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f205175 | 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 86 87 88 89 90 91 92 93 | import Mathlib
import OAI.Probability.CriticalPercolation.Growth.HomogeneousGeometry
namespace OAI
section
noncomputable section
open scoped Classical Pointwise BigOperators
namespace CriticalPercolation.HomogeneousAnalysis.Geometry
variable {A : Type*} (W : Geometry A)
def harmonicSpace : Submodule ℝ (A → ℝ) where
carrier := {u | W.IsHarmonic u}
zero_mem' := by intro x; simp
add_mem' := by
intro u v hu hv x
change (∑ s∈W.neighbors x, ((u x+v x)-(u s+v s)))=0
simp_rw [show ∀ s:A, (u x+v x)-(u s+v s)=(u x-u s)+(v x-v s) by intro s; ring]
rw [Finset.sum_add_distrib,hu x,hv x,add_zero]
smul_mem' := by
intro c u hu x
change (∑ s∈W.neighbors x, (c*u x-c*u s))=0
simp_rw [←mul_sub]
rw [←Finset.mul_sum,hu x,mul_zero]
def HasPolynomialGrowth (k : ℕ) (u : A → ℝ) : Prop :=
∃ C : ℝ, 0 ≤ C ∧ ∀ n, ∀ x∈W.ball n, |u x| ≤ C*((n:ℝ)+1)^k
def polynomialGrowthSpace (k : ℕ) : Submodule ℝ (A → ℝ) where
carrier := {u | W.HasPolynomialGrowth k u}
zero_mem' := ⟨0,le_rfl,by simp⟩
add_mem' := by
rintro u v ⟨C,hC,hu⟩ ⟨D,hD,hv⟩
refine ⟨C+D,add_nonneg hC hD,?_⟩
intro n x hx
change |u x+v x| ≤ _
calc
_ ≤ |u x|+|v x| := abs_add_le _ _
_ ≤ C*((n:ℝ)+1)^k+D*((n:ℝ)+1)^k := add_le_add (hu n x hx) (hv n x hx)
_ = _ := by ring
smul_mem' := by
rintro c u ⟨C,hC,hu⟩
refine ⟨|c| * C,mul_nonneg (abs_nonneg _) hC,?_⟩
intro n x hx
change |c*u x| ≤ _
rw [abs_mul,mul_assoc]
exact mul_le_mul_of_nonneg_left (hu n x hx) (abs_nonneg _)
def polynomialHarmonicSpace (k : ℕ) : Submodule ℝ (A → ℝ) :=
W.harmonicSpace ⊓ W.polynomialGrowthSpace k
lemma mem_polynomialHarmonicSpace (k : ℕ) (u : A → ℝ) :
u∈W.polynomialHarmonicSpace k ↔ W.IsHarmonic u ∧ W.HasPolynomialGrowth k u := Iff.rfl
lemma polynomial_growth_iff_length (k : ℕ) (u : A → ℝ) :
W.HasPolynomialGrowth k u ↔ ∃ C : ℝ, 0 ≤ C ∧
∀ x, |u x| ≤ C*((W.length x:ℝ)+1)^k := by
constructor
· rintro ⟨C,hC,hu⟩
exact ⟨C,hC,fun x => hu _ _ (W.mem_ball_length x)⟩
· rintro ⟨C,hC,hu⟩
refine ⟨C,hC,?_⟩
intro n x hx
apply (hu x).trans
apply mul_le_mul_of_nonneg_left _ hC
apply pow_le_pow_left₀ (by positivity)
exact_mod_cast Nat.add_le_add_right (W.length_le_iff.mpr hx) 1
lemma finite_family_growth_bound {ι : Type*} [Fintype ι] (k : ℕ) (v : ι → A → ℝ)
(hv : ∀ i, W.HasPolynomialGrowth k (v i)) :
∃ M : ℝ, 0 ≤ M ∧ ∀ n, 1 ≤ n → ∀ i, ∀ x∈W.ball n,
|v i x| ≤ M*(n:ℝ)^k := by
choose C hC hCv using hv
let M := (∑ i, C i)*2^k
have hm0 : 0 ≤ M := mul_nonneg (Finset.sum_nonneg (fun i _ => hC i)) (by positivity)
refine ⟨M,hm0,?_⟩
intro n hn i x hx
have hn' : (1:ℝ) ≤ n := by exact_mod_cast hn
have hCi : C i ≤ ∑ j, C j := Finset.single_le_sum (fun j _ => hC j) (Finset.mem_univ i)
calc
_ ≤ C i*((n:ℝ)+1)^k := hCv i n x hx
_ ≤ (∑ j, C j)*((n:ℝ)+1)^k := mul_le_mul_of_nonneg_right hCi (by positivity)
_ ≤ (∑ j, C j)*(2*(n:ℝ))^k := mul_le_mul_of_nonneg_left
(pow_le_pow_left₀ (by positivity) (by linarith) k) (Finset.sum_nonneg (fun j _ => hC j))
_ = M*(n:ℝ)^k := by dsimp [M]; rw [mul_pow]; ring
end CriticalPercolation.HomogeneousAnalysis.Geometry
end
end
end OAI
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