A non-concentration estimate for partially rectangular billiards

We consider quasimodes on planar domains with a partially rectangular boundary. We prove that for any $epsilon_0>0$, an $O(lambda^{-epsilon_0})$ quasimode must have $L^2$ mass in the wings bounded below by $lambda^{-2-delta}$ for any $delta>0$. The proof uses the authors recent work on 0-Gevrey smooth domains to approximate quasimodes on $C^{1,1}$ domains. There is an improvement for $C^{2,alpha}$ domains.