%X Let (M,g) be a smooth, compact Riemannian manifold, and let {φ_{h]} be an L² -normalized sequence of Laplace eigenfunctions, -h²Δ_{g}φ_{h} = φ_{h}. Given a smooth submanifold H ⊂ M of codimension K ≥ 1, we find conditions on the pair ({φ_{h}, H) for which |∫_{H}φ_{h} δσ_{H}| = o (h\frac{1-k}{2}), h→0âº. 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. %L discovery10087655 %I DUKE UNIV PRESS %O This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. %A Y Canzani %A J Galkowski %J Duke Mathematical Journal %V 168 %P 2991-3055 %T On the growth of eigenfunction averages: Microlocalization and geometry %N 16 %D 2019