Coarse-graining support: #
Support.Deterministic.CoarseCaccioppoli.SingleCubeToRaw.HarmonicQuantitativeCutoff.Standard
Imported from the Apache-2.0 CoarseGraining development at commit
c7ddd76c08ade64fed1b8d2ca51be14dfee8deb4.
theorem
HCPolySupport.coarseCaccioppoli_boundary_qone_of_quantitativeCutoff_of_of_heightChoice
{d : ℕ}
[NeZero d]
(Q : TriadicCube d)
(a : CoeffField d)
(s t C uL2Sq : ℝ)
{lam Lam : ℝ}
(k : ℝ → ℝ → ℕ)
{F : ℝ → ℝ}
(w : ℝ → ℝ → AHarmonicFunction a (openCubeSet Q))
(g : ℝ → ℝ → Vec d → ℝ)
(η : (ρ₁ ρ₂ : ℝ) → QuantitativeCubeCutoff Q ρ₁ ρ₂)
(Acirc1 AcircS U A1 AS : ℝ → ℝ → ℝ)
(hC : 0 ≤ C)
(hs : 0 < s)
(ht : 0 < t)
(hst : s + t < 1)
(hu : 0 ≤ uL2Sq)
(hnonneg : ∀ ⦃ρ : ℝ⦄, 1 / 3 ≤ ρ → ρ ≤ 1 → 0 ≤ F ρ)
(hbounded : CoarseCaccioppoliRadiusBoundedAbove F)
(hscale : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → CoarseCaccioppoliTriadicGapScaleChoice (k ρ₁ ρ₂) ρ₁ ρ₂)
(hlower :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
F ρ₁ ≤ cubeAverage Q fun (x : Vec d) => (η ρ₁ ρ₂).toFun x * scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x)
(hη_tsupport : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → tsupport (η ρ₁ ρ₂).toFun ⊆ openCubeSet Q)
(henergyAvg :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ → ρ₂ ≤ 1 → (cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x) = F ρ₂)
(hfluxMem :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
MeasureTheory.MemLp (fun (x : Vec d) => matVecMul (a x) ((w ρ₁ ρ₂).toH1.grad x)) 2 (normalizedCubeMeasure Q))
(huMem :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ → ρ₂ ≤ 1 → MeasureTheory.MemLp (fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) 2 (normalizedCubeMeasure Q))
(hgMem : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → MeasureTheory.MemLp (g ρ₁ ρ₂) 2 (normalizedCubeMeasure Q))
(hfluxEnergy :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
CoarseCaccioppoliFluxEnergyControls Q a s (fun (x : Vec d) => matVecMul (a x) ((w ρ₁ ρ₂).toH1.grad x))
fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x)
(hBgConst :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
0 ≤ coarseCaccioppoliConstantCutoffSize Q (fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x)
(scalarCutoffGradientField (η ρ₁ ρ₂).toFun) (coarseCaccioppoliQuantitativeCutoffHessianBound Q ρ₁ ρ₂))
(hBgCent :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
0 ≤ coarseCaccioppoliCenteredCutoffSize Q s (scalarCutoffGradientField (η ρ₁ ρ₂).toFun) (Acirc1 ρ₁ ρ₂)
(AcircS ρ₁ ρ₂) (√(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(coarseCaccioppoliQuantitativeCutoffHessianBound Q ρ₁ ρ₂) C)
(hAcirc1_nonneg : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → 0 ≤ Acirc1 ρ₁ ρ₂)
(hAcircS_nonneg : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → 0 ≤ AcircS ρ₁ ρ₂)
(hproj :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
CubeDescendantProjectedDualMeanZeroPoincareEstimate Q C
(cubeFluctuation Q fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) (g ρ₁ ρ₂) N)
(hgCirc1 :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
cubeBesovCircPartialNorm Q 1 2 1 N (g ρ₁ ρ₂) ≤ Acirc1 ρ₁ ρ₂ * √(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(hgCircS :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
