Abstract: The estimate for the Fourier transform of measure is a fundamental subject in Harmonic Analysis. When a measure is supported on a hypersurface whose Gaussian curvature is nonvanishing, the stationary phase method gives the optimal decay estimate. However, if hypersurfaces are degenerate, the Fourier decay worsens. For compensating this, Sogge--Stein introduced oscillatory integrals with suitable mitigating factors.

In this talk, we focus on oscillatory integrals with damping factor given by a power of the absolute value of the Gaussian curvature. Estimates for these integrals have several applications, including affine Fourier restriction estimates and maximal averages with damping factors. I will also present a sketch of the proof, which involves a modified version of stationary set method. 

Abstract: Characterising rectifiability using the existence of tangents is a very classical technique in GMT and positive results can be found even when using very weak notions of a tangent. This work builds on geometric ideas from Besicovitch in the plane to prove that 1-rectifiability in complete metric spaces is implied by the existence of connected tangents almost everywhere. This requires exploitation of the inherent 'gappiness' of purely 1-unrectifiable sets, which must first be quantified. The result is stated for metric spaces but is in fact also new in Euclidean space. 

Abstract: Multilinear versions of Fourier extension estimates have long been of interest to the harmonic analysis community, and have come to be useful tools in not just restriction theory, but a variety of other contexts including, but not limited to, decoupling, dispersive PDEs, and the study of maximal functions. In this talk, we present new multilinear bounds for Fourier extension operators associated to transversal and positively curved hypersurfaces via a modification of the polynomial partitioning method due to Guth, together with a certain localised refinement of the Bennett—Carbery—Tao multilinear extension estimates. 

Abstract: TBA