Decoupling for finite type phases in higher dimensions

Abstract

In this paper, we establish an ${\ell }^2$ decoupling inequality for the hypersurface \[\Big\{(ξ_1,...,ξ_{n-1},ξ_1^m+...+ξ_{n-1}^m): (ξ_1,...,ξ_{n-1}) \in [0,1]^{n-1}\Big\}\]associated with the decomposition adapted to hypersufaces of finite type, where $n\ge 2$ and $m\ge 4$ is an even number. The key ingredients of the proof include an ${\ell }^2$ decoupling inequality for the hypersurfaces \[\Big\{(ξ_1,...,ξ_{n-1},ϕ_1(ξ_1)+...+ϕ_s(ξ_s)+ξ_{s+1}^m+...+ξ_{n-1}^m): (ξ_1,...,ξ_{n-1}) \in [0,1]^{n-1}\Big\},\] $0\le s\le n-1$, with $\ddot{I}{•}_1,...,\ddot{I}{•}_s$ being $m$-nondegenerate.

11 pages