Marina Iliopoulou (Kent)

Sharp estimates for Hörmander-type operators

Abstract:
The restriction conjecture, lying at the heart of harmonic analysis, suggests that the Fourier transform of a function defined on a curved surface "behaves better" than if the domain surface was flat. Hörmander further suggested that oscillatory integrals more general than the Fourier transform should satisfy similar properties. However, Bourgain showed that this is false, as the mass of these more general oscillatory integrals may cluster too close to low-degree algebraic varieties, and thus, roughly speaking, can have too many peaks in little space. In this talk we describe the extent to which Hö​rmande​r's conjecture was correct, by providing sharp estimates for Hörmander-type oscillatory integrals that resemble the oscillatory integrals related to the restriction conjecture. This is joint work with L. Guth and J. Hickman.