| import OAI.Analysis.AncientFlat.Regularity.HolderContinuity |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter Topology Metric |
| open scoped ContDiff NNReal |
| namespace ParabolicLocalization |
| open ParabolicInterior |
| variable {V : Type} {E : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [NormedAddCommGroup E] [NormedSpace ℝ E] |
| omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] in |
| lemma norm_pair_sub_le_pdist (p q : V × ℝ) (hd : pdist p q≤1) : ‖p-q‖≤pdist p q := by |
| have ht : |p.2-q.2|^((1:ℝ)/4)≤1 := (le_add_of_nonneg_left (norm_nonneg _)).trans hd |
| have htr : |p.2-q.2|≤1 := by |
| by_contra h |
| have hh := Real.one_lt_rpow (lt_of_not_ge h) (by norm_num : (0:ℝ)<1/4) |
| linarith |
| have htime := Real.self_le_rpow_of_le_one (abs_nonneg (p.2-q.2)) htr (by norm_num : (1:ℝ)/4≤1) |
| rw [Prod.norm_def] |
| exact max_le (le_add_of_nonneg_right (Real.rpow_nonneg (abs_nonneg _) _)) |
| (by simpa only [Real.norm_eq_abs, Prod.snd_sub, pdist] using htime.trans (le_add_of_nonneg_left (norm_nonneg (p.1-q.1)))) |
| omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [NormedSpace ℝ E] in |
| lemma holder_of_bounded_lipschitz (α B : ℝ) (hα : 0<α) (hα1 : α≤1) (hB : 0≤B) |
| (L : ℝ≥0) (f : V × ℝ → E) (hf : LipschitzWith L f) |
| (hb : ∀ p, ‖f p‖≤B) : HolderBound α univ f (2*B+L) := by |
| refine ⟨fun p _ => (hb p).trans (by linarith [L.coe_nonneg] : B≤2*B+(L:ℝ)),?_⟩ |
| intro p _ q _ |
| have hd : 0≤pdist p q := pdist_nonneg p q |
| by_cases h : pdist p q≤1 |
| · have hpow := Real.self_le_rpow_of_le_one hd h hα1 |
| have hl := hf.norm_sub_le p q |
| calc |
| _≤(L:ℝ)*‖p-q‖ := hl |
| _≤(L:ℝ)*pdist p q := mul_le_mul_of_nonneg_left (norm_pair_sub_le_pdist p q h) L.coe_nonneg |
| _≤(L:ℝ)*pdist p q^α := mul_le_mul_of_nonneg_left hpow L.coe_nonneg |
| _≤_ := mul_le_mul_of_nonneg_right (by linarith) (Real.rpow_nonneg hd _) |
| · have hpow : 1≤pdist p q^α := Real.one_le_rpow (le_of_not_ge h) hα.le |
| have hb' := (norm_sub_le (f p) (f q)).trans (add_le_add (hb p) (hb q)) |
| nlinarith [mul_nonneg L.coe_nonneg (Real.rpow_nonneg hd α),mul_nonneg hB (sub_nonneg.mpr hpow)] |
| lemma compact_lipschitz {W G : Type*} [NormedAddCommGroup W] [NormedSpace ℝ W] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] (f : W → G) |
| (hf : ContDiff ℝ 1 f) (hs : HasCompactSupport f) : |
| ∃ L : ℝ≥0, LipschitzWith L f := by |
| obtain ⟨B,hB⟩ := (hs.fderiv ℝ).exists_bound_of_continuous (hf.continuous_fderiv (by norm_num)) |
| let L : ℝ≥0 := ⟨max B 0,le_max_right _ _⟩ |
| refine ⟨L,lipschitzWith_of_nnnorm_fderiv_le (𝕜 := ℝ) (hf.differentiable (by norm_num)) ?_⟩ |
| intro p |
| change ‖fderiv ℝ f p‖₊≤L |
| exact_mod_cast (hB p).trans (le_max_left B 0) |
| |
| omit [FiniteDimensional ℝ V] in |
| lemma compact_contDiff_holder (α : ℝ) (hα : 0<α) (hα1 : α≤1) |
| (f : V × ℝ → E) (hf : ContDiff ℝ 1 f) (hs : HasCompactSupport f) : |
| ∃ C : ℝ, 0<C ∧ HolderBound α univ f C := by |
| obtain ⟨B,hB⟩ := hs.exists_bound_of_continuous hf.continuous |
| obtain ⟨L,hL⟩ := compact_lipschitz f hf hs |
| have hh := holder_of_bounded_lipschitz α (max B 0) hα hα1 (le_max_right _ _) L f hL |
| (fun p => (hB p).trans (le_max_left _ _)) |
| refine ⟨2*max B 0+L+1,by positivity,hh.mono (by linarith)⟩ |
| end ParabolicLocalization |
| |
| end |
| |
| end OAI |
| |
