Initial ball from the Gaussian estimate

The box estimate in lem:uc-gaussian and continuity at time zero force the trace to vanish on its output ball.

theorem ESS.uc_initial_ball_vanishing (R T : ℝ) (hR : 0 < R) (hT : 0 < T) (x₀ : CKN.Foundation.Parabolic.Vec3) (w : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (Dw : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3) (D2w : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3) (Dtw : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (hcont : ContinuousOn w (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball x₀ R) (Set.Ico 0 T))) (hweak : CKN.HasSpaceTimeWeakDerivs (CKN.Foundation.Parabolic.vec3Ball x₀ R) (Set.Ioo 0 T) w Dw D2w Dtw) (hL2 : ∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball x₀ R) (Set.Ioo 0 T), ‖w z‖ₑ ^ 2 + ‖Dw z‖ₑ ^ 2 + ‖D2w z‖ₑ ^ 2 + ‖Dtw z‖ₑ ^ 2 < ⊤) (hzero : w (x₀, 0) = 0) (hgauss : ∃ (γ : ℝ) (C : ℝ), 0 < γ ∧ γ < 3 / 16 ∧ 0 < C ∧ ∀ (x : CKN.Foundation.Parabolic.Vec3) (t : ℝ), 0 < t → t ≤ γ * T → CKN.Foundation.Parabolic.vec3EuclideanNorm (x - x₀) ≤ ucRadiusFraction * R → 16 / 100 * t ≤ CKN.Foundation.Parabolic.vec3EuclideanNorm (x - x₀) ^ 2 → ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball x √(2 * t)) (Set.Ioo t (2 * t)), CKN.Foundation.Parabolic.vec3EuclideanNorm (w z) ^ 2 ≤ C * ucLocalEnergy x₀ R T w Dw * Real.exp (-CKN.Foundation.Parabolic.vec3EuclideanNorm (x - x₀) ^ 2 / (2 * t))) (x : CKN.Foundation.Parabolic.Vec3) : x ∈ CKN.Foundation.Parabolic.vec3Ball x₀ (ucRadiusFraction * R) → w (x, 0) = 0

A Gaussian box estimate propagates a zero initial trace from its center to the entire inner output ball (thm:uc).