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67aee1d | 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 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.AllowedProjection
import OAI.Analysis.HardSphere.Cutoffs.RegularizationWeight
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal Topology BigOperators
open scoped BigOperators ComplexConjugate
open MeasureTheory ProbabilityTheory Filter Set
open scoped ENNReal NNReal Topology BigOperators
open MeasureTheory ProbabilityTheory Real Complex
open scoped ENNReal ComplexConjugate
namespace HardSphere
open ContinuousLinearMap Filter
open scoped Topology
variable {E F : Type*} [NormedAddCommGroup E] [NormedAddCommGroup F]
[InnerProductSpace ℂ E] [InnerProductSpace ℂ F] [CompleteSpace E] [CompleteSpace F] [Nontrivial F]
local instance graphLiftHeatBoundRealAlgebra : Algebra ℝ (F →L[ℂ] F) :=
Algebra.restrictScalars ℝ ℂ _
lemma exists_graph_lift_of_heat_bound (A : E →L[ℂ] F) (hA : ‖A‖≤1)
(hdense : DenseRange (A.comp A.adjoint)) (u : F) (C : ℝ)
(hb : ∀ n : ℕ, (1 / ((n:ℝ)+1))⁻¹ *
(‖u‖^2-(inner ℂ u ((cfc (p := IsSelfAdjoint) (heatScalar (1/((n:ℝ)+1)))
(A.comp A.adjoint)) u)).re) ≤ C) :
∃ v : E, A v=u ∧ ‖v‖^2≤‖u‖^2+C := by
let d : ℕ→ℝ := fun n ↦ 1/((n:ℝ)+1)
have hd (n : ℕ) : 0<d n := by dsimp [d]; positivity
let R := A.comp A.adjoint
let v : ℕ→E := fun n ↦ A.adjoint ((cfc (p := IsSelfAdjoint) (regularizationWeight (d n)) R) u)
have hv (n : ℕ) : ‖v n‖^2≤‖u‖^2+C :=
(regularized_lift_norm_sq A hA (hd n) u).trans (add_le_add_right (hb n) _)
have hB : 0≤‖u‖^2+C := (sq_nonneg ‖v 0‖).trans (hv 0)
have hRp : R.IsPositive := by simpa [R] using isPositive_adjoint_comp_self A.adjoint
have hRn : ‖R‖≤1 := (opNorm_comp_le _ _).trans (by rw [adjoint.norm_map]; nlinarith [norm_nonneg A])
have ht : Tendsto (fun n ↦ A (v n)) atTop (𝓝 u) := by
have hh := regularizationOperator_tendsto hd tendsto_one_div_add_atTop_nhds_zero_nat hRp hRn hdense u
convert hh using 1
funext n
rw [regularizationOperator_eq hRp hRn (hd n)]
rfl
obtain ⟨w,hw,hwu⟩ := exists_bounded_lift_of_tendsto A v
(C := Real.sqrt (‖u‖^2+C)) (fun n ↦ (Real.le_sqrt (norm_nonneg _) hB).mpr (hv n)) ht
refine ⟨w,hwu,?_⟩
calc
‖w‖^2 ≤ (Real.sqrt (‖u‖^2+C))^2 := pow_le_pow_left₀ (norm_nonneg _) hw 2
_ = _ := Real.sq_sqrt hB
end HardSphere
end
end OAI
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