| import OAI.Geometry.RCD.Coordinates.TransverseFrameData |
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| namespace OAI |
|
|
| /-! # Bounded axial and transverse potentials before imposing any Gram identities -/ |
|
|
| open Set Filter MeasureTheory Finset |
| open scoped NNReal InnerProductSpace |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] |
| (μ : Measure X) (Q : QuadraticCheegerEnergy μ) |
|
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| theorem bounded_transverse_potentials {n : ℕ} (u : Fin n → SobolevSpace μ Q) |
| (A : Fin n → ℝ≥0) (hA : ∀ i, ∀ᵐ x ∂μ, |SobolevSpace.toLp μ Q (u i) x|≤A i) |
| (Ω : Set X) (v : X → Euclidean n) |
| (hu : ∀ i, SobolevSpace.toLp μ Q (u i) =ᵐ[μ.restrict Ω] fun x => v x i) |
| (hH : ∀ i, IsSobolevHarmonicOn μ Q Ω (u i)) |
| (a : Euclidean n) : |
| ∃ (w s : SobolevSpace μ Q) (B D : ℝ≥0), |
| (∀ᵐ x ∂μ, |SobolevSpace.toLp μ Q w x|≤B) ∧ |
| (∀ᵐ x ∂μ, |SobolevSpace.toLp μ Q s x|≤D) ∧ |
| (SobolevSpace.toLp μ Q w =ᵐ[μ.restrict Ω] fun x => ⟪v x,a⟫_ℝ) ∧ |
| (SobolevSpace.toLp μ Q s =ᵐ[μ.restrict Ω] fun x => ‖v x‖^2-⟪v x,a⟫_ℝ^2) ∧ |
| IsSobolevHarmonicOn μ Q Ω w ∧ |
| w=sobolevColumnMap μ Q u a ∧ |
| s=(∑ i, SobolevSpace.square μ Q (u i) (A i))-SobolevSpace.square μ Q w B := by |
| let w := sobolevColumnMap μ Q u a |
| let B : ℝ≥0 := ∑ i, ‖a i‖₊*A i |
| have hwglobal : SobolevSpace.toLp μ Q w =ᵐ[μ] |
| fun x => ∑ i, a i*SobolevSpace.toLp μ Q (u i) x := by |
| have h := sobolevColumnMap_representative μ Q u univ |
| (fun x => WithLp.toLp 2 (fun i => SobolevSpace.toLp μ Q (u i) x)) |
| (fun _ => EventuallyEq.rfl) a |
| simpa only [Measure.restrict_univ,PiLp.inner_apply,RCLike.inner_apply,conj_trivial] using h |
| have hwB : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ Q w x|≤B := by |
| filter_upwards [hwglobal,ae_all_iff.mpr hA] with x hx hb |
| rw [hx] |
| calc |
| _ ≤ ∑ i, |a i*SobolevSpace.toLp μ Q (u i) x| := Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ i, |a i| * (A i:ℝ) := sum_le_sum (fun i _ => by |
| rw [abs_mul]; exact mul_le_mul_of_nonneg_left (hb i) (abs_nonneg _)) |
| _ = B := by simp [B,Real.norm_eq_abs] |
| let z := ∑ i, SobolevSpace.square μ Q (u i) (A i) |
| have hz : SobolevSpace.toLp μ Q z =ᵐ[μ] |
| fun x => ∑ i, (SobolevSpace.toLp μ Q (u i) x)^2 := by |
| change SobolevSpace.toLpCLM μ Q (∑ i, SobolevSpace.square μ Q (u i) (A i)) =ᵐ[μ] _ |
| rw [map_sum] |
| change (∑ i, SobolevSpace.toLp μ Q (SobolevSpace.square μ Q (u i) (A i))) =ᵐ[μ] _ |
| filter_upwards [Lp.coeFn_finsetSum univ (fun i => SobolevSpace.toLp μ Q |
| (SobolevSpace.square μ Q (u i) (A i))),ae_all_iff.mpr |
| (fun i => SobolevSpace.square_coe μ Q (u i) (A i) (hA i))] with x hx hs |
| rw [hx] |
| simp only [Finset.sum_apply,hs] |
| let s := z-SobolevSpace.square μ Q w B |
| have hs : SobolevSpace.toLp μ Q s =ᵐ[μ] |
| fun x => (∑ i, (SobolevSpace.toLp μ Q (u i) x)^2)-(SobolevSpace.toLp μ Q w x)^2 := by |
| filter_upwards [hz,SobolevSpace.square_coe μ Q w B hwB, |
| Lp.coeFn_sub (SobolevSpace.toLp μ Q z) (SobolevSpace.toLp μ Q (SobolevSpace.square μ Q w B))] |
| with x hx hy hsub |
| change (SobolevSpace.toLp μ Q z-SobolevSpace.toLp μ Q (SobolevSpace.square μ Q w B)) x=_ |
| simp only [hsub,Pi.sub_apply,hx,hy] |
| let D : ℝ≥0 := (∑ i, A i^2)+B^2 |
| have hsD : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ Q s x|≤D := by |
| filter_upwards [hs,ae_all_iff.mpr hA,hwB] with x hx hb hw |
| rw [hx] |
| calc |
| _ ≤ |∑ i, (SobolevSpace.toLp μ Q (u i) x)^2|+|(SobolevSpace.toLp μ Q w x)^2| := abs_sub _ _ |
| _ = (∑ i, |SobolevSpace.toLp μ Q (u i) x|^2)+|SobolevSpace.toLp μ Q w x|^2 := by |
| rw [abs_of_nonneg (sum_nonneg (fun _ _ => sq_nonneg _))] |
| simp only [abs_pow,sq_abs] |
| _ ≤ (∑ i, (A i:ℝ)^2)+(B:ℝ)^2 := add_le_add |
| (sum_le_sum (fun i _ => pow_le_pow_left₀ (abs_nonneg _) (hb i) 2)) |
| (pow_le_pow_left₀ (abs_nonneg _) hw 2) |
| _ = D := by simp [D] |
| have hw := sobolevColumnMap_representative μ Q u Ω v hu a |
| refine ⟨w,s,B,D,hwB,hsD,hw,?_,IsSobolevHarmonicOn.linearCombination μ Q hH a,rfl,rfl⟩ |
| · filter_upwards [ae_restrict_of_ae hs,ae_all_iff.mpr hu,hw] with x hx hux hwx |
| rw [hx,hwx,EuclideanSpace.real_norm_sq_eq] |
| simp only [hux] |
|
|
| end RegularPointCoordinates |
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| end |
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| end OAI |
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