| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Transport.RadialMoments |
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|
| namespace OAI |
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| section |
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| open Set Filter MeasureTheory |
| open scoped Topology NNReal |
|
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| namespace CAT0Fillings.RadialSobolev |
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| section |
| variable {n : ℕ} {ω p R D : ℝ} {f g : ℝ → ℝ} |
| (hR : 0 ≤ R) (hD : 0 < D) (hn : 0 < n) (hω : 0 < ω) |
| (hf : Continuous f) (hg : Continuous g) (hp : 0 < p) |
| (hfp : ∀ r ∈ Ioo 0 R, 0 < f r) (hgp : ∀ r ∈ Ioo 0 D, 0 < g r) |
| (hmass : radialCDF n ω f p R = radialCDF n ω g p D) |
| include hR hf hfp hmass |
|
|
| lemma radialTransport_integral_moment_unscaled (H : ℝ → ℝ) : |
| (∫ s in (0:ℝ)..D, s^(n-1)*(g s)^p*H s) = |
| ∫ r in (0:ℝ)..R, r^(n-1)*(f r)^p* |
| H (radialTransport (f := f) hD.le hn hω hg hp hgp r) := by |
| apply mul_left_cancel₀ (ne_of_gt (mul_pos (Nat.cast_pos.mpr hn) hω)) |
| simpa only [←intervalIntegral.integral_const_mul,mul_assoc] using |
| radialTransport_integral_moment hR hD hn hω hf hg hp hfp hgp hmass H |
|
|
| lemma radialTransport_integral_power (q : ℝ) : |
| (∫ s in (0:ℝ)..D, s^(n-1)*(g s)^q) = |
| ∫ r in (0:ℝ)..R, r^(n-1)*(f r)^p* |
| (g (radialTransport (f := f) hD.le hn hω hg hp hgp r))^(q-p) := by |
| rw [←radialTransport_integral_moment_unscaled hR hD hn hω hf hg hp hfp hgp hmass (fun s => (g s)^(q-p))] |
| apply intervalIntegral.integral_congr_Ioo_of_le hD.le |
| intro s hs |
| dsimp only |
| rw [mul_assoc,←Real.rpow_add (hgp s hs)] |
| congr 2 |
| ring |
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|
| end |
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| lemma radial_divergence_algebra {n : ℕ} {r t d : ℝ} (hn : 2 ≤ n) (hr : r ≠ 0) : |
| r^(n-1)*(d+(n-1:ℕ)*t/r) = (n-1:ℕ)*r^(n-2)*t+r^(n-1)*d := by |
| have he : n-1 = n-2+1 := by omega |
| rw [he,pow_succ] |
| field_simp |
| ring |
|
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| lemma weighted_transport_derivative_integrable {n : ℕ} {f t : ℝ → ℝ} {R q : ℝ} |
| (hR : 0 ≤ R) (hf : Continuous f) (hq : 0 ≤ q) |
| (ht : AbsolutelyContinuousOnInterval t 0 R) |
| (hdt : ∀ r ∈ Ioo 0 R, DifferentiableAt ℝ t r) : |
| IntervalIntegrable (fun r => (f r)^q* |
| (((n-1:ℕ):ℝ)*r^(n-2)*t r+r^(n-1)*deriv t r)) volume 0 R := by |
| have hv : AbsolutelyContinuousOnInterval (fun r => r^(n-1)*t r) 0 R := |
| (contDiff_id.pow (n-1)).contDiffOn.absolutelyContinuousOnInterval.fun_mul ht |
| have hh := hv.intervalIntegrable_deriv.continuousOn_mul |
| (hf.rpow_const (fun _ => Or.inr hq)).continuousOn |
| apply hh.congr_uIoo |
| simp only [uIoo_of_le hR] |
| intro r hr |
| dsimp only |
| rw [(weightedTransport_hasDerivAt (n := n) (hdt r hr).hasDerivAt).deriv] |
|
|
| lemma finite_transport_ibp {n : ℕ} {ω p q R D : ℝ} {f g : ℝ → ℝ} {K : ℝ≥0} |
