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arXiv · 2607.28882

Nonconcentration of eigenfunctions in Microlocal Kakeya-Nikodym norms: a phase space approach

Matthew D. Blair
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Abstract

Previous works of the author and Sogge [BS17], [BS18] showed the significance of microlocal Kakeya-Nikodym averages in improving $L^p$ bounds on (approximate) eigenfunctions of the Laplacian in the high frequency limit. These averages are formed by taking the $L^2$ norm of an eigenfunction when localized in phase space to a small, frequency-dependent tube about a geodesic segment via a pseudodifferential operator. The former work showed that for values of $p$ beneath the Stein-Tomas exponent, $L^p$ norms are controlled by a supremum over these averages. The latter work then showed that when $(M,g)$ has nonpositive sectional curvatures, there is a logarithmic gain in the averages. In combination, these two works improved the $L^p$ theory for eigenfunctions over the universal bounds of Sogge in this geometric setting. In the present work, we develop sufficient conditions for improving these averages which are more general than nonpositive curvature. Instead our sufficient conditions are rooted in the dynamics of the geodesic flow on the tangent bundle, considering cases where the flow expands and contracts tangent vectors in at least some directions, e.g. partially hyperbolic flows. We make use of Gaussian wave packet (phase space) transforms on the manifold in order to fully appreciate the gain these hypotheses impart on the microlocal averages. In the process, we further develop Gaussian beam approximations to the wave equation in a coordinate invariant manner.

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Matthew D. Blair. 2026-07-30. Nonconcentration of eigenfunctions in Microlocal Kakeya-Nikodym norms: a phase space approach. https://arxiv.org/abs/2607.28882

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