cubeBesovCircPartialNorm Q (1 - s) 2 1 N (g ρ₁ ρ₂) ≤ AcircS ρ₁ ρ₂ * √(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(hU :
∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → (cubeLpNorm Q 2 fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) ≤ U ρ₁ ρ₂)
(hA1 : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → Acirc1 ρ₁ ρ₂ ≤ A1 ρ₁ ρ₂)
(hAS : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → AcircS ρ₁ ρ₂ ≤ AS ρ₁ ρ₂)
(hcoeff :
CoarseCaccioppoliBoundaryRadiusEnergyBridgeCanonicalCoefficientBounds Q a s C uL2Sq (fun (ρ₁ ρ₂ : ℝ) => ↑(k ρ₁ ρ₂))
(coarseCaccioppoliBoundaryLocalizedExplicitHeightOfScaleChoice Q a s t C k) U
(coarseCaccioppoliQuantitativeCutoffGradientBound Q) (coarseCaccioppoliQuantitativeCutoffHessianBound Q) A1 AS)
(hEll : IsEllipticFieldOn lam Lam (openCubeSet Q) a)
(hData :
∀ j ≤ Q.scale,
∀ R ∈ descendantsAtScale Q j,
∃ (sigmaR : Mat d) (sigmaStarR : Mat d) (kappaR : Mat d),
IsCoarseBlockMatrix (openCubeSet R) a (deterministicCoarseBlockMatrix (openCubeSet R) a) ∧ IsSigmaStarCoarse (openCubeSet R) a sigmaStarR ∧ IsKappaCoarse (openCubeSet R) a sigmaStarR kappaR ∧ IsSigmaCoarse (openCubeSet R) a sigmaR sigmaStarR kappaR ∧ IsUnit (Matrix.det sigmaStarR))
(hBsum_s :
Summable fun (n : ℕ) => geometricWeight s 1 n * (maxDescendantBBlockNormAtScale Q (Q.scale - ↑n) a).rpow (1 / 2))
(hSigmaSum_t :
Summable fun (n : ℕ) =>
geometricWeight t 1 n * (maxDescendantSigmaStarInvNormAtScale Q (Q.scale - ↑n) a).rpow (1 / 2))
:
Boundary coarse Caccioppoli for the actual harmonic family, with the testing inequality discharged by the weak-testing bridge and a quantitative cutoff family.
theorem
HCPolySupport.coarseCaccioppoli_boundary_qone_of_quantitativeCutoff_of_of_outerRadiusBelowOne
{d : ℕ}
[NeZero d]
(Q : TriadicCube d)
(a : CoeffField d)
(s t C uL2Sq : ℝ)
{lam Lam : ℝ}
(k : ℝ → ℝ → ℕ)
{F : ℝ → ℝ}
(w : ℝ → ℝ → AHarmonicFunction a (openCubeSet Q))
(g : ℝ → ℝ → Vec d → ℝ)
(η : (ρ₁ ρ₂ : ℝ) → QuantitativeCubeCutoff Q ρ₁ ρ₂)
(Acirc1 AcircS U A1 AS : ℝ → ℝ → ℝ)
(hC : 0 ≤ C)
(hs : 0 < s)
(ht : 0 < t)
(hst : s + t < 1)
(hu : 0 ≤ uL2Sq)
(hnonneg : ∀ ⦃ρ : ℝ⦄, 1 / 3 ≤ ρ → ρ ≤ 1 → 0 ≤ F ρ)
(hbounded : CoarseCaccioppoliRadiusBoundedAbove F)
(hscale : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → CoarseCaccioppoliTriadicGapScaleChoice (k ρ₁ ρ₂) ρ₁ ρ₂)
(hlower :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
F ρ₁ ≤ cubeAverage Q fun (x : Vec d) => (η ρ₁ ρ₂).toFun x * scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x)
(houter : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → ρ₂ < 1)
(henergyAvg :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ → ρ₂ ≤ 1 → (cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x) = F ρ₂)
(hfluxMem :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
MeasureTheory.MemLp (fun (x : Vec d) => matVecMul (a x) ((w ρ₁ ρ₂).toH1.grad x)) 2 (normalizedCubeMeasure Q))
(huMem :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ → ρ₂ ≤ 1 → MeasureTheory.MemLp (fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) 2 (normalizedCubeMeasure Q))
