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We prove (adjoint) bilinear restriction estimates for general phases atdifferent scales in the full non-endpoint mixed norm range, and give boundswith a sharp and explicit dependence on the phases. These estimates haveapplications to high-low frequency interactions for solutions to partialdifferential equations, as well as to the linear restriction problem forsurfaces with degenerate curvature. As a consequence, we obtain new bilinearrestriction estimates for elliptic phases and wave/Klein-Gordon interactions inthe full bilinear range, and give a refined Strichartz inequality for theKlein-Gordon equation. In addition, we extend these bilinear estimates to holdin adapted function spaces by using a transference type principle which holdsfor vector valued waves.

Multi-scale bilinear restriction estimates for general phases