| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Bubble.Moments |
| import OAI.Geometry.CAT0Fillings.Transport.Integrals |
| import OAI.Geometry.CAT0Fillings.Transport.Cauchy |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology NNReal |
|
|
| namespace CAT0Fillings.RadialSobolev |
|
|
| lemma finite_transport_estimate {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) (hexp2 : 2*(q-1)=p) : |
| (n:ℝ)^2*(∫ s in (0:ℝ)..D,s^(n-1)*(g s)^q)^2 ≤ |
| q^2*(∫ r in (0:ℝ)..R,r^(n-1)*(deriv f r)^2)* |
| (∫ s in (0:ℝ)..D,s^(n-1)*(g s)^p*s^2) := 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 hz : Continuous (fun r => (f r)^(q-1)*t r) := |
| (hf.continuous.rpow_const (fun _ => Or.inr (sub_nonneg.mpr hq))).mul ht |
| have hcs := weighted_interval_cs hR (continuous_id.pow (n-1)) hz hf |
| (fun r hr => pow_nonneg hr.1 (n-1)) |
| have hI : (∫ r in (0:ℝ)..R,r^(n-1)*deriv f r*((f r)^(q-1)*t r)) = |
| ∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(q-1)*deriv f r*t r := by |
| apply intervalIntegral.integral_congr |
| intro r hr |
| ring |
| have hM : (∫ r in (0:ℝ)..R,r^(n-1)*((f r)^(q-1)*t r)^2) = |
| ∫ s in (0:ℝ)..D,s^(n-1)*(g s)^p*s^2 := by |
| rw [radialTransport_integral_moment_unscaled hR hD hn0 hω hf.continuous hg hp hfp |
| (fun r _ => hgp r) hmass (fun s => s^2)] |
| apply intervalIntegral.integral_congr_Ioo_of_le hR |
| intro r hr |
| dsimp only |
| rw [mul_pow,←Real.rpow_mul_natCast (hfp r hr).le] |
| have he : (q-1)*(2:ℝ)=p := by linarith |
| rw [Nat.cast_ofNat,he] |
| ring |
| dsimp only [Pi.pow_apply, id_eq] at hcs |
| rw [hI,hM] at hcs |
| have hibp := finite_transport_ibp hR hD hn hω hf hg hp hq hfp hgp hfR hmass hexp |
| have hJ0 : 0 ≤ (n:ℝ)*(∫ s in (0:ℝ)..D,s^(n-1)*(g s)^q) := by |
| apply mul_nonneg (Nat.cast_nonneg n) |
| apply intervalIntegral.integral_nonneg hD.le |
| intro s hs |
| exact mul_nonneg (pow_nonneg hs.1 _) (Real.rpow_nonneg (hgp s).le _) |
| have hsq := (sq_le_sq₀ hJ0 (hJ0.trans hibp)).mpr hibp |
| have hsq' := mul_le_mul_of_nonneg_left hcs (sq_nonneg q) |
| calc |
| _ ≤ (-q*(∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(q-1)*deriv f r*t r))^2 := by |
| simpa only [mul_pow] using hsq |
| _ = q^2*(∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(q-1)*deriv f r*t r)^2 := by ring |
| _ ≤ _ := by simpa only [mul_assoc] using hsq' |
|
|
| end CAT0Fillings.RadialSobolev |
| end |
|
|
| section |
|
|
| open Real Set Filter MeasureTheory |
| open scoped Topology NNReal |
|
|
| namespace CAT0Fillings.RadialSobolev |
|
|
| lemma radial_integral_smul (n : ℕ) (p A D : ℝ) (g : ℝ → ℝ) |
| (hA : 0 ≤ A) (hg : ∀ r, 0 ≤ g r) : |
| (∫ r in (0:ℝ)..D,r^(n-1)*(A*g r)^p) = |
| A^p*∫ r in (0:ℝ)..D,r^(n-1)*(g r)^p := by |
| rw [←intervalIntegral.integral_const_mul] |
| apply intervalIntegral.integral_congr |
| intro r hr |
| dsimp only |
| rw [Real.mul_rpow hA (hg r)] |
| ring |
|
|
| lemma radial_second_moment_smul (n : ℕ) (p A D : ℝ) (g : ℝ → ℝ) |
| (hA : 0 ≤ A) (hg : ∀ r, 0 ≤ g r) : |
| (∫ r in (0:ℝ)..D,r^(n-1)*(A*g r)^p*r^2) = |
| A^p*∫ r in (0:ℝ)..D,r^(n-1)*(g r)^p*r^2 := by |
| rw [←intervalIntegral.integral_const_mul] |
| apply intervalIntegral.integral_congr |
| intro r hr |
| dsimp only |
| rw [Real.mul_rpow hA (hg r)] |
| ring |
|
|
| lemma radial_integral_pos {n : ℕ} {p R : ℝ} {f : ℝ → ℝ} |