(hgMem : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → MeasureTheory.MemLp (g ρ₁ ρ₂) 2 (normalizedCubeMeasure Q))
(hfluxEnergy :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
CoarseCaccioppoliFluxEnergyControls Q a s (fun (x : Vec d) => matVecMul (a x) ((w ρ₁ ρ₂).toH1.grad x))
fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x)
(hBgConst :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
0 ≤ coarseCaccioppoliConstantCutoffSize Q (fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x)
(scalarCutoffGradientField (η ρ₁ ρ₂).toFun) (coarseCaccioppoliQuantitativeCutoffHessianBound Q ρ₁ ρ₂))
(hBgCent :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
0 ≤ coarseCaccioppoliCenteredCutoffSize Q s (scalarCutoffGradientField (η ρ₁ ρ₂).toFun) (Acirc1 ρ₁ ρ₂)
(AcircS ρ₁ ρ₂) (√(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(coarseCaccioppoliQuantitativeCutoffHessianBound Q ρ₁ ρ₂) C)
(hAcirc1_nonneg : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → 0 ≤ Acirc1 ρ₁ ρ₂)
(hAcircS_nonneg : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → 0 ≤ AcircS ρ₁ ρ₂)
(hproj :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
CubeDescendantProjectedDualMeanZeroPoincareEstimate Q C
(cubeFluctuation Q fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) (g ρ₁ ρ₂) N)
(hgCirc1 :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
cubeBesovCircPartialNorm Q 1 2 1 N (g ρ₁ ρ₂) ≤ Acirc1 ρ₁ ρ₂ * √(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(hgCircS :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
cubeBesovCircPartialNorm Q (1 - s) 2 1 N (g ρ₁ ρ₂) ≤ AcircS ρ₁ ρ₂ * √(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(hU :
∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → (cubeLpNorm Q 2 fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) ≤ U ρ₁ ρ₂)
(hA1 : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → Acirc1 ρ₁ ρ₂ ≤ A1 ρ₁ ρ₂)
(hAS : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → AcircS ρ₁ ρ₂ ≤ AS ρ₁ ρ₂)
(hcoeff :
CoarseCaccioppoliBoundaryRadiusEnergyBridgeCanonicalCoefficientBounds Q a s C uL2Sq (fun (ρ₁ ρ₂ : ℝ) => ↑(k ρ₁ ρ₂))
(coarseCaccioppoliBoundaryLocalizedExplicitHeightOfScaleChoice Q a s t C k) U
(coarseCaccioppoliQuantitativeCutoffGradientBound Q) (coarseCaccioppoliQuantitativeCutoffHessianBound Q) A1 AS)
(hEll : IsEllipticFieldOn lam Lam (openCubeSet Q) a)
(hData :
∀ j ≤ Q.scale,
∀ R ∈ descendantsAtScale Q j,
∃ (sigmaR : Mat d) (sigmaStarR : Mat d) (kappaR : Mat d),
IsCoarseBlockMatrix (openCubeSet R) a (deterministicCoarseBlockMatrix (openCubeSet R) a) ∧ IsSigmaStarCoarse (openCubeSet R) a sigmaStarR ∧ IsKappaCoarse (openCubeSet R) a sigmaStarR kappaR ∧ IsSigmaCoarse (openCubeSet R) a sigmaR sigmaStarR kappaR ∧ IsUnit (Matrix.det sigmaStarR))
(hBsum_s :
Summable fun (n : ℕ) => geometricWeight s 1 n * (maxDescendantBBlockNormAtScale Q (Q.scale - ↑n) a).rpow (1 / 2))
(hSigmaSum_t :
Summable fun (n : ℕ) =>
geometricWeight t 1 n * (maxDescendantSigmaStarInvNormAtScale Q (Q.scale - ↑n) a).rpow (1 / 2))
:
Boundary coarse Caccioppoli for the actual harmonic family, with cutoff
support discharged from the strict outer-radius condition ρ₂ < 1.
