| import OAI.Geometry.RCD.Heat.CompactHeatMeasure |
| import OAI.Geometry.RCD.Sobolev.BoundedLpSquare |
|
|
| namespace OAI |
|
|
| /-! # Compact continuous approximation preserving a uniform amplitude -/ |
|
|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped NNReal ENNReal Topology CompactlySupported |
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|
| noncomputable section |
| namespace RegularPointCoordinates |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [ProperSpace X] (μ : Measure X) [μ.Regular] |
|
|
| /-- A bounded L² class has uniformly bounded compact continuous approximants. -/ |
| theorem exists_bounded_continuous_L2_approximation (f : Lp ℝ 2 μ) (C : ℝ≥0) |
| (hf : ∀ᵐ x ∂μ, |f x| ≤ C) : |
| ∃ F : ℕ → C_c(X, ℝ), (∀ i x, |F i x| ≤ C) ∧ |
| Tendsto (fun i => compactTestToL2 μ (F i)) atTop (𝓝 f) := by |
| have hex (i : ℕ) : ∃ F : C_c(X, ℝ), |
| ‖compactTestToL2 μ F - f‖ ≤ 1 / ((i : ℝ) + 1) := by |
| let ε := 1 / ((i : ℝ) + 1) |
| have hε : 0 < ε := by dsimp [ε]; positivity |
| obtain ⟨g, hgs, hge, hgc, hgm⟩ := (Lp.memLp f).exists_hasCompactSupport_eLpNorm_sub_le |
| (by norm_num : (2 : ℝ≥0∞) ≠ ∞) ((ENNReal.ofReal_pos.mpr hε).ne') |
| let F : C_c(X, ℝ) := ⟨⟨g, hgc⟩, hgs⟩ |
| have he : (fun x => (f - compactTestToL2 μ F) x) =ᵐ[μ] (fun x => f x - g x) := by |
| filter_upwards [Lp.coeFn_sub f (compactTestToL2 μ F), compactTestToL2_coe μ F] with x hx hy |
| rw [hx] |
| change f x - compactTestToL2 μ F x = f x - g x |
| rw [hy] |
| rfl |
| refine ⟨F, ?_⟩ |
| rw [norm_sub_rev, Lp.norm_def, eLpNorm_congr_ae he] |
| exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hge).trans_eq |
| (ENNReal.toReal_ofReal hε.le) |
| choose G hG using hex |
| have hlim : Tendsto (fun i => compactTestToL2 μ (G i)) atTop (𝓝 f) := by |
| rw [tendsto_iff_norm_sub_tendsto_zero] |
| exact squeeze_zero (fun i => norm_nonneg _) hG tendsto_one_div_add_atTop_nhds_zero_nat |
| let F (i : ℕ) : C_c(X, ℝ) := |
| ⟨⟨fun x => clipReal C (G i x), (clipReal_lipschitz C).continuous.comp (G i).continuous⟩, |
| (G i).hasCompactSupport.comp_left (clipReal_zero C)⟩ |
| let P : Lp ℝ 2 μ → Lp ℝ 2 μ := (clipReal_lipschitz C).compLp (clipReal_zero C) |
| have hPf : P f = f := by |
| apply Lp.ext |
| filter_upwards [(clipReal_lipschitz C).coeFn_compLp (clipReal_zero C) f, hf] with x hx hb |
| exact hx.trans (clipReal_eq_self C hb) |
| have hFG (i : ℕ) : compactTestToL2 μ (F i) = P (compactTestToL2 μ (G i)) := by |
| apply Lp.ext |
| filter_upwards [compactTestToL2_coe μ (F i), compactTestToL2_coe μ (G i), |
| (clipReal_lipschitz C).coeFn_compLp (clipReal_zero C) (compactTestToL2 μ (G i))] |
| with x hx hy hz |
| rw [hx, hz] |
| change clipReal C (G i x) = clipReal C (compactTestToL2 μ (G i) x) |
| rw [hy] |
| refine ⟨F, fun i x => clipReal_abs_le C (G i x), ?_⟩ |
| have hh := ((clipReal_lipschitz C).continuous_compLp (clipReal_zero C)).tendsto f |>.comp hlim |
| change Tendsto (fun i => P (compactTestToL2 μ (G i))) atTop (𝓝 (P f)) at hh |
| rw [hPf] at hh |
| exact hh.congr' (Eventually.of_forall fun i => (hFG i).symm) |
|
|
| end RegularPointCoordinates |
|
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| end |
|
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| end OAI |
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|
