| import OAI.Analysis.AncientFlat.Evolution.TimeOneTails | |
| import OAI.Analysis.AncientFlat.Evolution.C0Ascoli | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Metric | |
| open scoped Topology ContDiff ZeroAtInfty | |
| namespace FlatSimilarity | |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] | |
| theorem compact_time_one_difference (c : ℝ) (Af Bf : V → ℝ) | |
| (h : SymbolBounds c Af Bf) | |
| (P Pc : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ)) | |
| (hP : TimeOneMap Af Bf P) | |
| (hPc : TimeOneMap (fun _ => c^2) (fun _ => 0) Pc) : | |
| IsCompactOperator (P-Pc) := by | |
| obtain ⟨K,hK,hKb⟩ := hP.uniform_lipschitz h | |
| obtain ⟨Kc,hKc,hKcb⟩ := hPc.uniform_lipschitz (SymbolBounds.constant h.1) | |
| let Q := P-Pc | |
| have hh : ∀ d : C₀(V,ℂ), d∈closedBall 0 1 → ‖d‖≤1 := by | |
| intro d hd | |
| simpa only [mem_closedBall,dist_zero_right] using hd | |
| have ht : TotallyBounded (Q '' closedBall 0 1) := by | |
| apply FlatCompactness.totallyBounded_of_lipschitz_tails _ ‖Q‖ (K+Kc) (norm_nonneg Q) (add_nonneg hK hKc) | |
| · rintro f ⟨d,hd,rfl⟩ x | |
| exact (C0Convolution.apply_norm_le (Q d) x).trans (((Q.le_opNorm d).trans | |
| (mul_le_mul_of_nonneg_left (hh d hd) (norm_nonneg Q))).trans_eq (mul_one _)) | |
| · rintro f ⟨d,hd,rfl⟩ x y | |
| have he : Q d x-Q d y=(P d x-P d y)-(Pc d x-Pc d y) := by | |
| simp only [Q,sub_apply,ZeroAtInftyContinuousMap.sub_apply] | |
| ring | |
| rw [he] | |
| calc | |
| _ ≤ ‖P d x-P d y‖+‖Pc d x-Pc d y‖ := norm_sub_le _ _ | |
| _ ≤ K*‖d‖*‖x-y‖+Kc*‖d‖*‖x-y‖ := add_le_add (hKb d x y) (hKcb d x y) | |
| _ = (K+Kc)*‖d‖*‖x-y‖ := by ring | |
| _ ≤ (K+Kc)*1*‖x-y‖ := mul_le_mul_of_nonneg_right | |
| (mul_le_mul_of_nonneg_left (hh d hd) (add_nonneg hK hKc)) (norm_nonneg _) | |
| _ = _ := by rw [mul_one] | |
| · intro ε hε | |
| obtain ⟨R,hR,hRb⟩ := hP.uniform_tails_difference h hPc ε hε | |
| refine ⟨R,hR,?_⟩ | |
| rintro f ⟨d,hd,rfl⟩ x hx | |
| exact hRb d (hh d hd) x hx | |
| apply (isCompactOperator_iff_isCompact_closure_image_closedBall Q.toLinearMap zero_lt_one).mpr | |
| exact ht.closure.isCompact_of_isClosed isClosed_closure | |
| end FlatSimilarity | |
| end | |
| end OAI | |