Bilinear Kakeya–Nikodym averages of eigenfunctions on compact Riemannian surfaces

We obtain an improvement of the bilinear estimates of Burq, Gérard and Tzvetkov [6] in the spirit of the refined Kakeya–Nikodym estimates [2] of Blair and the second author. We do this by using microlocal techniques and a bilinear version of Hörmander's oscillatory integral theorem in [7].