Topics:
1. Interpolation theorems in generality of Marcinkiewicz, Riesz-Thorin and Stein.
2. Riesz potentials, Hardy-Littlewood-Sobolev fractional integration theorem. Poincare
and local Sobolev inequalities. Gagliardo-Nirenberg theorem, Moser-Trudinger inequality.
3. Calderon-Zygmund singular integrals the L2 theory. The non-convolution case Cotlar’s Lemma.
4. The Lp theory of singular integrals, the Calderon-Zygmund decomposition.
5. Maximal singular integrals.
6. BMO, the Spanne-Stein theorem, the John-Nirenberg inequality.
7. The Fefferman-Stein sharp function.
8. H¨ormander’s multiplier theorem.
9. Khinchine’s inequality/ large deviation estimate and the Littlewood-Paley theorem.
10. Restriction theory of the Fourier transform.
11. Bochner-Riesz theorem of Fefferman and Strichartz inequality.
12. The theory of Ap weights and Muckenhoupt’s theory for the Hardy-Littlewood maximal
function and weighted inequalities for singular integrals.
13. (Time permitting) Carleson measures and Fefferman’s theorem that BMO is the dual
of Hardy space.
There is no textbook for this course. Lecture notes for each Lecture will be posted on Canvas.
501 and 503