@article{discovery10087655, publisher = {DUKE UNIV PRESS}, note = {This version is the author accepted manuscript. For information on re-use, please refer to the publisher's terms and conditions.}, pages = {2991--3055}, journal = {Duke Mathematical Journal}, month = {November}, title = {On the growth of eigenfunction averages: Microlocalization and geometry}, number = {16}, volume = {168}, year = {2019}, url = {http://doi.org/10.1215/00127094-2019-0020}, author = {Canzani, Y and Galkowski, J}, abstract = {Let (M,g) be a smooth, compact Riemannian manifold, and let \{{\ensuremath{\phi}}\_\{h]\} be an L2 -normalized sequence of Laplace eigenfunctions, -h2{\ensuremath{\Delta}}\_\{g\}{\ensuremath{\phi}}\_\{h\} = {\ensuremath{\phi}}\_\{h\}. Given a smooth submanifold H {$\subset$} M of codimension K {$\ge$} 1, we find conditions on the pair (\{{\ensuremath{\phi}}\_\{h\}, H) for which {\ensuremath{|}}{$\int$}\_\{H\}{\ensuremath{\phi}}\_\{h\} {\ensuremath{\delta}}{\ensuremath{\sigma}}\_\{H\}{\ensuremath{|}} = o (h{$\backslash$}frac\{1-k\}\{2\}), h{$\rightarrow$}0{$^+$}. One such condition is that the set of conormal directions to H that are recurrent has measure 0. In particular, we show that the upper bound holds for any H if (M, g) is a surface with Anosov geodesic flow or a manifold of constant negative curvature. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages.} }