ℓ² Decoupling in ℝ² for Curves with Vanishing Curvature

We expand the class of curves ( φ 1 ( t ) , φ 2 ( t ) ) , t ∈ [ 0 , 1 ] (\varphi _1(t),\varphi _2(t)),\ t\in [0,1] for which the ℓ 2 \ell ^2 decoupling conjecture holds for 2 ≤ p ≤ 6 2\leq p\leq 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 ) (t,t^{1+\nu }) for ν ∈ ( 0 , ∞ ) \nu \in (0,\infty ) .