Math 254B - Topics in real analysis Restriction theorems, Besicovitch sets, and applications to PDE Terence Tao

Time and place: MS 5233, MWF 11-12

Required: 245A, in particular familiarity with Fourier series and Banach spaces. Taking 254A last quarter will be helpful not necessary.

Let S be a (mostly) smooth hypersurface in R^{n}, such as a sphere, paraboloid, or cone, wth surface measure dσ. We are interested in studying the behaviour of Fourier transforms of measures on S, i.e. quantities such as

f̂dσ(x) = ∫_{S} e^{−2πix·ξ}f(x) dσ(ξ)

where f is a function on S in some L^{p} class (not necessarily smooth). Such functions arise in the study of certain PDE such as the Schrödinger equation and the wave equation, and also in the classical theory of summation of Fourier series.

A current area of research in harmonic analysis is the restriction conjecture, which asserts that if S is (for instance) the unit sphere S^{n−1}, one has the bounds

∥f̂dσ∥_{L^{p}(R^{n})} ≲ ∥f∥_{L^{q}(S)}

if and only if 1 ≤ q ≤ ∞, p > 2n/(n − 1) and p ≥ \frac{n+1}{n−1}q′. This conjecture is not fully solved, but partial progress has been made, and we will show how this progress has been used to show existence and stability for certain non-linear wave and Schrödinger equations.

More recent progress by Bourgain, Wolff, and others has shown that these problems are somewhat surprisingly linked to a purely geometrical conjecture, known as the Kakeya conjecture:

If E ⊂ R^{n} contains a line segment in every dimension, then the Hausdorff dimension of E must be exactly equal to n.

This problem originated in the classical Kakeya needle problem posed in the 1900s (what is the smallest area you need in order to turn a unit needle around by 180 degrees)? but has since been shown to be linked to a large number of other unsolved problems.

We will discuss as many of the above results and connections as time allows. Many of the results are very recent (1991-1999).

There is no textbook, although some of the material can be found in Stein’s ”Harmonic Analysis”.

First class is on Monday April 5, 11am, in MS 5233.