Sobre el problema de restricción de la transformada de Fourier y sus aplicaciones (2018)
Abstract: Since the seventies, the interest in understanding the mapping properties of restricting the Fourier transform of a function to a manifold, has triggered important new lines of research in analysis. In this thesis we focus on the multilinear theory of restriction, in particular, we extend to the hyperbolic paraboloid a theorem of Ramos about elliptic surfaces in R3, who got the sharp dependence on transversality in the multilinear inequality of Bennett, Carbery and Tao. Furthermore, we point to a possible route towards the proof of Ramos’ theorem in higher dimensions. We show also an application of restriction theory to Falconer’s conjecture, a problem in geometric measure theory. This problem relates to the rate of decay of spherical means of the Fourier transform of compactly supported measures. We exhibit measures whose Fourier transform decays slowly in the whole space, in contrast to previous results.