MIGSAA Harmonic Analysis (Semester 2, 2017)

Weeklies

Week 1: We discussed the aims of this course and introduced the space of test functions, continuous linear functionals and operators on it. This allowed us to define distributions as the continuous linear functionals on the space of test functions. Locally integrable functions, point evaluations (Dirac delta distributions) and more general Radon measures are distributions, as are their distributional derivatives.

Week 2: We discussed the proof that every locally integrable function can be identified with a unique distribution. We then considered operations with distributions, such as derivatives, multiplication with a smooth function, convolution, and discussed them in examples.

Week 3: We briefly introduced the Schwartz space of rapidly decaying smooth functions, its dual space of tempered distributions and the Fourier transform. As main theorems, we proved that the Fourier transform is an isomorphism on each of these spaces, and it restricts to an isometric isomorphism on the space L2 of square-integrable functions.

Week 4: We covered the Malgrange-Ehrenpreis theorem, which says that every differential operator with constant coefficients has a fundamental solution, and stated the Paley-Wiener theorems for the range of the Fourier transform on the space of test functions or compactly supported distributions. Then we covered Chapter 1 towards the interpolation theorem of Riesz-Thorin.

Week 5: We proved the interpolation theorem of Riesz-Thorin, before we study elementary properties of integral operators, following Chapter 2.

Week 6: We discussed further properties of integral operators (as in Chapter 2), in particular what the boundedness of an integral operator says or does not say about the boundedness of an integral operator with a smaller kernel. We then discussed weak-Lp spaces and the Marcinkiewicz interpolation theorem.

Week 7: We covered Chapter 3: the Hardy-Littlewood Maximal inequality based on Wiener's Vitaly-type covering lemma and the Lebesgue differentiation theorem.

Week 8: We discussed the Calderon-Zygmund decomposition and the Hilbert transform as in Chapters 3 and 4.

Week 9: We discuss Calderon-Zygmund theory and give a quick introduction to Fourier multipliers and pseudodifferential operators. The material follows Chapters 4 and 5.

Week 10: We discuss pseudodifferential operators and their use in function spaces and elliptic PDE.