Speaker:

Shuanglin Shao, University of Minnesota

Title:

Existence of extremals for a Fourier restriction inequality for S^2

Abstract:

The adjoint Fourier restriction operator for the sphere maps L^2(S^2) to
L^4(R^3). We prove that there exists an extremal for this fundamental inequality
in the sense that any extremising sequence of non-negative functions is
precompact in L^2(S^2).  I will start with some computation, which compares the
sphere with the paraboloid, and a formal concentration-compactness argument,
which relies on a refinement of this inequality. They help us identify three
essential difficulties for an extremising sequence: possibility of convergence
to Dirac masses, lack of regularity, and lack of exact scaling symmetry. The
argument involves a delicate analysis on a bilinear convolution operator and
Fourier integral operator.  symmetrization plays a crucial role in the
argument.  If time permitted, I will end it to show that constant functions are
local maxima by use of spherical harmonics and Gegenbauer polynomials.

This is a joint work with Michael Christ.