Square function estimates for the Fourier extension of orthonormal systems on space curves
Jacob Denson (University of Edinburgh)
Thursday 29th October 15:00-16:00
Maths 311B
Abstract
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 Fourier 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 dime
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