Caltech/UCLA Joint Analysis Seminar

I will consider a question of Mattila regarding the rate of decay of the Fourier transform of fractal measures. One cannot hope for decay in every direction; instead we consider the rate of decay after averaging over the directions. These estimates were first considered in connection with Falconer's conjecture (a continuous analogue of the Erd\"os distinct distances problem), and have since found application elsewhere. I will present a new bound which takes advantage of the Bennett--Carbery--Tao multilinear extension estimates and a pointwise estimate of Bourgain and Guth. This is joint work with Renato Luc\`a.