Jacob Denson

Square function estimates for the Fourier extension of orthonormal systems on space curves

How can we optimally exploit the geometric properties of the Fourier decay of a non-degenerate curve?

In this talk, we study a square-function variant of the L2-Lq restriction inequality for non-degenerate curves. To understand this problem, we are required to develop a Hilbert-space approach to studying Fourier restriction for curves.

The Hilbert-space theory of restriction for non-degenerate curves is made more complicated than the hypersurface case, where a robust theory is provided by the methods of Elias M. Stein and Peter A. Tomas, because the Fourier transform of a measure supported on a curve decays non-isotropically. Optimally exploiting the geometry of the regions upon which curves exhibit different Fourier decay has proved challenging.

We develop such an approach, finding a novel reduction to the study of Lp-improving estimates for averaging operators associated with high-codimension cones generated by the curve. We obtain optimal Lp-Lp' bounds for such operators using a variant of the high-low method. This yields the best possible square-function estimates, up to the endpoint, in three dimensions.