| import OAI.Analysis.AncientFlat.Regularity.CompactHolder |
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| namespace OAI |
| noncomputable section |
| open Set Filter |
| namespace ParabolicInterior |
| open ParabolicLocalization |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
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| omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [NormedSpace ℝ E] in |
| lemma compact_local_holder (α : ℝ) (hα : 0≤α) (K : Set (V × ℝ)) (hK : IsCompact K) |
| (f : V × ℝ → E) |
| (hf : ∀ p∈K, ∃ U : Set (V × ℝ), IsOpen U ∧ p∈U ∧ |
| ∃ C : ℝ, 0≤C ∧ HolderBound α U f C) : |
| ∃ C : ℝ, 0≤C ∧ HolderBound α K f C := by |
| classical |
| choose U hU hp C hC hB using fun p : K => hf p p.property |
| have hcover : K⊆⋃ p : K, U p := by |
| intro p hp' |
| exact mem_iUnion.mpr ⟨⟨p,hp'⟩,hp ⟨p,hp'⟩⟩ |
| obtain ⟨s,hs⟩ := hK.elim_finite_subcover U hU hcover |
| let W : {p : K // p∈s} → Set (V × ℝ) := fun p => U p.val |
| have hW : K⊆⋃ p, W p := by |
| intro p hp' |
| obtain ⟨q,hq⟩ := mem_iUnion.mp (hs hp') |
| obtain ⟨hq',hpq⟩ := mem_iUnion.mp hq |
| exact mem_iUnion.mpr ⟨⟨q,hq'⟩,hpq⟩ |
| obtain ⟨δ,hδ,hδcover⟩ := lebesgue_number_lemma_of_metric hK (fun p : {p : K // p∈s} => hU p.val) hW |
| let B := ∑ p∈s, C p |
| have hB0 : 0≤B := Finset.sum_nonneg fun p _ => hC p |
| have hCB (p : K) (hp' : p∈s) : C p≤B := Finset.single_le_sum (fun q _ => hC q) hp' |
| have hb (p) (hp' : p∈K) : ‖f p‖≤B := by |
| obtain ⟨q,hq⟩ := mem_iUnion.mp (hW hp') |
| exact ((hB q.val).1 p hq).trans (hCB q.val q.property) |
| let d := min δ 1 |
| have hd : 0<d := lt_min hδ zero_lt_one |
| have hdp : 0<d^α := Real.rpow_pos_of_pos hd _ |
| let A := B+2*B/d^α |
| have hA : 0≤A := by dsimp [A]; positivity |
| refine ⟨A,hA,fun p hp' => (hb p hp').trans (by dsimp [A]; exact le_add_of_nonneg_right (div_nonneg (by positivity) hdp.le)),?_⟩ |
| intro p hp' q hq' |
| by_cases hn : pdist p q<d |
| · obtain ⟨i,hi⟩ := hδcover p hp' |
| have hpU : p∈W i := hi (Metric.mem_ball_self hδ) |
| have hqU : q∈W i := hi (by |
| rw [Metric.mem_ball,dist_eq_norm, norm_sub_rev] |
| exact (norm_pair_sub_le_pdist p q (hn.le.trans (min_le_right _ _))).trans_lt |
| (hn.trans_le (min_le_left _ _))) |
| exact ((hB i.val).2 p hpU q hqU).trans |
| (mul_le_mul_of_nonneg_right ((hCB i.val i.property).trans (by dsimp [A]; exact le_add_of_nonneg_right (div_nonneg (by positivity) hdp.le))) |
| (Real.rpow_nonneg (pdist_nonneg _ _) _)) |
| · have hdq : d^α≤pdist p q^α := Real.rpow_le_rpow hd.le (not_lt.mp hn) hα |
| have hfar : 2*B≤(2*B/d^α)*pdist p q^α := by |
| calc |
| _ = (2*B/d^α)*d^α := (div_mul_cancel₀ _ hdp.ne').symm |
| _ ≤ _ := mul_le_mul_of_nonneg_left hdq (by positivity) |
| exact ((norm_sub_le _ _).trans (by linarith [hb p hp',hb q hq'])).trans |
| (hfar.trans (mul_le_mul_of_nonneg_right (by dsimp [A]; linarith) (Real.rpow_nonneg (pdist_nonneg _ _) _))) |
| end ParabolicInterior |
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| end |
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| end OAI |
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