Microlocal decoupling inequalities and the distance problem on Riemannian manifolds

Abstract

We study the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. Key new ingredients include a family of refined microlocal decoupling inequalities, which are related to the work of Beltran-Hickman-Sogge on Wolff-type inequalities, and an analog of Orponen's radial projection lemma which has proved quite useful in recent work on distance sets.

arXiv comment36 pages, 2 figures, Referee's suggestions incorporated

References in corpus (6)

• Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds
• On the dimension and smoothness of radial projections
• Weighted restriction estimates and application to Falconer distance set problem
• Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions
• New bounds on the dimensions of planar distance sets
• Sharp $L^2$ estimate of Schrödinger maximal function in higher dimensions

Cited by in corpus (2)

• On smoothing estimates in modulation spaces and the nonlinear Schrödinger equation with slowly decaying initial data
• Local smoothing and Hardy spaces for Fourier integral operators on manifolds