| import OAI.Analysis.RCDHeat.ExistingAnalytic |
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| namespace OAI |
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| /-! # Passing an additive transport error from averages to continuous values -/ |
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| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [SecondCountableTopology X] {μ : Measure X} [IsLocallyFiniteMeasure μ] |
| [IsUnifLocDoublingMeasure μ] [μ.IsOpenPosMeasure] |
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| theorem quadratic_comparison_of_average_bound_add |
| (u v U V : X → ℝ) (hu : LocallyIntegrable u μ) (hv : LocallyIntegrable v μ) |
| (hU : Continuous U) (hV : Continuous V) (heU : u =ᵐ[μ] U) (heV : v =ᵐ[μ] V) |
| (c e : ℝ) (h : ∀ x y : X, ∀ r : ℝ, 0 < r → |
| (⨍z in closedBall x r, u z ∂μ) - (⨍z in closedBall y r, v z ∂μ) ≤ |
| c*(dist x y + 2*r)^2+e) : |
| ∀ x y, U x-V y ≤ c*dist x y^2+e := by |
| let S := {x : X | Tendsto (fun r : ℝ => ⨍z in closedBall x r, u z ∂μ) |
| (𝓝[>] 0) (𝓝 (U x))} |
| let T := {x : X | Tendsto (fun r : ℝ => ⨍z in closedBall x r, v z ∂μ) |
| (𝓝[>] 0) (𝓝 (V x))} |
| have hS : ∀ᵐ x ∂μ, x ∈ S := by |
| filter_upwards [IsUnifLocDoublingMeasure.ae_tendsto_average μ hu (1:ℝ),heU] with x hx he |
| change Tendsto _ _ (𝓝 (U x)) |
| rw [←he] |
| exact hx (fun _ : ℝ => x) id tendsto_id (by |
| filter_upwards [self_mem_nhdsWithin] with r hr |
| simp only [mem_closedBall,dist_self,one_mul] |
| exact le_of_lt hr) |
| have hT : ∀ᵐ x ∂μ, x ∈ T := by |
| filter_upwards [IsUnifLocDoublingMeasure.ae_tendsto_average μ hv (1:ℝ),heV] with x hx he |
| change Tendsto _ _ (𝓝 (V x)) |
| rw [←he] |
| exact hx (fun _ : ℝ => x) id tendsto_id (by |
| filter_upwards [self_mem_nhdsWithin] with r hr |
| simp only [mem_closedBall,dist_self,one_mul] |
| exact le_of_lt hr) |
| have hST : ∀ x ∈ S, ∀ y ∈ T, U x-V y ≤ c*dist x y^2+e := by |
| intro x hx y hy |
| have hc : Tendsto (fun r : ℝ => c*(dist x y+2*r)^2+e) (𝓝[>] 0) |
| (𝓝 (c*dist x y^2+e)) := by |
| have hh : Continuous (fun r : ℝ => c*(dist x y+2*r)^2+e) := by fun_prop |
| simpa using (hh.continuousAt (x := 0)).tendsto.mono_left nhdsWithin_le_nhds |
| have hb : ∀ᶠ r in 𝓝[>] (0:ℝ), |
| (⨍z in closedBall x r, u z ∂μ) - (⨍z in closedBall y r, v z ∂μ) ≤ |
| c*(dist x y+2*r)^2+e := by |
| filter_upwards [self_mem_nhdsWithin] with r hr using h x y r hr |
| exact le_of_tendsto_of_tendsto (hx.sub hy) hc hb |
| have hd : Dense (S ×ˢ T) := (Measure.dense_of_ae hS).prod (Measure.dense_of_ae hT) |
| have hc : IsClosed {z : X×X | U z.1-V z.2 ≤ c*dist z.1 z.2^2+e} := |
| isClosed_le ((hU.comp continuous_fst).sub (hV.comp continuous_snd)) (by fun_prop) |
| have hs : closure (S ×ˢ T) ⊆ {z : X×X | U z.1-V z.2 ≤ c*dist z.1 z.2^2+e} := |
| hc.closure_subset_iff.mpr (fun z hz => hST z.1 hz.1 z.2 hz.2) |
| intro x y |
| exact hs (hd (x,y)) |
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| end RegularPointCoordinates |
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| end OAI |
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