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Kirsti Biggs (Uppsala University)

Title: A counting approach to square functions over local fields

Abstract: A general philosophy in harmonic analysis is that square function estimates imply decoupling estimates, which imply discrete restriction estimates, and these in turn imply the mean value estimates used to count solutions to Diophantine equations. However, sometimes this philosophy can be turned on its head: Gressman et al. (2021) proved square function estimates for extension operators on non-degenerate real curves by counting near-solutions to a related system of equations. We extend these results to the case of sufficiently smooth real planar curves which are not completely flat, and, in the case of polynomial curves, we prove such an estimate over general local fields. This is joint work with Julia Brandes and Kevin Hughes.