HARMONIC ANALYSIS

Second cycle degree in MATHEMATICS

Language: English

Teaching period: Second Semester

Lecturer: PAOLO CIATTI

Number of ECTS credits allocated: 6

Syllabus
 Prerequisites: Real Analysis. Some knowledge of Complex Analysis in one variable could be useful. Examination methods: Oral exam Course unit contents: A basic problem in Harmonic Analysis is to determine all of the L^p-estimates that the Fourier transform obey: to find all the inequalities of the form ||F(f)||_p < C ||f||_q, where F(f) is the Fourier transform of a function f. This problem was solved in the early 20th century. The Hausdorffâ€”Young inequalities, that we will prove and discuss, give all the inequalities of this form. The restriction problem concerns a generalisation of this problem where we replace the L^q(E) norm on the right hand side with an L^q norm on a surface S in E. In the sixties Elias M. Stein discovered several interesting things about this question. One important discovery is that the curvature of the surface S matters. The unit sphere in the Euclidean space obeys inequalities that are qualitative different from the inequalities that obeys a unit flat disk of the same dimension. When the surface S is the unit sphere, Stein made a conjecture about all the restriction estimates associated to the sphere and proved some nontrivial cases. The two dimensional case in this conjecture was then proved by Charles Fefferman. We will prove the theorems of Stein and Fefferman and then state and discuss the conjecture. In dimension three and higher, this conjecture is wide open and looks very difficult.