| import OAI.Analysis.AncientFlat.Weighted.WeightedCharts | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff ENNReal | |
| namespace FlatWeighted | |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] | |
| def finiteLocalDomain (T a : ℝ) (p : V × ℝ) : Set (V × ℝ) := | |
| {q | ‖q.1‖<a ∧ -a^4<q.2 ∧ q.2<a^4 ∧ -T≤(chart p q).2 ∧ (chart p q).2<0} | |
| def finiteYNorm (α w a T : ℝ) (f : V × ℝ → ℂ) : ℝ≥0∞ := | |
| ⨆ p : {p : V × ℝ // -T≤p.2 ∧ p.2<0}, | |
| holderNorm α (finiteLocalDomain T a p.val) (pullback (w-4) p.val f) | |
| def innerExtension (ψ : ℝ → ℝ) (T : ℝ) (f : V × ℝ → ℂ) (p : V × ℝ) : ℂ := | |
| if -T≤p.2 then f p else | |
| ψ ((-p.2-T)/(‖p.1‖^2+Real.sqrt T)^2) • f (p.1,-T) | |
| def ExtensionCutoff (ψ : ℝ → ℝ) : Prop := | |
| ContDiff ℝ ∞ ψ ∧ (∀ s : ℝ, 0 ≤ s → 0≤ψ s ∧ ψ s≤1) ∧ | |
| (∃ ε : ℝ, 0<ε ∧ ∀ s ∈ Icc 0 ε, ψ s=1) ∧ (∀ s : ℝ, 1 ≤ s → ψ s=0) | |
| end FlatWeighted | |
| end | |
| end OAI | |