| (hR : 0 ≤ R) (hD : 0 < D) (hn : 2 ≤ n) (hω : 0 < ω) |
| (hf : LipschitzWith K f) (hg : Continuous g) (hp : 0 < p) (hq : 1 ≤ q) |
| (hfp : ∀ r ∈ Ioo 0 R, 0 < f r) (hgp : ∀ r, 0 < g r) (hfR : f R = 0) |
| (hmass : radialCDF n ω f p R = radialCDF n ω g p D) |
| (hexp : q+p/(n:ℝ)=p) : |
| (n:ℝ)*(∫ s in (0:ℝ)..D,s^(n-1)*(g s)^q) ≤ |
| -q*∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(q-1)*deriv f r* |
| radialTransport (f := f) hD.le (by omega : 0 < n) hω hg hp (fun r _ => hgp r) r := by |
| have hn0 : 0 < n := by omega |
| let t := radialTransport (f := f) hD.le hn0 hω hg hp (fun r _ => hgp r) |
| have ht : Continuous t := radialTransport_continuous _ _ _ hf.continuous hg hp _ |
| have hta : AbsolutelyContinuousOnInterval t 0 R := |
| radialTransport_ac hR hD hn0 hω hf.continuous hg hp hfp (fun r _ => hgp r) hmass |
| have hdt : ∀ r ∈ Ioo 0 R, DifferentiableAt ℝ t r := fun r hr => |
| (radialTransport_hasDerivAt hD.le hn0 hω hf.continuous hg hp hfp (fun r _ => hgp r) hmass hr).differentiableAt |
| have hA := weighted_transport_derivative_integrable (n := n) hR hf.continuous (by linarith : 0 ≤ q) hta hdt |
| have hJ : Continuous (fun r => r^(n-1)*(f r)^p*(g (t r))^(q-p)) := |
| ((continuous_id.pow _).mul (hf.continuous.rpow_const (fun _ => Or.inr hp.le))).mul |
| ((hg.comp ht).rpow_const (fun r => Or.inl (hgp (t r)).ne')) |
| rw [radialTransport_integral_power hR hD hn0 hω hf.continuous hg hp hfp (fun r _ => hgp r) hmass q, |
| ←intervalIntegral.integral_const_mul] |
| calc |
| (∫ r in (0:ℝ)..R,(n:ℝ)*(r^(n-1)*(f r)^p*(g (t r))^(q-p))) ≤ |
| ∫ r in (0:ℝ)..R,(f r)^q*((n-1:ℕ)*r^(n-2)*t r+r^(n-1)*deriv t r) := by |
| apply intervalIntegral.integral_mono_on_of_le_Ioo hR |
| (hJ.intervalIntegrable _ _ |>.const_mul _) hA |
| intro r hr |
| have hpos := radialTransport_deriv_pos hD hn0 hω hf.continuous hg hp hfp (fun r _ => hgp r) hmass hr |
| have htp := radialTransport_interior hD.le hn0 hω hf.continuous hg hp hfp (fun r _ => hgp r) hmass hr |
| have hagm := transport_pointwise_agm hn0 (hfp r hr) (hgp (t r)) hpos.le |
| (div_nonneg htp.1.le hr.1.le) hexp |
| (radialTransport_jacobian hD hn0 hω hf.continuous hg hp hfp (fun r _ => hgp r) hmass hr) |
| have hh := mul_le_mul_of_nonneg_left hagm (pow_nonneg hr.1.le (n-1)) |
| have he := radial_divergence_algebra (t := t r) (d := deriv t r) hn hr.1.ne' |
| change (n:ℝ)*(r^(n-1)*(f r)^p*(g (t r))^(q-p)) ≤ _ |
| calc |
| _ = r^(n-1)*((n:ℝ)*(f r)^p*(g (t r))^(q-p)) := by ring |
| _ ≤ r^(n-1)*((f r)^q*(deriv t r+(n-1:ℕ)*(t r/r))) := hh |
| _ = (f r)^q*(r^(n-1)*(deriv t r+(n-1:ℕ)*t r/r)) := by ring |
| _ = _ := by rw [he] |
| _ = -q*∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(q-1)*deriv f r*t r := |
| radial_integration_by_parts hn hR hf hq hfR hta hdt |
|
|
| end CAT0Fillings.RadialSobolev |
| end |
|
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| end OAI |
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