Decoupling Inequality for Paraboloid Under Shell Type Restriction and Its Application to the Periodic Zakharov System

Abstract

AbstractIn this paper, we establish local well-posedness for the Zakharov system on $${\mathbb {T}}^d$$ T d , $$d\ge 3$$ d ≥ 3 in a low regularity setting. Our result improves the work of Kishimoto (J Anal Math 119:213–253, 2013). Moreover, the result is sharp up to $$\varepsilon $$ ε -loss of regularity when $$d=3$$ d = 3 and $$d\ge 5$$ d ≥ 5 as long as one utilizes the iteration argument. We introduce ideas from recent developments of the Fourier restriction theory. The key element in the proof of our well-posedness result is a new trilinear discrete Fourier restriction estimate involving paraboloid and cone. We prove this trilinear estimate by improving Bourgain–Demeter’s range of exponent for the linear decoupling inequality for paraboloid (Bourgain and Demeter in Ann Math 182:351–389) under the constraint that the input space-time function f satisfies $$\textrm{supp}\, {\hat{f}} \subset \{ (\xi ,\tau ) \in {\mathbb {R}}^{d+1}: 1- \frac{1}{N} \le |\xi | \le 1 + \frac{1}{N},\; |\tau - |\xi |^2| \le \frac{1}{N^{2}} \} $$ supp f ^ ⊂ { ( ξ , τ ) ∈ R d + 1 : 1 - 1 N ≤ | ξ | ≤ 1 + 1 N , | τ - | ξ | 2 | ≤ 1 N 2 } for large $$N\ge 1$$ N ≥ 1 .