| import Mathlib |
| import OAI.Geometry.PrescribedPotential.GlobalSobolev |
| import OAI.Geometry.PrescribedRicci.KahlerCofactorDivergence |
|
|
| namespace OAI |
|
|
| /-! Complex Integration By Parts. -/ |
|
|
| noncomputable section |
| open Matrix Filter Set Topology MeasureTheory |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise |
| namespace Anticanonical.SourceSmooth |
| namespace KaehlerMetric |
| variable {d : ℕ} |
|
|
| def barDeriv (f : Coordinates d → ℂ) (z : Coordinates d) (j : Fin d) : ℂ := |
| (2 : ℂ)⁻¹ * (fderiv ℝ f z (coordinateVector j) + |
| Complex.I * fderiv ℝ f z (Complex.I • coordinateVector j)) |
|
|
| def holDeriv (f : Coordinates d → ℂ) (z : Coordinates d) (j : Fin d) : ℂ := |
| (2 : ℂ)⁻¹ * (fderiv ℝ f z (coordinateVector j) - |
| Complex.I * fderiv ℝ f z (Complex.I • coordinateVector j)) |
|
|
| lemma contDiff_deriv_direction {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} |
| (hf : ContDiff ℝ ∞ f) (v : E) : ContDiff ℝ ∞ (fun z => fderiv ℝ f z v) := |
| (hf.fderiv_right (by simp)).clm_apply contDiff_const |
|
|
| lemma contDiff_barDeriv {f : Coordinates d → ℂ} (hf : ContDiff ℝ ∞ f) (j : Fin d) : |
| ContDiff ℝ ∞ (fun z => barDeriv f z j) := |
| contDiff_const.mul ((contDiff_deriv_direction hf _).add |
| (contDiff_const.mul (contDiff_deriv_direction hf _))) |
|
|
| lemma contDiff_holDeriv {f : Coordinates d → ℂ} (hf : ContDiff ℝ ∞ f) (j : Fin d) : |
| ContDiff ℝ ∞ (fun z => holDeriv f z j) := |
| contDiff_const.mul ((contDiff_deriv_direction hf _).sub |
| (contDiff_const.mul (contDiff_deriv_direction hf _))) |
|
|
| lemma compactSupport_barDeriv {f : Coordinates d → ℂ} (hf : HasCompactSupport f) |
| (j : Fin d) : HasCompactSupport (fun z => barDeriv f z j) := by |
| exact ((hf.fderiv_apply ℝ (coordinateVector j)).add |
| (hf.fderiv_apply ℝ (Complex.I • coordinateVector j)).mul_left).mul_left |
|
|
| lemma integrable_mul_deriv {f g : Coordinates d → ℂ} |
| (hf : ContDiff ℝ ∞ f) (hg : ContDiff ℝ ∞ g) (hc : HasCompactSupport f) |
| (v : Coordinates d) : Integrable (fun z => f z * fderiv ℝ g z v) := |
| (hf.continuous.mul (contDiff_deriv_direction hg v).continuous).integrable_of_hasCompactSupport |
| hc.mul_right |
|
|
| lemma integrable_deriv_mul {f g : Coordinates d → ℂ} |
| (hf : ContDiff ℝ ∞ f) (hg : ContDiff ℝ ∞ g) (hc : HasCompactSupport f) |
| (v : Coordinates d) : Integrable (fun z => fderiv ℝ f z v * g z) := |
| ((contDiff_deriv_direction hf v).continuous.mul hg.continuous).integrable_of_hasCompactSupport |
| (hc.fderiv_apply ℝ v).mul_right |
|
|
| lemma contDiffAt_deriv_direction {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {z : E} |
| (hf : ContDiffAt ℝ ∞ f z) (v : E) : ContDiffAt ℝ ∞ (fun y => fderiv ℝ f y v) z := |
| (hf.fderiv_right (by simp)).clm_apply contDiffAt_const |
|
|
| lemma continuous_mul_on_tsupport {E : Type*} [TopologicalSpace E] {f g : E → ℂ} |
| (hf : Continuous f) (hg : ∀ z ∈ tsupport f, ContinuousAt g z) : |
| Continuous (fun z => f z * g z) := by |
| rw [continuous_iff_continuousAt] |
| intro z |
| by_cases hz : z ∈ tsupport f |
| · exact hf.continuousAt.mul (hg z hz) |
| · apply (continuousAt_const (y := (0 : ℂ))).congr |
| filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hz] with y hy |
