ℓ2 Decoupling for Certain Surfaces of Finite Type in ℝ3

In this article, we establish an ℓ2 decoupling inequality for the surface F42:={(ξ1,ξ2,ξ14+ξ24):(ξ1,ξ2)∈[0,1]2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_4^2: = \{ ({\xi _1},{\xi _2},\xi _1^4 + \xi _2^4):({\xi _1},{\xi _2}) \in {[0,1]^2}\} $$\end{document} associated with the decomposition adapted to finite type geometry from our previous work [Li, Z., Miao, C., Zheng, J.: A restriction estimate for a certain surface of finite type in ℝ3. J. Fourier Anal. Appl., 27(4), Paper No. 63, 24 pp. (2021)]. The key ingredients of the proof include the so-called generalized rescaling technique, an ℓ2 decoupling inequality for the surfaces {(ξ1,ξ2,ϕ1(ξ1)+ξ24):(ξ1,ξ2)∈[0,1]2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ ({\xi _1},{\xi _2},{\phi _1}({\xi _1}) + \xi _2^4):({\xi _1},{\xi _2}) \in {[0,1]^2}\} $$\end{document} with ϕ1 being non-degenerate, reduction of dimension arguments and induction on scales.