| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Compactness.Ascoli |
| import OAI.Geometry.CAT0Fillings.Compactness.Atlas |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal |
|
|
| namespace CAT0Fillings.JointBV |
| variable {α : Type*} [MeasurableSpace α] {μ : Measure α} {f : α → ℝ} |
|
|
| lemma lintegral_enorm_rpow_eq (hf : AEStronglyMeasurable f μ) {p : ℝ} (hp : 0 < p) : |
| (∫⁻ x, ‖f x‖ₑ^p ∂μ) = eLpNorm f (ENNReal.ofReal p) μ ^ p := by |
| rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by positivity) ENNReal.ofReal_ne_top hf, |
| ENNReal.toReal_ofReal hp.le,←ENNReal.rpow_mul] |
| simp [hp.ne'] |
|
|
| lemma eLpNorm_two_interpolation (hf : AEStronglyMeasurable f μ) {p : ℝ} (hp : 2 < p) : |
| eLpNorm f 2 μ ^ (2:ℝ) ≤ eLpNorm f 1 μ ^ ((p-2)/(p-1)) * |
| eLpNorm f (ENNReal.ofReal p) μ ^ (p/(p-1)) := by |
| have hp1 : 0 < p-1 := by linarith |
| let a := (p-2)/(p-1) |
| let b := 1/(p-1) |
| have ha : 0 ≤ a := div_nonneg (by linarith) hp1.le |
| have hb : 0 ≤ b := div_nonneg zero_le_one hp1.le |
| have hab : a+b=1 := by dsimp [a,b]; field_simp; ring |
| have hsum : a+p*b=2 := by dsimp [a,b]; field_simp; ring |
| have hh := ENNReal.lintegral_mul_norm_pow_le hf.aemeasurable.enorm |
| (hf.aemeasurable.enorm.pow_const p) ha hb hab |
| have he : (fun x => ‖f x‖ₑ^a*(‖f x‖ₑ^p)^b) = fun x => ‖f x‖ₑ^(2:ℝ) := by |
| ext x |
| rw [←ENNReal.rpow_mul,←ENNReal.rpow_add_of_nonneg a (p*b) ha (mul_nonneg (by linarith) hb),hsum] |
| rw [he,lintegral_enorm_rpow_eq hf (by norm_num : (0:ℝ)<2), |
| lintegral_enorm_rpow_eq hf (by linarith : 0<p),←eLpNorm_one_eq_lintegral_enorm hf, |
| ←ENNReal.rpow_mul] at hh |
| simpa only [ENNReal.ofReal_ofNat,a,b,mul_one_div] using hh |
|
|
| lemma lpNorm_two_sq_interpolation (h1 : MemLp f 1 μ) {p : ℝ} (hp : 2 < p) |
| (hpMem : MemLp f (ENNReal.ofReal p) μ) : |
| lpNorm f 2 μ ^ 2 ≤ lpNorm f 1 μ ^ ((p-2)/(p-1)) * |
| lpNorm f (ENNReal.ofReal p) μ ^ (p/(p-1)) := by |
| have hh := eLpNorm_two_interpolation h1.aestronglyMeasurable hp |
| have hp1 : 0 < p-1 := by linarith |
| have hp2 : 0 < p-2 := by linarith |
| have hp0 : 0 < p := by linarith |
| have hr : eLpNorm f 1 μ ^ ((p-2)/(p-1)) * eLpNorm f (ENNReal.ofReal p) μ ^ (p/(p-1)) ≠ ∞ := by |
| apply ENNReal.mul_ne_top |
| · apply ENNReal.rpow_ne_top_of_nonneg (by positivity) h1.eLpNorm_ne_top |
| · apply ENNReal.rpow_ne_top_of_nonneg (by positivity) hpMem.eLpNorm_ne_top |
| have hh' := ENNReal.toReal_mono hr hh |
| simpa only [ENNReal.toReal_mul,←ENNReal.toReal_rpow,toReal_eLpNorm, |
| Real.rpow_two] using hh' |
|
|
| end CAT0Fillings.JointBV |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| open JointBV Foundations |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| omit [Nonempty X] in |
| lemma dist_value [Nonempty X] {u v : X → ℝ} {K J : ℝ≥0} |
| (hu : LipschitzWith K u) (hv : LipschitzWith J v) : |
| dist (value (hT := hT) hu) (value hv) = |
| lpNorm (fun x => u x-v x) 2 (MassMeasure.currentMassMeasure hT) := by |
| rw [dist_eq_norm,value,value,←MemLp.toLp_sub ((boundedLip_of_lipschitz hu).memLp 2) |
| ((boundedLip_of_lipschitz hv).memLp 2),Lp.norm_toLp,toReal_eLpNorm] |
| rfl |
|
|
| lemma totallyBounded_value_of_lpBound {ι : Type*} {p : ℝ} (hp : 2 < p) |
