Vol. 8, No. 3, 2015

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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
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