theorem
HCPolySupport.coarseCaccioppoli_interior_qone_of_quantitativeCutoff_of_of_radiusAgreement
{d : ℕ}
[NeZero d]
(Q : TriadicCube d)
(a : CoeffField d)
(s t C uL2Sq : ℝ)
{lam Lam : ℝ}
(k : ℝ → ℝ → ℕ)
{F G₀ : ℝ → ℝ}
(w : ℝ → ℝ → AHarmonicFunction a (openCubeSet Q))
(g : ℝ → ℝ → Vec d → ℝ)
(η : (ρ₁ ρ₂ : ℝ) → QuantitativeCubeCutoff Q ρ₁ ρ₂)
(Acirc1 AcircS U A1 AS : ℝ → ℝ → ℝ)
(hC : 0 ≤ C)
(hs : 0 < s)
(ht : 0 < t)
(hst : s + t < 1)
(hu : 0 ≤ uL2Sq)
(hagree : CoarseCaccioppoliRadiusAgreement F G₀)
(hG_nonneg : ∀ ⦃ρ : ℝ⦄, 1 / 3 ≤ ρ → ρ ≤ 1 → 0 ≤ G₀ ρ)
(hG_bounded : CoarseCaccioppoliRadiusBoundedAbove G₀)
(hscale : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → CoarseCaccioppoliTriadicGapScaleChoice (k ρ₁ ρ₂) ρ₁ ρ₂)
(hlower :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
G₀ ρ₁ ≤ cubeAverage Q fun (x : Vec d) => (η ρ₁ ρ₂).toFun x * scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x)
(hη_tsupport : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → tsupport (η ρ₁ ρ₂).toFun ⊆ openCubeSet Q)
(henergyAvg :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ → ρ₂ ≤ 1 → (cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x) = G₀ ρ₂)
(hfluxMem :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
MeasureTheory.MemLp (fun (x : Vec d) => matVecMul (a x) ((w ρ₁ ρ₂).toH1.grad x)) 2 (normalizedCubeMeasure Q))
(huMem :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ → ρ₂ ≤ 1 → MeasureTheory.MemLp (fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) 2 (normalizedCubeMeasure Q))
(hgMem : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → MeasureTheory.MemLp (g ρ₁ ρ₂) 2 (normalizedCubeMeasure Q))
(hfluxEnergy :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
CoarseCaccioppoliFluxEnergyControls Q a s (fun (x : Vec d) => matVecMul (a x) ((w ρ₁ ρ₂).toH1.grad x))
fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x)
(hBgConst :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
0 ≤ coarseCaccioppoliConstantCutoffSize Q (fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x)
(scalarCutoffGradientField (η ρ₁ ρ₂).toFun) (coarseCaccioppoliQuantitativeCutoffHessianBound Q ρ₁ ρ₂))
(hBgCent :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
0 ≤ coarseCaccioppoliCenteredCutoffSize Q s (scalarCutoffGradientField (η ρ₁ ρ₂).toFun) (Acirc1 ρ₁ ρ₂)
(AcircS ρ₁ ρ₂) (√(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(coarseCaccioppoliQuantitativeCutoffHessianBound Q ρ₁ ρ₂) C)
(hAcirc1_nonneg : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → 0 ≤ Acirc1 ρ₁ ρ₂)
(hAcircS_nonneg : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → 0 ≤ AcircS ρ₁ ρ₂)
(hproj :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
CubeDescendantProjectedDualMeanZeroPoincareEstimate Q C
(cubeFluctuation Q fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) (g ρ₁ ρ₂) N)
(hgCirc1 :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
cubeBesovCircPartialNorm Q 1 2 1 N (g ρ₁ ρ₂) ≤ Acirc1 ρ₁ ρ₂ * √(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(hgCircS :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
