A note on bilinear wave-Schr\"odinger interactions

TL;DR

This paper establishes sharp bilinear restriction estimates for wave-Schrödinger interactions, identifies conditions for their validity, and discusses failures and extensions to adapted function spaces, advancing understanding of wave and Schrödinger equation interactions.

Contribution

It provides a sharp condition for bilinear restriction estimates, constructs a counter-example showing potential failure, and introduces a transference principle to extend estimates to $U^2$ spaces.

Findings

Sharp bilinear restriction condition for wave-Schrödinger interactions.

Counter-example demonstrating failure of estimates due to cone curvature issues.

Extension of estimates to $U^2$ function spaces via transference principle.

Abstract

We consider bilinear restriction estimates for wave-Schr\"odinger interactions and provided a sharp condition to ensure that the product belongs to $L^q_t L^r_x$ in the full bilinear range $\frac{2}{q} + \frac{d+1}{r} < d+1$, $1 \leqslant q, r \leqslant 2$. Moreover, we give a counter-example which shows that the bilinear restriction estimate can fail, even in the transverse setting. This failure is closely related to the lack of curvature of the cone. Finally we mention extensions of these estimates to adapted function spaces. In particular we give a general transference type principle for $U^2$ type spaces that roughly implies that if an estimate holds for homogeneous solutions, then it also holds in $U^2$. This transference argument can be used to obtain bilinear and multilinear estimates in $U^2$ from the corresponding bounds for homogeneous solutions.