File size: 4,032 Bytes
2fca478 | 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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | import OAI.Geometry.Perimeter.Regularity.HeightCompactness
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped Topology ENNReal
namespace RPISingularHarnack
open RPI.Boundary
variable {n : ℕ}
def compactLpExtension (K : Set (Euc n))
(g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) : Euc n → ℝ :=
Function.extend (Subtype.val : K → Euc n) g (fun _ ↦ 0)
@[simp] theorem compactLpExtension_coe (K : Set (Euc n))
(g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) (x : K) :
compactLpExtension K g x = g x :=
Subtype.val_injective.extend_apply _ _ x
theorem compactLpExtension_zero (K : Set (Euc n))
(g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) {x : Euc n} (hx : x ∉ K) :
compactLpExtension K g x = 0 := by
apply Function.extend_apply'
rintro ⟨y,rfl⟩; exact hx y.property
theorem compactLpExtension_integrable (K : Set (Euc n)) (hK : IsCompact K)
(g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) :
Integrable (compactLpExtension K g) volume := by
have hi : IntegrableOn (compactLpExtension K g) K volume := by
rw [integrableOn_iff_comap_subtypeVal hK.measurableSet]
simpa only [Function.comp_def,compactLpExtension_coe] using
(memLp_one_iff_integrable.mp (Lp.memLp g))
exact hi.integrable_of_forall_notMem_eq_zero (fun _ hx ↦ compactLpExtension_zero K g hx)
theorem compactLpExtension_error (K : Set (Euc n)) (hK : IsCompact K)
(g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n)))
(f : Euc n → ℝ) (hi : Integrable f volume) (hs : Function.support f ⊆ K) :
(∫⁻ x, ‖f x-compactLpExtension K g x‖ₑ) = edist (compactFunctionLp K hK f hi) g := by
rw [Lp.edist_def,eLpNorm_one_eq_lintegral_enorm
((Lp.aestronglyMeasurable (compactFunctionLp K hK f hi)).sub (Lp.aestronglyMeasurable g))]
have hz : Function.support (fun x ↦ ‖f x-compactLpExtension K g x‖ₑ) ⊆ K := by
intro x hx
by_contra hnot
have hf : f x=0 := Function.notMem_support.mp (fun h ↦ hnot (hs h))
simp [hf,compactLpExtension_zero K g hnot] at hx
rw [← setLIntegral_eq_of_support_subset hz,← lintegral_subtype_comap hK.measurableSet]
apply lintegral_congr_ae
filter_upwards [compactFunctionLp_ae K hK f hi] with x hx
simp only [compactLpExtension_coe,Pi.sub_apply,hx]
theorem compactLpExtension_tendsto (K : Set (Euc n)) (hK : IsCompact K)
(f : ℕ → Euc n → ℝ) (hi : ∀ i, Integrable (f i) volume)
(hs : ∀ i, Function.support (f i) ⊆ K)
(g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n)))
(hlim : Tendsto (fun i ↦ compactFunctionLp K hK (f i) (hi i)) atTop (𝓝 g)) :
Tendsto (fun i ↦ ∫⁻ x, ‖f i x-compactLpExtension K g x‖ₑ) atTop (𝓝 0) := by
have he (i : ℕ) := compactLpExtension_error K hK g (f i) (hi i) (hs i)
simp_rw [he]
simpa only [edist_self] using hlim.edist (tendsto_const_nhds : Tendsto (fun _ : ℕ ↦ g) atTop (𝓝 g))
theorem compact_support_L1_subsequence
(K : Set (Euc n)) (hK : IsCompact K) (f : ℕ → Euc n → ℝ)
(hm : ∀ i, Measurable (f i)) (hi : ∀ i, Integrable (f i) volume)
(hs : ∀ i, Function.support (f i) ⊆ K)
(A : ℝ) (hA : 0 ≤ A) (hb : ∀ i, (∫ x, ‖f i x‖) ≤ A)
(D : ℝ≥0∞) (hD : D ≠ ⊤)
(ht : ∀ i w, (∫⁻ x, ‖f i (x+w)-f i x‖ₑ) ≤ ‖w‖ₑ * D) :
∃ g : Euc n → ℝ, Integrable g volume ∧ Function.support g ⊆ K ∧
∃ σ : ℕ → ℕ, StrictMono σ ∧
Tendsto (fun i ↦ ∫⁻ x, ‖f (σ i) x-g x‖ₑ) atTop (𝓝 0) := by
obtain ⟨g,σ,hσ,hg⟩ := compactFunctionLp_subsequence_of_translation K hK f hm hi A hA hb D hD ht
refine ⟨compactLpExtension K g,compactLpExtension_integrable K hK g,?_,σ,hσ,?_⟩
· intro x hx
by_contra hnot
exact hx (compactLpExtension_zero K g hnot)
· exact compactLpExtension_tendsto K hK (fun i ↦ f (σ i)) (fun i ↦ hi (σ i))
(fun i ↦ hs (σ i)) g hg
end RPISingularHarnack
end
end OAI
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