| import OAI.Analysis.AncientFlat.Evolution.DilationCore |
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|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped Topology ZeroAtInfty BoundedContinuousFunction |
| namespace ParametrixDilation |
| variable {V : Type} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| lemma dist_dil_le (t : ℝ) (f g : C₀(V,ℂ)) : dist (dil t f) (dil t g)≤dist f g := by |
| let : NormedAddCommGroup C₀(V,ℂ) := inferInstance |
| let : NormedSpace ℂ C₀(V,ℂ) := inferInstance |
| rw [dist_eq_norm,dist_eq_norm,←map_sub] |
| exact norm_dil_le t (f-g) |
| lemma tendsto_dil_apply {ι : Type*} {l : Filter ι} {r : ι → ℝ} {u : ι → C₀(V,ℂ)} |
| {r₀ : ℝ} {u₀ : C₀(V,ℂ)} (hr : Tendsto r l (𝓝 r₀)) (hu : Tendsto u l (𝓝 u₀)) : |
| Tendsto (fun i => dil (r i) (u i)) l (𝓝 (dil r₀ u₀)) := by |
| apply tendsto_iff_dist_tendsto_zero.mpr |
| have h₁ := tendsto_iff_dist_tendsto_zero.mp hu |
| have h₂ := tendsto_iff_dist_tendsto_zero.mp (((continuous_dil u₀).tendsto r₀).comp hr) |
| apply squeeze_zero (fun i => dist_nonneg) (fun i => ?_) (by simpa using h₁.add h₂) |
| exact (dist_triangle _ (dil (r i) u₀) _).trans |
| (add_le_add (dist_dil_le (r i) (u i) u₀) le_rfl) |
| lemma continuousOn_dil_apply {u : ℝ → C₀(V,ℂ)} {I : Set ℝ} |
| (hu : ContinuousOn u I) : ContinuousOn (fun t => dil t (u t)) I := by |
| intro t ht |
| exact tendsto_dil_apply continuousWithinAt_id (hu t ht) |
| end ParametrixDilation |
|
|
| end |
|
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| end OAI |
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