statement of Mihlin-Hörmander theorem.
The Hilbert transform and the harmonic conjugate function.
Singular integrals of convolution type and statement of boundedness properties.
Applications of the Riesz and Hilbert transforms.
The Calderón-Zygmund decomposition
Proof of the boundedness of singular integrals.