Maximal functions, Nikodym sets and local smoothing estimates

par Chen Mingfeng (IHES)

vendredi 9 oct. 2026, 14:00 → 15:00 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Séminaire d'Analyse

I will discuss some recent works related to maximal operators and Nikodym sets in the plane. Nikodym constructed a measure-zero Borel set in the plane that contains a line segment passing through each point in the unit square. Fenton and Falconer asked whether a similar construction is possible when “lines through a point” are replaced by ”circles with a prescribed center.” This was answered in the negative by Bourgain and Marstrand. We study a generalization of this question, in which “lines through a point” or “circles with prescribed center” are replaced by general analytic families of plane curves. 
Using decouping, we prove new maximal function and local smoothing estimates for families of curves satisfying a multi-parameter analogue of Sogge's cinematic curvature condition. Combining with a recent nonlinear generalization of Falconer's prescribed projection theorem to construct Nikodym-type sets, we characterize when Nikodym-type sets exist for analytic families of plane curves.

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