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

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 ${\mathcal{S}}_4^3:=\{({\xi }_1,{\xi }_2,{\xi }_3,{\xi }_1^4+{\xi }_2^4+{\xi }_3^4):0\le {\xi }_j\le 1,\text{for}j=1,2,3\}$ in ${\mathbb{R}}^4$ and study decoupling estimates.

15 pages. The author is thankful to Zhuoran Li for bringing into the author's knowledge that the main result of this article is closely related to the paper of C. Gao et. al. (arXiv: 2202.11326) after the submission of this article in arXiv