| 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 |
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| end |
|
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| end OAI |
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