| simp only [hy, zero_mul, Pi.zero_apply] |
|
|
| lemma contDiff_mul_on_tsupport {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {f g : E → ℂ} (hf : ContDiff ℝ ∞ f) |
| (hg : ∀ z ∈ tsupport f, ContDiffAt ℝ ∞ g z) : |
| ContDiff ℝ ∞ (fun z => f z * g z) := by |
| rw [contDiff_iff_contDiffAt] |
| intro z |
| by_cases hz : z ∈ tsupport f |
| · exact hf.contDiffAt.mul (hg z hz) |
| · apply (contDiffAt_const (c := (0 : ℂ))).congr_of_eventuallyEq |
| filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hz] with y hy |
| simp only [hy, zero_mul, Pi.zero_apply] |
|
|
| lemma integrable_mul_deriv_local {f g : Coordinates d → ℂ} |
| (hf : ContDiff ℝ ∞ f) (hg : ∀ z ∈ tsupport f, ContDiffAt ℝ ∞ g z) |
| (hc : HasCompactSupport f) (v : Coordinates d) : |
| Integrable (fun z => f z * fderiv ℝ g z v) := |
| (continuous_mul_on_tsupport hf.continuous (fun z hz => |
| (contDiffAt_deriv_direction (hg z hz) v).continuousAt)).integrable_of_hasCompactSupport |
| hc.mul_right |
|
|
| lemma integrable_deriv_mul_local {f g : Coordinates d → ℂ} |
| (hf : ContDiff ℝ ∞ f) (hg : ∀ z ∈ tsupport f, ContDiffAt ℝ ∞ g z) |
| (hc : HasCompactSupport f) (v : Coordinates d) : |
| Integrable (fun z => fderiv ℝ f z v * g z) := |
| (continuous_mul_on_tsupport (contDiff_deriv_direction hf v).continuous (fun z hz => |
| (hg z (tsupport_fderiv_apply_subset ℝ v hz)).continuousAt)).integrable_of_hasCompactSupport |
| (hc.fderiv_apply ℝ v).mul_right |
|
|
| lemma integral_mul_barDeriv_local {f g : Coordinates d → ℂ} |
| (hf : ContDiff ℝ ∞ f) (hg : ∀ z ∈ tsupport f, ContDiffAt ℝ ∞ g z) |
| (hc : HasCompactSupport f) (j : Fin d) : |
| (∫ z, f z * barDeriv g z j) = -(∫ z, barDeriv f z j * g z) := by |
| have hv (v : Coordinates d) : (∫ z, f z * fderiv ℝ g z v) = |
| -(∫ z, fderiv ℝ f z v * g z) := |
| integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable |
| (integrable_deriv_mul_local hf hg hc v) (integrable_mul_deriv_local hf hg hc v) |
| ((continuous_mul_on_tsupport hf.continuous (fun z hz => (hg z hz).continuousAt)).integrable_of_hasCompactSupport hc.mul_right) |
| (fun z _ => hf.differentiable (by simp) z) |
| (fun z hz => (hg z hz).differentiableAt (by simp)) |
| have he (z : Coordinates d) : f z * barDeriv g z j = |
| (2 : ℂ)⁻¹ * (f z * fderiv ℝ g z (coordinateVector j)) + |
| ((2 : ℂ)⁻¹ * Complex.I) * (f z * fderiv ℝ g z (Complex.I • coordinateVector j)) := by |
| unfold barDeriv |
| ring |
| have he' (z : Coordinates d) : barDeriv f z j * g z = |
| (2 : ℂ)⁻¹ * (fderiv ℝ f z (coordinateVector j) * g z) + |
| ((2 : ℂ)⁻¹ * Complex.I) * (fderiv ℝ f z (Complex.I • coordinateVector j) * g z) := by |
| unfold barDeriv |
| ring |
| simp_rw [he, he'] |
| rw [integral_add ((integrable_mul_deriv_local hf hg hc _).const_mul _) ((integrable_mul_deriv_local hf hg hc _).const_mul _), |
| integral_add ((integrable_deriv_mul_local hf hg hc _).const_mul _) ((integrable_deriv_mul_local hf hg hc _).const_mul _)] |
| simp only [integral_const_mul, hv] |
| ring |
|
|
| end KaehlerMetric |
| end Anticanonical.SourceSmooth |
|
|
| end |
|
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| end OAI |
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