File size: 3,674 Bytes
7f37411 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Graphs.NormalizedAngularWord
import OAI.MeasureTheory.FalconerCompletion.Graphs.SmoothFinsetProduct2
namespace OAI
/-! Graph energies and pair estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace Attempt4PrefixShell
open Set
open scoped ContDiff
variable {V α : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
noncomputable def normalizedAngularWordND (D : ℝ) (L : List (V →L[ℝ] V))
(f : SmoothFunctionND V) : SmoothFunctionND V :=
((D:ℂ)^L.length)⁻¹ • angularWordOperatorND L f
lemma normalizedAngularWordND_extract {D : ℝ} (hD : D ≠ 0) (L : List (V →L[ℝ] V))
(f : SmoothFunctionND V) :
(D:ℂ)^L.length • normalizedAngularWordND D L f = angularWordOperatorND L f := by
rw [normalizedAngularWordND,smul_smul,mul_inv_cancel₀ (pow_ne_zero _ (Complex.ofReal_ne_zero.mpr hD)),one_smul]
lemma normalizedAngularWordND_support (D : ℝ) (L : List (V →L[ℝ] V))
(f : SmoothFunctionND V) : tsupport (normalizedAngularWordND D L f).1 ⊆ tsupport f.1 := by
apply subset_trans _ (angularWordOperatorND_support L f)
change tsupport (fun x => ((D:ℂ)^L.length)⁻¹ * (angularWordOperatorND L f).1 x) ⊆ _
apply closure_mono
intro x hx hz
exact hx (by simp [hz])
lemma smoothFinsetProductND_smul (S : Finset α) (c : α → ℂ) (f : α → SmoothFunctionND V) :
smoothFinsetProductND S (fun i => c i • f i) = (∏ i ∈ S, c i) • smoothFinsetProductND S f := by
apply Subtype.ext
funext x
change (∏ i ∈ S, c i*(f i).1 x)=(∏ i ∈ S,c i)*(∏ i ∈ S,(f i).1 x)
exact Finset.prod_mul_distrib
lemma assignedAngularProductND_normalized (S : Finset α) (D : α → ℝ)
(k : α → List (V →L[ℝ] V)) (f : α → SmoothFunctionND V) (hD : ∀ i ∈ S, D i ≠ 0) :
smoothFinsetProductND S (fun i => angularWordOperatorND (k i) (f i)) =
((∏ i ∈ S, D i ^ (k i).length : ℝ):ℂ) •
smoothFinsetProductND S (fun i => normalizedAngularWordND (D i) (k i) (f i)) := by
simp only [Complex.ofReal_prod,Complex.ofReal_pow]
rw [← smoothFinsetProductND_smul]
apply Subtype.ext
funext x
apply Finset.prod_congr rfl
intro i hi
exact congrArg (fun g : SmoothFunctionND V => g.1 x) (normalizedAngularWordND_extract (hD i hi) (k i) (f i)).symm
lemma assignedAngularScaleND_le (S : Finset α) (D : α → ℝ)
(k : α → List (V →L[ℝ] V)) {M : ℝ} {b : ℕ} (hM : 1 ≤ M)
(hD₀ : ∀ i ∈ S, 0 ≤ D i) (hD : ∀ i ∈ S, D i ≤ M)
(hk : ∑ i ∈ S, (k i).length ≤ b) :
∏ i ∈ S, D i ^ (k i).length ≤ M^b := by
calc
_ ≤ ∏ i ∈ S, M^(k i).length := Finset.prod_le_prod₀
(fun i hi => pow_nonneg (hD₀ i hi) _) (fun i hi => pow_le_pow_left₀ (hD₀ i hi) (hD i hi) _)
_ = M^(∑ i ∈ S,(k i).length) := by rw [Finset.prod_pow_eq_pow_sum]
_ ≤ M^b := pow_le_pow_right₀ hM hk
lemma assignedAngularScaleND_paid (S : Finset α) (D : α → ℝ)
(k : α → List (V →L[ℝ] V)) {M r : ℝ} {j : ℕ} (hM : 1 ≤ M)
(hr : 0 ≤ r) (hpaid : r*M^2 ≤ 1)
(hD₀ : ∀ i ∈ S, 0 ≤ D i) (hD : ∀ i ∈ S, D i ≤ M)
(hk : ∑ i ∈ S, (k i).length ≤ 2*j) :
r^j * (∏ i ∈ S, D i ^ (k i).length) ≤ 1 := by
calc
_ ≤ r^j * M^(2*j) := mul_le_mul_of_nonneg_left
(assignedAngularScaleND_le S D k hM hD₀ hD hk) (pow_nonneg hr _)
_ = (r*M^2)^j := by rw [mul_pow,pow_mul]
_ ≤ 1 := by simpa using pow_le_pow_left₀ (by positivity) hpaid j
end Attempt4PrefixShell
end GraphLiteralSupport
end
end
end OAI
|