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import OAI.Geometry.PrescribedPotential.ConstantProjection
import OAI.Geometry.PrescribedPotential.PatchCutoffs
import OAI.Geometry.PrescribedPotential.NormalizedPoisson
namespace OAI
/-! Bounded Poisson. -/
section
noncomputable section
open Set Filter Topology
open scoped ContDiff Classical
namespace GlobalElliptic
namespace LinearNormalization
variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℝ H]
def coefficient (C : ℂ →L[ℝ] H) (hC : Function.Injective C) (P : H →L[ℝ] H)
(hP : ∀ u, P u ∈ (C : ℂ →ₗ[ℝ] H).range) : H →L[ℝ] ℂ := by
let S := (C : ℂ →ₗ[ℝ] H).range
letI : FiniteDimensional ℝ S := LinearMap.finiteDimensional_range _
let e : ℂ ≃ₗ[ℝ] S := LinearEquiv.ofInjective (C : ℂ →ₗ[ℝ] H) hC
let l : S →L[ℝ] ℂ := LinearMap.toContinuousLinearMap e.symm.toLinearMap
exact l ∘L P.codRestrict S hP
lemma coefficient_spec (C : ℂ →L[ℝ] H) (hC : Function.Injective C) (P : H →L[ℝ] H)
(hP : ∀ u, P u ∈ (C : ℂ →ₗ[ℝ] H).range) (u : H) :
C (coefficient C hC P hP u) = P u :=
LinearEquiv.ofInjective_symm_apply (C : ℂ →ₗ[ℝ] H) (h := hC) ⟨P u, hP u⟩
end LinearNormalization
open Anticanonical SourceSmooth EllipticKernel SobolevChart
variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
[ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
namespace Localizers
variable (D : Localizers A ι)
def constantsOrder (s : ℝ) : ℂ →L[ℝ] D.Sobolev s :=
LinearMap.toContinuousLinearMap ((D.embed s).comp Smooth.constLinear)
omit [ConnectedSpace X] in
@[simp] lemma constantsOrder_apply (s : ℝ) (c : ℂ) :
D.constantsOrder s c = D.embed s (Smooth.const c) := rfl
lemma constants_injective : Function.Injective D.constants := by
intro c e h
have hh := D.embed_injective 0 h
let x : X := Classical.arbitrary X
exact congrArg (fun u : Smooth A => u x) hh
variable {D}
def ConstantProjection.coefficient (P : D.ConstantProjection) : D.Sobolev 0 →L[ℝ] ℂ :=
LinearNormalization.coefficient D.constants D.constants_injective P.proj (by
intro u
obtain ⟨c, hc⟩ := P.value u
exact ⟨c, hc.symm⟩)
lemma ConstantProjection.coefficient_spec (P : D.ConstantProjection) (u : D.Sobolev 0) :
D.constants (P.coefficient u) = P.proj u :=
LinearNormalization.coefficient_spec _ _ _ _ u
lemma ConstantProjection.coefficient_constants (P : D.ConstantProjection) (c : ℂ) :
P.coefficient (D.constants c) = c := by
apply D.constants_injective
rw [P.coefficient_spec]
exact P.fixed c
end Localizers
namespace GluingData
variable {g : KaehlerMetric A} (D : GluingData g ι)
lemma fredholm_injective (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
(P : D.localizers.ConstantProjection) :
Function.Injective (1 - D.fredholmOperator m hm he P :
D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 0) := by
intro u v huv
have hz : (1 - D.fredholmOperator m hm he P) (u-v) = 0 := by
rw [map_sub, huv, sub_self]
have hh : D.fredholmOperator m hm he P (u-v) = u-v := by
exact (sub_eq_zero.mp hz).symm
exact sub_eq_zero.mp (D.fredholm_kernel m hm he P (u-v) hh)
def fredholmEquiv (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
(P : D.localizers.ConstantProjection) :
D.localizers.Sobolev 0 ≃L[ℝ] D.localizers.Sobolev 0 :=
ContinuousLinearEquiv.ofBijective (1 - D.fredholmOperator m hm he P)
(LinearMap.ker_eq_bot.mpr (D.fredholm_injective m hm he P))
(LinearMap.range_eq_top.mpr (D.fredholm_surjective m hm he P))
@[simp] lemma fredholmEquiv_apply (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
(P : D.localizers.ConstantProjection) (u : D.localizers.Sobolev 0) :
D.fredholmEquiv m hm he P u =
u - ((m^2 : ℝ) • D.resolventZero m hm he u - P.proj u) := rfl
def poissonInput (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
(P : D.localizers.ConstantProjection) : D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 0 :=
- (D.fredholmEquiv m hm he P).symm.toContinuousLinearMap
lemma poissonInput_equation (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
(P : D.localizers.ConstantProjection) (f : D.localizers.Sobolev 0) :
D.poissonInput m hm he P f = -f - P.proj (D.poissonInput m hm he P f) +
(m^2 : ℝ) • D.resolventZero m hm he (D.poissonInput m hm he P f) := by
have h : D.fredholmEquiv m hm he P (D.poissonInput m hm he P f) = -f := by
change D.fredholmEquiv m hm he P (- (D.fredholmEquiv m hm he P).symm f) = -f
rw [map_neg, ContinuousLinearEquiv.apply_symm_apply]
rw [D.fredholmEquiv_apply] at h
calc
_ = -f + ((m^2 : ℝ) • D.resolventZero m hm he (D.poissonInput m hm he P f) -
P.proj (D.poissonInput m hm he P f)) := sub_eq_iff_eq_add.mp h
_ = _ := by abel
def poissonRaw (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
(P : D.localizers.ConstantProjection) : D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 2 :=
D.completedResolvent m hm he ∘L D.poissonInput m hm he P
def poissonResidue (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
(P : D.localizers.ConstantProjection) : D.localizers.Sobolev 0 →L[ℝ] ℂ :=
P.coefficient ∘L D.poissonInput m hm he P
lemma poissonRaw_equation (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
(P : D.localizers.ConstantProjection) (f : D.localizers.Sobolev 0) :
D.completedL (D.poissonRaw m hm he P f) = f + D.localizers.constants (D.poissonResidue m hm he P f) := by
have h := D.completedResolvent_equation m hm he (D.poissonInput m hm he P f)
have hw := D.poissonInput_equation m hm he P f
change (m^2 : ℝ) • D.resolventZero m hm he (D.poissonInput m hm he P f) -
D.completedL (D.poissonRaw m hm he P f) = D.poissonInput m hm he P f at h
change _ = f + D.localizers.constants (P.coefficient (D.poissonInput m hm he P f))
rw [P.coefficient_spec]
have hh := h.trans hw
apply sub_right_inj.mp
calc
(m^2 : ℝ) • D.resolventZero m hm he (D.poissonInput m hm he P f) -
D.completedL (D.poissonRaw m hm he P f) = _ := hh
_ = (m^2 : ℝ) • D.resolventZero m hm he (D.poissonInput m hm he P f) -
(f + P.proj (D.poissonInput m hm he P f)) := by abel
end GluingData
end GlobalElliptic
end
end
end OAI
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