Sharp null form estimates on endline geometric conditions of the cone

TL;DR

This paper establishes sharp null form estimates on the endline of the cone geometry for wave solutions, extending previous endpoint results to mixed norms and achieving global results for variable coefficient equations.

Contribution

It extends endpoint null form estimates to mixed norms, improves local results to global, and introduces a new uniform bilinear restriction estimate approach.

Findings

Extended endpoint null form estimates to mixed norms.

Achieved global in time estimates for variable coefficient wave equations.

Developed a new induction on scale method for uniform bilinear restriction estimates.

Abstract

We prove $\mathcal{H}^{\alpha_1}\times\mathcal{H}^{\alpha_2}\to L^q_tL^r_x$ null form estimates for solutions to homogeneous wave equations with $(q,r)$ on the endline of the condition concerning geometry of the cone, except the critical index. This extends the previous endpoint result of Tao, Math. Z. 238, no. 2, 215-268, (2001) in symmetric norms to mixed norms and improves the local in time result of Tataru, MR1979953, to be global in the setting of constant variable coefficient equations, as well as the sharp off-endline estimates established by Lee and Vargas, Amer. J. Math 130 (2008), no. 5, 1279-1326, to the borderline with respect to the cone condition. Our proof is based on the multiplier theory in mixed norms, which ultimately reduces the question to a uniform endline bilinear restriction estimates including high-low frequency interactions for a family of conic type surfaces depending on a parameter $\sigma$, which converges to the oblique cone as $\sigma\to 0$. We prove this uniform estimate by using the enhanced version of the induction on scale method.