A sharp square function estimate for the cone in $\mathbb{R}^3$
Larry Guth, Hong Wang, Ruixiang Zhang
Abstract
We prove a sharp square function estimate for the cone in R 3 and consequently the local smoothing conjecture for the wave equation in 2 + 1 dimensions.2 To be more specific, Sogge originally made the conjecture for α in the range α > 1 2 -2 p and Wolff confirmed Sogge's conjecture for p ≥ 74 and α in this range.Later in the work [15] of Heo, Nazarov and Seeger it was conjectured further that when p > 4 the conjecture should hold for α ≥ 1 2 -2 p . 3 Such kind of "locally constant" heuristic will be used a few times in the current paper.To justify this intuition one can use Corollary 4.3 in [3].See also Lemma 6.1 and Lemma 6.2 in Section 6 of the current paper.4 This definition works best if τ is honestly tiled by θ.In general we abuse the notation a bit: Throughout this paper, by writing "summing over θ ⊂ τ ", we really mean "summing over all θ ∈ A(τ )" where the collection A(τ ) is chosen as follows: Each A(τ ) only contains those θ's who intersect τ , and all A(τ ) form a disjoint union {θ} = τ A(τ ).