cubeBesovCircPartialNorm Q (1 - s) 2 1 N (g ρ₁ ρ₂) ≤ AcircS ρ₁ ρ₂ * √(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(hU :
∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → (cubeLpNorm Q 2 fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) ≤ U ρ₁ ρ₂)
(hA1 : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → Acirc1 ρ₁ ρ₂ ≤ A1 ρ₁ ρ₂)
(hAS : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → AcircS ρ₁ ρ₂ ≤ AS ρ₁ ρ₂)
(hcoeff :
CoarseCaccioppoliBoundaryRadiusEnergyBridgeCanonicalCoefficientBounds Q a s C uL2Sq (fun (ρ₁ ρ₂ : ℝ) => ↑(k ρ₁ ρ₂))
(coarseCaccioppoliBoundaryLocalizedExplicitHeightOfScaleChoice Q a s t C k) U
(coarseCaccioppoliQuantitativeCutoffGradientBound Q) (coarseCaccioppoliQuantitativeCutoffHessianBound Q) A1 AS)
(hEll : IsEllipticFieldOn lam Lam (openCubeSet Q) a)
(hData :
∀ j ≤ Q.scale,
∀ R ∈ descendantsAtScale Q j,
∃ (sigmaR : Mat d) (sigmaStarR : Mat d) (kappaR : Mat d),
IsCoarseBlockMatrix (openCubeSet R) a (deterministicCoarseBlockMatrix (openCubeSet R) a) ∧ IsSigmaStarCoarse (openCubeSet R) a sigmaStarR ∧ IsKappaCoarse (openCubeSet R) a sigmaStarR kappaR ∧ IsSigmaCoarse (openCubeSet R) a sigmaR sigmaStarR kappaR ∧ IsUnit (Matrix.det sigmaStarR))
(hBsum_s :
Summable fun (n : ℕ) => geometricWeight s 1 n * (maxDescendantBBlockNormAtScale Q (Q.scale - ↑n) a).rpow (1 / 2))
(hSigmaSum_t :
Summable fun (n : ℕ) =>
geometricWeight t 1 n * (maxDescendantSigmaStarInvNormAtScale Q (Q.scale - ↑n) a).rpow (1 / 2))
:
Interior coarse Caccioppoli for the actual harmonic family, transported across the radius agreement used for the centered quantity.
theorem
HCPolySupport.coarseCaccioppoli_interior_qone_of_quantitativeCutoff_of_of_outerRadiusBelowOne
{d : ℕ}
[NeZero d]
(Q : TriadicCube d)
(a : CoeffField d)
(s t C uL2Sq : ℝ)
{lam Lam : ℝ}
(k : ℝ → ℝ → ℕ)
{F G₀ : ℝ → ℝ}
(w : ℝ → ℝ → AHarmonicFunction a (openCubeSet Q))
(g : ℝ → ℝ → Vec d → ℝ)
(η : (ρ₁ ρ₂ : ℝ) → QuantitativeCubeCutoff Q ρ₁ ρ₂)
(Acirc1 AcircS U A1 AS : ℝ → ℝ → ℝ)
(hC : 0 ≤ C)
(hs : 0 < s)
(ht : 0 < t)
(hst : s + t < 1)
(hu : 0 ≤ uL2Sq)
(hagree : CoarseCaccioppoliRadiusAgreement F G₀)
(hG_nonneg : ∀ ⦃ρ : ℝ⦄, 1 / 3 ≤ ρ → ρ ≤ 1 → 0 ≤ G₀ ρ)
(hG_bounded : CoarseCaccioppoliRadiusBoundedAbove G₀)
(hscale : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → CoarseCaccioppoliTriadicGapScaleChoice (k ρ₁ ρ₂) ρ₁ ρ₂)
(hlower :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
G₀ ρ₁ ≤ cubeAverage Q fun (x : Vec d) => (η ρ₁ ρ₂).toFun x * scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x)
(houter : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → ρ₂ < 1)
(henergyAvg :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ → ρ₂ ≤ 1 → (cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x) = G₀ ρ₂)
(hfluxMem :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
MeasureTheory.MemLp (fun (x : Vec d) => matVecMul (a x) ((w ρ₁ ρ₂).toH1.grad x)) 2 (normalizedCubeMeasure Q))
(huMem :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ → ρ₂ ≤ 1 → MeasureTheory.MemLp (fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) 2 (normalizedCubeMeasure Q))
