In the first part of the course, we will study basic concepts in Harmonic Analysis, specifically, interpolation theorems, Hardy-Littlewood maximal function and the Hardy-Littlewood-Sobolev fractional integration theorem. This will be followed by the Littlewood-Paley theory and their applications in r-variational estimates as well as the Calderon-Zygmund theory of singular integrals and Radon operators. In the second part of the course we will study restriction theorems and Kakeya maximal functions, we will also construct the Besicovitch set. This will be a starting point of the decoupling theory, which will be used to prove the Vinogadov mean value conjecture from number theory. Finally, we will prove the Carleson theorem, which asserts that partial sums of Fourier series of a square integrable function converge pointwise almost everywhere.
No textbook
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