Sharp bounds for multilinear curved Kakeya, restriction and oscillatory integral estimates away from the endpoint

TL;DR

This paper improves multilinear Kakeya, restriction, and oscillatory integral estimates by removing losses away from the endpoint using an advanced heat flow method, achieving sharper bounds without curvature assumptions.

Contribution

It develops a refined heat flow approach to eliminate losses in multilinear estimates away from the endpoint, extending results to curved Kakeya and restriction settings without curvature conditions.

Findings

Eliminated scale-dependent losses in multilinear estimates

Established global restriction estimates away from the endpoint

Extended methods to curved Kakeya and restriction problems

Abstract

We revisit the multilinear Kakeya, curved Kakeya, restriction, and oscillatory integral estimates that were obtained in paper of Bennett, Carbery, and the author using a heat flow monotonicity method applied to a fractional Cartesian product, together with induction on scales arguments. Many of these estimates contained losses of the form $R^\varepsilon$ (or $\log^{O(1)} R$) for some scale factor $R$. By further developing the heat flow method, and applying it directly for the first time to the multilinear curved Kakeya and restriction settings, we are able to eliminate these losses, as long as the exponent $p$ stays away from the endpoint. In particular, we establish global multilinear restriction estimates away from the endpoint, without any curvature hypotheses on the hypersurfaces.