${\ell }^2$ decoupling for certain surfaces of finite type in ${\mathbb{R}}^3$

Abstract

In this article, we establish an ${\ell }^2$ decoupling inequality for the surface $$F_4^2:=\{({\xi }_1,{\xi }_2,{\xi }_1^4+{\xi }_2^4):({\xi }_1,{\xi }_2)\in [0,1{]}^2\}$$ associated with the decomposition adapted to finite type geometry from our previous work. The key ingredients of the proof include the so-called generalized rescaling technique, an ${\ell }^2$ decoupling inequality for the surfaces $$\{({\xi }_1,{\xi }_2,{\varphi }_1({\xi }_1)+{\xi }_2^4):({\xi }_1,{\xi }_2)\in [0,1{]}^2\}$$ with ${\varphi }_1$ being non-degenerate, reduction of dimension arguments and induction on scales.

17 pages, comments are welcome