ℓ² decoupling in ℝ² for curves with vanishing curvature

Abstract

We expand the class of curves (ϕ 1 (t), ϕ 2 (t)), t ∈ [0, 1] for which the ℓ 2 decoupling conjecture holds for 2 ≤ p ≤ 6. Our class of curves includes all real-analytic regular curves with isolated points of vanishing curvature and all curves of the form (t, t 1+ν ) for ν ∈ (0, ∞).

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

“…Although a projection forgets certain geometry of the surface, the decoupling of the projected sets, as a subset of R 2 , is easier to study. See [1,7,22].…”

“…Let T ′ and P be as in Step B1, and let |θ| ≤ π/4 and ρ be the counterclockwise rotation by θ. Denote Q = P • ρ. Then we have (1) We have max{Q x , Q y } σ 2 over ρ −1 (T ′ ).…”

“…Yet in R 2 , a smooth hypersurface reduces to a smooth curve, and its decoupling theory is relatively well-understood. As an initiation of this subject, a decoupling inequality for every analytic function defined on a compact interval is proved in [1] . Later in Section 12.6 of [7], Demeter proved a slightly more refined decoupling inequality for every analytic function defined on a compact interval, with the partition chosen to be adapted to the curvature of the function.…”

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

“…Bourgain and Demeter's 2015 breakthrough l 2 -decoupling theorem for the truncated elliptic paraboloid in R n [2] is of substantial importance in harmonic analysis, and has been generalised in various directions since then. In [1], the authors studied the l 2 -decoupling inequality on R 2 for general real analytic phase functions over a compact interval. Following this, Demeter [6] improved upon [1] by choosing partitions of the unit interval that fit the curvature of each analytic phase function.…”

“…In [1], the authors studied the l 2 -decoupling inequality on R 2 for general real analytic phase functions over a compact interval. Following this, Demeter [6] improved upon [1] by choosing partitions of the unit interval that fit the curvature of each analytic phase function. Later, the second author [17] further proved a uniform decoupling inequality for all polynomial phase functions with a given bound on the degree of the polynomials.…”