Decoupling for mixed-homogeneous polynomials in $\mathbb R^3$

Abstract

We prove decoupling inequalities for mixed-homogeneous bivariate polynomials, which partially answers a conjecture of Bourgain, Demeter and Kemp.

Decoupling for smooth surfaces in $\mathbb{R}^3$

“…In the cases considered in [13] and [15], {| det D 2 φ| < δ} is a neighborhood of the graph of some polynomial. Both papers used projections to decouple such sets in the setting of R 2 .…”

“…Later Kemp [14] proved decoupling inequalities for surfaces with constantly zero Gaussian curvature but without umbilical points, and in a recent work [13] he is also able to prove decoupling inequalities for a broad class of C 5 surfaces in R 3 lacking planar points. Also recently, the authors proved in [15] a decoupling inequality for every mixed-homogeneous polynomial in R 3 . Note that none of the previous partial results implies one another.…”

“…For some elementary properties of δ-flatness, the reader may refer to the appendix of [15]. It is worth noting that δ-flat sets are also considered in [5,17].…”