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12dee50 | 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 | 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
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