Maximal Operators and Restriction Bounds for Weyl Sums

Date: 2021-06-01

Time: 11:00 - 11:40

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We give a survey of recent results on so called maximal operators  on Weyl sums
S(u;N) =sum_{1le n le N} exp(2 pi i (u_1n+…+u_dn^d)),
where u = (u_1,…,u_d) in [0,1)^d. Namely, given a partition     I cup J subseteq {1,…,,d}, we define the map
(u_i)_{i in I}  mapsto  sup_{u_j,, j   in J} |S(u;N)|
which corresponds to the maximal operator on the Weyl sums associated with the components u_j, j  in J, of u.
We are interested in understanding this map for almost all (u_i)_{i in I}  and also in
the various norms of these operators.  Questions like this have several surprising applications, including outside of number theory,
and are also related to restriction theorems for Weyl sums.

Based on joint work with R. Baker and C. Chen and also with J. Brandes.