File size: 3,157 Bytes
b37e936 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | 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
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
end
end OAI
|