| import OAI.Geometry.Kahler.BasePotentialSmooth |
| |
| namespace OAI |
|
|
| universe uKahler10148_1 uKahler10148_2 uKahler10162_1 uKahler10162_2 uKahler10184_1 uKahler10184_2 |
|
|
| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open scoped ContDiff ENNReal |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| open Set Filter Topology |
| open scoped ContDiff |
| noncomputable section |
|
|
| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
| |
| lemma local_jet_series_derivative_on {E : Type uKahler10148_1} {F : Type uKahler10148_2} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] |
| {f : ℕ → E → F} {s : Set E} (hf : ∀ j y, y ∈ s → ContDiffAt ℝ ∞ (f j) y) |
| (hs : IsOpen s) (hc : IsPreconnected s) {x : E} (hx : x ∈ s) |
| {v : ℕ → ℕ → ℝ} (hv : ∀ n, Summable (v n)) |
| (hb : ∀ n j y, y ∈ s → ‖iteratedFDeriv ℝ n (f j) y‖ ≤ v n j) (n : ℕ) : |
| HasFDerivAt (fun y => ∑' j, iteratedFDeriv ℝ n (f j) y) |
| (∑' j, fderiv ℝ (iteratedFDeriv ℝ n (f j)) x) x := by |
| apply hasFDerivAt_tsum_of_isPreconnected (hv (n+1)) hs hc |
| (fun j y hy => ((hf j y hy).differentiableAt_iteratedFDeriv (by exact_mod_cast (ENat.natCast_lt_top n))).hasFDerivAt) |
| (fun j y hy => ?_) hx |
| (Summable.of_norm_bounded (hv n) (fun j => hb n j x hx)) hx |
| simpa only [norm_fderiv_iteratedFDeriv] using hb (n+1) j y hy |
|
|
| lemma local_iteratedFDeriv_tsum_on {E : Type uKahler10162_1} {F : Type uKahler10162_2} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] |
| {f : ℕ → E → F} {s : Set E} (hf : ∀ j y, y ∈ s → ContDiffAt ℝ ∞ (f j) y) |
| (hs : IsOpen s) (hc : IsPreconnected s) |
| {v : ℕ → ℕ → ℝ} (hv : ∀ n, Summable (v n)) |
| (hb : ∀ n j y, y ∈ s → ‖iteratedFDeriv ℝ n (f j) y‖ ≤ v n j) (n : ℕ) : |
| EqOn (iteratedFDeriv ℝ n (fun y => ∑' j, f j y)) |
| (fun y => ∑' j, iteratedFDeriv ℝ n (f j) y) s := by |
| induction n with |
| | zero => |
| intro x hx |
| simp_rw [iteratedFDeriv_zero_eq_comp] |
| exact (continuousMultilinearCurryFin0 ℝ E F).symm.toContinuousLinearEquiv.map_tsum |
| | succ n ih => |
| intro x hx |
| have he : iteratedFDeriv ℝ n (fun y => ∑' j, f j y) =ᶠ[𝓝 x] |
| (fun y => ∑' j, iteratedFDeriv ℝ n (f j) y) := |
| Filter.eventuallyEq_of_mem (hs.mem_nhds hx) (fun y hy => ih hy) |
| simp_rw [iteratedFDeriv_succ_eq_comp_left,Function.comp_apply] |
| rw [he.fderiv_eq,(local_jet_series_derivative_on hf hs hc hx hv hb n).fderiv] |
| exact (continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (n+1) => E) F).symm.toContinuousLinearEquiv.map_tsum |
|
|
| lemma local_contDiff_tsum_on {E : Type uKahler10184_1} {F : Type uKahler10184_2} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] |
| {f : ℕ → E → F} {s : Set E} (hf : ∀ j y, y ∈ s → ContDiffAt ℝ ∞ (f j) y) |
| (hs : IsOpen s) (hc : IsPreconnected s) |
| {v : ℕ → ℕ → ℝ} (hv : ∀ n, Summable (v n)) |
| (hb : ∀ n j y, y ∈ s → ‖iteratedFDeriv ℝ n (f j) y‖ ≤ v n j) : |
| ContDiffOn ℝ ∞ (fun y => ∑' j, f j y) s := by |
| apply contDiffOn_of_differentiableOn |
| intro n hn x hx |
| have he : iteratedFDerivWithin ℝ n (fun y => ∑' j, f j y) s =ᶠ[𝓝 x] |
| (fun y => ∑' j, iteratedFDeriv ℝ n (f j) y) := by |
| filter_upwards [hs.mem_nhds hx] with y hy |
| rw [iteratedFDerivWithin_of_isOpen n hs hy] |
| exact local_iteratedFDeriv_tsum_on hf hs hc hv hb n hy |
| exact ((local_jet_series_derivative_on hf hs hc hx hv hb n).differentiableAt.congr_of_eventuallyEq he).differentiableWithinAt |
| |
| end PinchedHartogs.BaseConstruction |
| |
| end |
| |
| end OAI |
| |