| (hn : 0 < n) (hp : 0 < p) (hR : 0 < R) (hf : Continuous f) |
| (hfp : ∀ r ∈ Ioo 0 R, 0 < f r) : |
| 0 < ∫ r in (0:ℝ)..R,r^(n-1)*(f r)^p := by |
| have h := radialCDF_strictMonoOn (ω := 1) hn zero_lt_one hf hp hfp |
| (show (0:ℝ) ∈ Icc 0 R from ⟨le_rfl,hR.le⟩) (show R ∈ Icc 0 R from ⟨hR.le,le_rfl⟩) hR |
| rw [radialCDF_zero,radialCDF, mul_one] at h |
| exact (mul_pos_iff.mp h).resolve_right (by rintro ⟨hneg,_⟩; exact (not_lt_of_ge (Nat.cast_nonneg _)) hneg) |>.2 |
|
|
| lemma normalization_power {p F I : ℝ} (hp : 0 < p) (hF : 0 < F) (hI : 0 < I) : |
| ((F/I)^(1/p))^p = F/I := by |
| rw [←Real.rpow_mul (div_nonneg hF.le hI.le),one_div_mul_cancel hp.ne',Real.rpow_one] |
| |
| lemma normalize_transport_estimate {n : ℕ} {R D : ℝ} {f : ℝ → ℝ} {K : ℝ≥0} |
| (hn : 2 < n) (hR : 0 < R) (hD : 0 < D) (hf : LipschitzWith K f) |
| (hfp : ∀ r ∈ Ioo 0 R, 0 < f r) (hfR : f R = 0) : |
| let F := ∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(sobolevP n) |
| let I := ∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevP n) |
| let J := ∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevQ n) |
| let M := ∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevP n)*r^2 |
| (n:ℝ)^2*J^2*(F/I)^(2/sobolevP n) ≤ |
| (sobolevQ n)^2*(∫ r in (0:ℝ)..R,r^(n-1)*(deriv f r)^2)*M := by |
| dsimp only |
| let F := ∫ r in (0:ℝ)..R,r^(n-1)*(f r)^(sobolevP n) |
| let I := ∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevP n) |
| let A := (F/I)^(1/sobolevP n) |
| have hs := sobolev_exponents hn |
| have hF : 0 < F := radial_integral_pos (by omega) hs.1 hR hf.continuous hfp |
| have hI : 0 < I := radial_integral_pos (by omega) hs.1 hD (bubble_continuous n) (fun r _ => bubble_pos n r) |
| have hA : 0 < A := Real.rpow_pos_of_pos (div_pos hF hI) _ |
| have hAp : A^(sobolevP n) = F/I := normalization_power hs.1 hF hI |
| have hm : radialCDF n 1 f (sobolevP n) R = |
| radialCDF n 1 (fun r => A*bubble n r) (sobolevP n) D := by |
| rw [radialCDF,radialCDF,radial_integral_smul n _ A D _ hA.le (fun r => (bubble_pos n r).le),hAp] |
| change (n:ℝ)*1*F = (n:ℝ)*1*(F/I*I) |
| rw [div_mul_cancel₀ _ hI.ne'] |
| have ht := finite_transport_estimate hR.le hD (by omega) zero_lt_one hf |
| ((bubble_continuous n).const_mul A) hs.1 hs.2.1 hfp |
| (fun r => mul_pos hA (bubble_pos n r)) hfR hm hs.2.2.1 hs.2.2.2.1 |
| rw [radial_integral_smul _ _ _ _ _ hA.le (fun r => (bubble_pos n r).le), |
| radial_second_moment_smul _ _ _ _ _ hA.le (fun r => (bubble_pos n r).le),mul_pow] at ht |
| have he : (A^(sobolevQ n))^2 = A^(sobolevP n)*A^2 := by |
| rw [←Real.rpow_mul_natCast hA.le, ←Real.rpow_natCast, ←Real.rpow_add hA] |
| congr 1 |
| norm_num only [Nat.cast_ofNat] |
| linarith [hs.2.2.2.2] |
| have heA : A^2 = (F/I)^(2/sobolevP n) := by |
| rw [←Real.rpow_mul_natCast (div_nonneg hF.le hI.le)] |
| change (F/I)^((1/sobolevP n)*(2:ℝ)) = _ |
| congr 1 |
| ring |
| rw [he] at ht |
| have hout := (mul_le_mul_iff_right₀ (Real.rpow_pos_of_pos hA (sobolevP n))).mp |
| (show A^(sobolevP n)*((n:ℝ)^2* |
| (∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevQ n))^2*A^2) ≤ |
| A^(sobolevP n)*((sobolevQ n)^2*(∫ r in (0:ℝ)..R,r^(n-1)*(deriv f r)^2)* |
| (∫ r in (0:ℝ)..D,r^(n-1)*(bubble n r)^(sobolevP n)*r^2)) by convert ht using 1 <;> first | rfl | ring) |
| rwa [heA] at hout |
|
|
| end CAT0Fillings.RadialSobolev |
| end |
|
|
| end OAI |
|
|
