Refined and microlocal Kakeya–Nikodym bounds for eigenfunctions in two dimensions

Blair, Matthew; Sogge, Christopher

doi:10.2140/apde.2015.8.747

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Refined and microlocal Kakeya–Nikodym bounds for eigenfunctions in two dimensions

Matthew D. Blair and Christopher D. Sogge

Vol. 8 (2015), No. 3, 747–764

DOI: 10.2140/apde.2015.8.747

Abstract

We obtain some improved essentially sharp Kakeya–Nikodym estimates for eigenfunctions in two dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to the geodesic flow.

Keywords

eigenfunctions, Kakeya averages

Mathematical Subject Classification 2010

Primary: 58J51

Secondary: 35L20, 42B37

Milestones

Received: 12 September 2014

Revised: 31 December 2014

Accepted: 9 February 2015

Published: 3 June 2015

Authors

Matthew D. Blair
Department of Mathematics and Statistics University of New Mexico Albuquerque, NM 87131 United States

Christopher D. Sogge
Department of Mathematics Johns Hopkins University 3400 N. Charles Street Baltimore, MD 21218-2689 United States

Vol. 8, No. 3, 2015