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

Jianhui Li, Tongou Yang

Abstract

For each $d\geq 0$, we prove decoupling inequalities in $\mathbb R^3$ for the graphs of all bivariate polynomials of degree at most $d$ with bounded coefficients, with the decoupling constant depending uniformly in $d$ but not the coefficients of each individual polynomial. As a consequence, we prove a decoupling inequality for (a compact piece of) every smooth surface in $\mathbb{R}^3$, which in particular solves a conjecture of Bourgain, Demeter and Kemp.

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

Abstract

For each , we prove decoupling inequalities in for the graphs of all bivariate polynomials of degree at most with bounded coefficients, with the decoupling constant depending uniformly in but not the coefficients of each individual polynomial. As a consequence, we prove a decoupling inequality for (a compact piece of) every smooth surface in , which in particular solves a conjecture of Bourgain, Demeter and Kemp.

Paper Structure

This paper contains 58 sections, 24 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Main idea of proof
  5. Passing to polynomials and projecting to $\mathbb{R}^2$.
  6. Generalised 2D uniform decoupling.
  7. Small Hessian determinant and cylindrical decoupling.
  8. Induction on scales
  9. Open problems
  10. Notation.
  11. Proof of Conjecture
  12. $\varepsilon$-dependent partitions
  13. Removal of $\varepsilon$-dependence of partitions
  14. Theorem \ref{['thm_main_uniform_decoupling_eps']} implies Theorem \ref{['thm_conj_eps']}
  15. Ingredients of the proof of Theorem \ref{['thm_main_uniform_decoupling_eps']}
  16. ...and 43 more sections

Key Result

Theorem 1.3

Let $\phi:[-1,1]^2\to \mathbb{R}$ be a smooth function. Then for every $0<\delta<1$, there is a family $\mathcal{P}_\delta=\mathcal{P}_\delta(\phi)$ of rectangles $T$ covering $[-1,1]^2$, such that the following statements hold:

Figures (3)

  • Figure 1: Illustration of the structure proposition
  • Figure 2: The case $m \leq b$
  • Figure 3: The case $m \geq b$

Theorems & Definitions (48)

  • Definition 1.1: $\delta$-flatness
  • Conjecture 1.2: Bourgain-Demeter-Kemp BDK2019, Conjecture 7.4
  • Theorem 1.3: Solution to Conjecture \ref{['conj_BDK']}
  • Theorem 1.4: Uniform decoupling theorem
  • Theorem 1.5: Bourgain-Demeter, BD2015BD2017
  • Theorem 2.1: Theorem \ref{['thm_conj']} with $\varepsilon$-dependent partitions
  • Theorem 2.2: Theorem \ref{['thm_main_uniform_decoupling']} with $\varepsilon$-dependent partitions
  • proof
  • proof
  • Theorem 3.1
  • ...and 38 more