| (u : ι → X → ℝ) (L : ι → ℝ≥0) (hu : ∀ j, LipschitzWith (L j) (u j)) |
| (h1 : TotallyBounded (range fun j => value1 (hT := hT) (hu j))) |
| (C : ℝ≥0) (hC : ∀ j, lpNorm (u j) (ENNReal.ofReal p) (MassMeasure.currentMassMeasure hT) ≤ C) : |
| TotallyBounded (range fun j => value (hT := hT) (hu j)) := by |
| let μ := MassMeasure.currentMassMeasure hT |
| let a := (p-2)/(p-1) |
| let b := p/(p-1) |
| have ha : 0 < a := div_pos (by linarith) (by linarith) |
| have hb : 0 ≤ b := div_nonneg (by linarith) (by linarith) |
| have hp' : 1 ≤ ENNReal.ofReal p := by rw [←ENNReal.ofReal_one]; exact ENNReal.ofReal_le_ofReal (by linarith) |
| apply totallyBounded_range_of_control h1 |
| intro ε hε |
| have ht : Tendsto (fun t : ℝ => t^a*(2*(C:ℝ))^b) (𝓝 0) (𝓝 0) := by |
| simpa only [id_eq,Real.zero_rpow ha.ne',zero_mul] using |
| ((continuous_id.tendsto (0:ℝ)).rpow_const (Or.inr ha.le)).mul_const ((2*(C:ℝ))^b) |
| obtain ⟨δ,hδ,hd⟩ := Metric.mem_nhds_iff.mp (ht (Iio_mem_nhds (sq_pos_of_pos hε))) |
| refine ⟨δ,hδ,?_⟩ |
| intro i j hij |
| have hmem1 : MemLp (fun x => u i x-u j x) 1 μ := |
| ((boundedLip_of_lipschitz (hu i)).memLp 1).sub ((boundedLip_of_lipschitz (hu j)).memLp 1) |
| have hmemp : MemLp (fun x => u i x-u j x) (ENNReal.ofReal p) μ := |
| ((boundedLip_of_lipschitz (hu i)).memLp _).sub ((boundedLip_of_lipschitz (hu j)).memLp _) |
| have hh := lpNorm_two_sq_interpolation hmem1 hp hmemp |
| have he1 : lpNorm (fun x => u i x-u j x) 1 μ = dist (value1 (hT := hT) (hu i)) (value1 (hu j)) := by |
| rw [dist_value1,lpNorm_one_eq_integral_norm hmem1.aestronglyMeasurable] |
| rfl |
| have hep : lpNorm (fun x => u i x-u j x) (ENNReal.ofReal p) μ ≤ 2*(C:ℝ) := by |
| have hh := lpNorm_sub_le (g := u j) ((boundedLip_of_lipschitz (hu i)).memLp (μ := μ) (ENNReal.ofReal p)) hp' |
| change lpNorm (u i-u j) (ENNReal.ofReal p) μ ≤ 2*(C:ℝ) |
| exact hh.trans ((add_le_add (hC i) (hC j)).trans_eq (by ring)) |
| have hmono := Real.rpow_le_rpow lpNorm_nonneg hep hb |
| have hsmall : (dist (value1 (hT := hT) (hu i)) (value1 (hu j)))^a*(2*(C:ℝ))^b < ε^2 := by |
| apply hd |
| simpa only [mem_ball,Real.dist_eq,sub_zero,abs_of_nonneg dist_nonneg] using hij |
| rw [he1] at hh |
| have hh' := hh.trans (mul_le_mul_of_nonneg_left hmono (Real.rpow_nonneg dist_nonneg a)) |
| have hd2 := (hh'.trans_lt hsmall) |
| rw [←dist_value (hT := hT) (hu i) (hu j)] at hd2 |
| nlinarith [dist_nonneg (x := value (hT := hT) (hu i)) (y := value (hT := hT) (hu j))] |
|
|
| theorem totallyBounded_value {ι : Type*} |
| (h : Slicing.NormalApprox (k+1) T) (hz : boundarySucc T = 0) (hX : IsCAT0 X) |
| (u : ι → X → ℝ) (L U : ι → ℝ≥0) (hu : ∀ j, LipschitzWith (L j) (u j)) |
| (hU : ∀ j x, |u j x| ≤ U j) |
| (M : ℝ≥0) (hMv : ∀ j, ‖value (hT := hT) (hu j)‖ ≤ M) |
| (hMg : ∀ j, ‖q.gradient (hu j)‖ ≤ M) |
| {p : ℝ} (hp : 2 < p) (C : ℝ≥0) |
| (hC : ∀ j, lpNorm (u j) (ENNReal.ofReal p) (MassMeasure.currentMassMeasure hT) ≤ C) : |
| TotallyBounded (range fun j => value (hT := hT) (hu j)) := |
| totallyBounded_value_of_lpBound (hT := hT) hp u L hu |
| (q.totallyBounded_value1 h hz hX u L U hu hU M hMv hMg) C hC |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| end OAI |
|
|
