Microlocal decoupling inequalities and the distance problem on Riemannian manifolds

Alex Iosevich, Bochen Liu, Yakun Xi

Published 2019-09-11Version 1

In this paper, 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.