Title: Multi-scale bilinear restriction estimates for general phases,
Abstract: We prove (adjoint) bilinear restriction estimates for general phases at
different scales in the full non-endpoint mixed norm range, and give bounds
with a sharp and explicit dependence on the phases. These estimates have
applications to high-low frequency interactions for solutions to partial
differential equations, as well as to the linear restriction problem for
surfaces with degenerate curvature. As a consequence, we obtain new bilinear
restriction estimates for elliptic phases and wave/Klein-Gordon interactions in
the full bilinear range, and give a refined Strichartz inequality for the
Klein-Gordon equation. In addition, we extend these bilinear estimates to hold
in adapted function spaces by using a transference type principle which holds
for vector valued waves.