Harmonic Analysis

a)Remarks and Questions around the Localisation Problem for n-dimensional Fourier Integrals; b)Averages in Finite Fields

speaker: Anthony Carbery (University of Edinburgh)

abstract: Remarks and Questions around the Localisation Problem for n-dimensional Fourier Integrals: We surveyed the known results in this area, in particular how the problem relates to L2 weighted inequalities for the extension operator for the Fourier transform, and paying attention to the notion of Sets of Divergence for the Localisation Problem (SDLP's) and the related notion of tube-nullity. Many open questions were discussed. Most of the work was joint work with F.Soria and A. Vargas.

Averages in Finite Fields: In joint work with Jim Wright we examined analogues in the case of vector spaces over finite fields of the Lp - Lq improving and "spherical maximal" problems for averages in Rn. We gave necessary conditions on the exponents for these phenomena to hold, and, in the case of improving, demonstrated that for each k there is a class of surfaces of dimension k in Fn for which the optimal result holds. This was obtained by using A.Weil's estimates on exponential sums that were developed in his solution of the Riemann hypothesis for finite fields.

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