(hgMem : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → MeasureTheory.MemLp (g ρ₁ ρ₂) 2 (normalizedCubeMeasure Q))
(hfluxEnergy :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
CoarseCaccioppoliFluxEnergyControls Q a s (fun (x : Vec d) => matVecMul (a x) ((w ρ₁ ρ₂).toH1.grad x))
fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x)
(hBgConst :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
0 ≤ coarseCaccioppoliConstantCutoffSize Q (fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x)
(scalarCutoffGradientField (η ρ₁ ρ₂).toFun) (coarseCaccioppoliQuantitativeCutoffHessianBound Q ρ₁ ρ₂))
(hBgCent :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
0 ≤ coarseCaccioppoliCenteredCutoffSize Q s (scalarCutoffGradientField (η ρ₁ ρ₂).toFun) (Acirc1 ρ₁ ρ₂)
(AcircS ρ₁ ρ₂) (√(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(coarseCaccioppoliQuantitativeCutoffHessianBound Q ρ₁ ρ₂) C)
(hAcirc1_nonneg : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → 0 ≤ Acirc1 ρ₁ ρ₂)
(hAcircS_nonneg : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → 0 ≤ AcircS ρ₁ ρ₂)
(hproj :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
CubeDescendantProjectedDualMeanZeroPoincareEstimate Q C
(cubeFluctuation Q fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) (g ρ₁ ρ₂) N)
(hgCirc1 :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
cubeBesovCircPartialNorm Q 1 2 1 N (g ρ₁ ρ₂) ≤ Acirc1 ρ₁ ρ₂ * √(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(hgCircS :
∀ ⦃ρ₁ ρ₂ : ℝ⦄,
1 / 3 ≤ ρ₁ →
ρ₁ < ρ₂ →
ρ₂ ≤ 1 →
∀ (N : ℕ),
cubeBesovCircPartialNorm Q (1 - s) 2 1 N (g ρ₁ ρ₂) ≤ AcircS ρ₁ ρ₂ * √(cubeAverage Q fun (x : Vec d) => scalarVariationEnergyIntegrand a (w ρ₁ ρ₂) x))
(hU :
∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → (cubeLpNorm Q 2 fun (x : Vec d) => (w ρ₁ ρ₂).toH1.toFun x) ≤ U ρ₁ ρ₂)
(hA1 : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → Acirc1 ρ₁ ρ₂ ≤ A1 ρ₁ ρ₂)
(hAS : ∀ ⦃ρ₁ ρ₂ : ℝ⦄, 1 / 3 ≤ ρ₁ → ρ₁ < ρ₂ → ρ₂ ≤ 1 → AcircS ρ₁ ρ₂ ≤ AS ρ₁ ρ₂)
(hcoeff :
CoarseCaccioppoliBoundaryRadiusEnergyBridgeCanonicalCoefficientBounds Q a s C uL2Sq (fun (ρ₁ ρ₂ : ℝ) => ↑(k ρ₁ ρ₂))
(coarseCaccioppoliBoundaryLocalizedExplicitHeightOfScaleChoice Q a s t C k) U
(coarseCaccioppoliQuantitativeCutoffGradientBound Q) (coarseCaccioppoliQuantitativeCutoffHessianBound Q) A1 AS)
(hEll : IsEllipticFieldOn lam Lam (openCubeSet Q) a)
(hData :
∀ j ≤ Q.scale,
∀ R ∈ descendantsAtScale Q j,
∃ (sigmaR : Mat d) (sigmaStarR : Mat d) (kappaR : Mat d),
IsCoarseBlockMatrix (openCubeSet R) a (deterministicCoarseBlockMatrix (openCubeSet R) a) ∧ IsSigmaStarCoarse (openCubeSet R) a sigmaStarR ∧ IsKappaCoarse (openCubeSet R) a sigmaStarR kappaR ∧ IsSigmaCoarse (openCubeSet R) a sigmaR sigmaStarR kappaR ∧ IsUnit (Matrix.det sigmaStarR))
(hBsum_s :
Summable fun (n : ℕ) => geometricWeight s 1 n * (maxDescendantBBlockNormAtScale Q (Q.scale - ↑n) a).rpow (1 / 2))
(hSigmaSum_t :
Summable fun (n : ℕ) =>
geometricWeight t 1 n * (maxDescendantSigmaStarInvNormAtScale Q (Q.scale - ↑n) a).rpow (1 / 2))
:
Interior coarse Caccioppoli for the actual harmonic family, with cutoff
support discharged from the strict outer-radius condition ρ₂ < 1.