Sharp restriction theory: rigidity, stability, and symmetry breaking

Abstract: We report on recent progress concerning two distinct problems in sharp restriction theory to the unit sphere. Firstly, the classical estimate of Agmon-Hörmander for the adjoint restriction operator to the sphere is in general not saturated by constants. We describe the surprising intermittent behaviour exhibited by the optimal constant and the space of maximizers, both for the inequality itself and for a stable form thereof. Secondly, the Stein-Tomas inequality on the sphere is rigid in the following rather strong sense: constants continue to maximize the weighted inequality as long as the perturbation is sufficiently small and regular, in a precise sense to be discussed. We present several examples highlighting why such assumptions are natural, and describe some consequences to the (mostly unexplored) higher dimensional setting. This talk is based on joint work with E. Carneiro and G. Negro.

mathematical physicsanalysis of PDEsclassical analysis and ODEscombinatoricscomplex variablesfunctional analysisinformation theorymetric geometryoptimization and controlprobability

Audience: researchers in the topic

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