Refined and Microlocal Kakeya–Nikodym Bounds of Eigenfunctions in Higher Dimensions

Abstract

We prove a Kakeya–Nikodym bound on eigenfunctions and quasimodes, which sharpens a result of the authors (Blair and Sogge in Anal PDE 8:747–764, 2015) and extends it to higher dimensions. As in the prior work, the key intermediate step is to prove a microlocal version of these estimates, which involves a phase space decomposition of these modes that is essentially invariant under the bicharacteristic/geodesic flow. In a companion paper (Blair and Sogge in J Differ Geom, 2015), it will be seen that these sharpened estimates yield improved L q (M) bounds on eigenfunctions in the presence of nonpositive curvature when $${2 < q < \frac{2(d+1)}{d-1}}$$ .