Differential Operators and Function Spaces II (Block 4, 2012)

Weeklies

Week 1: We started with a discussion of the Fourier transform on various spaces of functions and distributions. This included characterizations of the Fourier image of the spaces of testfunctions and compactly supported distributions (Paley-Wiener theorems, in particular Theorems 1.7.5 and 1.7.7) as well as mapping properties between certain Lp spaces by means of the method of interpolation including the Riesz-Thorin theorem (in particular Lemma 2 and Theorems 1 and 3). We outlined the proof of Riesz-Thorin, but still need to finish it.

Week 2: We finished the proof of the Riesz-Thorin theorem and discussed the key parts of the proof of Marcinkiewicz's theorem (Theorems 3 and 4 in Tao's notes). Some applications of these theorems are covered in the exercise classes. Then we turned to integral operators defined by a measurable kernel function as in Exercise 6. The Schur test was recalled as a basic criterion for boundedness, and we studied how truncating the kernel affects boundedness, in particular the Christ-Kiselev Lemma and its maximal operator version (Section 8 of Chapter 2 in Tao's Graduate Fourier Analysis notes). We also applied Christ-Kiselev to almost-everywhere convergence of the Fourier transform (8.8-8.10 of Chapter 2)

Week 3: We briefly considered smooth truncations (around 8.16). Afterwards the Hardy-Littlewood maximal inequality and its application to the Lebesgue differentiation theorem were discussed (Chapter 3, Proposition 1.1, Theorem 2.1). We also discussed an alternative approach to the Hardy-Littlewood maximal inequality using the TT* method (still Section 1 of Chapter 3).

Week 4: We briefly discussed the Hardy-Littlewood maximal inequality for dyadic squares (still Section 1 of Chapter 3) and then proceeded to Calderon-Zygmund decompositions (Section 4 of Chapter 3). We studied the Hilbert transform as an example of a Calderon-Zygmund operator, and started with the general theory of such operators (Sections 1 and 2 of Chapter 4), up to the Calderon-Zygmund theorem, Cor. 2.9.

Week 5: We continued with Calderon-Zygmund theory, but skipped Section 3. Littlewood-Paley decompositions were introduced to show that pseudodifferential operators have singular kernels (see notes). The Hörmander-Mikhlin multiplier theorem, Thm. 4.4, was also mentioned, as were some simple applications.

Week 6: We briefly introduced Calderon-Zygmund theory for vector-valued operators (in particular the Littlewood-Paley inequality) and stated the composition formula for pseudodifferential operators.

Week 7: We proved the composition formula for pseudodifferential operators and deduced that pseudodifferential operators of order 0 are bounded on L2.

Week 8: We give some further applications of the calculus of pseudodifferential operators and then start with Fourier analysis on locally compact abelian groups, following Rudin.

Week 9: We finish our discussion of Fourier analysis on locally compact abelian groups.