Differential Operators and Function Spaces II (Block 4, 2011)

Weeklies

Week 1: We started with a discussion of the Fourier transform on various spaces of functions and distributions. This includes 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 covered most of this material except for Riesz-Thorin.

Week 2: We proved 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 and can also be turned into a project. Then we turned to integral operators defined by a measurable kernel function as in Exercise 2.2. The Schur test was recalled as a basic criterion for boundedness. We finally studied how truncating the kernel affects boundedness, in particular the Christ-Kiselev Lemma (Section 8 of Chapter 2 in Tao's Graduate Fourier Analysis notes).

Week 3: We started by applying Christ-Kiselev to almost-everywhere convergence of the Fourier transform (8.8-8.10 of Chapter 2 in Tao's Graduate Fourier Analysis notes) and 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).

Week 4: In the first lecture we briefly considered two alternative approaches and interpretations of the Hardy-Littlewood maximal inequality, namely conditional expectations and the TT* method (still Section 1 of Chapter 3). We then proceeded to Calderon-Zygmund decompositions (Section 4 of Chapter 3) and discussed the Hilbert transform and general Calderon-Zygmund operators (Sections 1 and 2 of Chapter 4) up to Lemma 2.7.

Week 5 We continued with Calderon-Zygmund theory (including the Calderon-Zygmund theorem, Cor. 2.9, and the Hörmander-Mikhlin multiplier theorem, Thm. 4.4) and skipped Section 3. In addition, we used Littlewood-Paley decompositions to show that pseudodifferential operators have singular kernels (see notes).

Week 6 We finished Calderon-Zygmund theory with remarks on applications and the extension to vector-valued operators, in particular the Littlewood-Paley inequality. Then we partially covered the proof that the composition of pseudodifferential operators is again a pseudodifferential operator.

Week 7 We finish the composition of pseudodifferential operators and consider some direct consequences (L2-boundedness, mapping properties between Sobolev spaces, Garding's inequality, elliptic regularity). The additional topics roughly follow Chapter 7 in M. E. Taylor, Partial Differential Equations, Vol. 2.

Week 8 Douglas talked about the convolution algebra L1(G) on an LCA group, the dual group of characters as well as simple properties of the Fourier transform. He got to Theorem 1.2.6 of the notes, which identifies the dual group as an LCA group.

Week 9 We used Bochner's theorem about positive definite functions to prove the inversion formula and Plancherel theorem for the Fourier transform as well as Pontryagin duality.