${l}^2$ Decoupling for certain degenerate surfaces in $\mathbb{R}^4$

TL;DR

This paper establishes decoupling inequalities for a specific degenerate hypersurface in four-dimensional space, extending decoupling theory to new classes of degenerate surfaces inspired by prior foundational work.

Contribution

It introduces novel decoupling estimates for a particular degenerate hypersurface in 4, expanding the scope of decoupling theory to include certain degenerate geometries.

Findings

Proved decoupling inequalities for 4 hypersurface _4.

Extended decoupling methods to degenerate surfaces.

Provided bounds that generalize previous non-degenerate cases.

Abstract

In this article, we aim to study decoupling inequality for a specific degenerate hypersurface in $\mathbb{R}^4$. Inspired by the work of Bourgain--Demeter and Li--Zheng, we consider the hypersurface $\mathcal{S}^3_{4}:=\{(\xi_1,\xi_2,\xi_3,\xi^4_1+\xi^4_2+\xi^4_3):0\leq \xi_j\leq1, \text{for}~j=1,2,3\}$ in $\mathbb{R}^4$ and study decoupling estimates.