openAI-math-dataset / lean /OAI /Geometry /SmoothYau /Smoothness /ContDiffPartialIteratedFDeriv.lean
| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Function | |
| open scoped Topology ContDiff Manifold SchwartzMap | |
| open FourierTransform TemperedDistribution MeasureTheory | |
| open scoped SchwartzMap ENNReal Real Laplacian BoundedContinuousFunction | |
| open MeasureTheory FourierTransform TemperedDistribution | |
| open scoped SchwartzMap BoundedContinuousFunction Real ENNReal ContDiff | |
| open MeasureTheory | |
| open scoped ENNReal | |
| open Set Filter | |
| open scoped SchwartzMap ContDiff Topology | |
| namespace YauCounterexamples | |
| section SmoothFamily | |
| variable {A E F : Type*} [NormedAddCommGroup A] [NormedSpace ℝ A] | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| lemma contDiff_partial_iteratedFDeriv (f : A → E → F) | |
| (hf : ContDiff ℝ ∞ (Function.uncurry f)) (n : ℕ) : | |
| ContDiff ℝ ∞ (fun p : A × E => iteratedFDeriv ℝ n (f p.1) p.2) := by | |
| induction n with | |
| | zero => | |
| simpa only [iteratedFDeriv_zero_eq_comp, Function.comp_apply, Function.uncurry_def, ContinuousLinearEquiv.coe_coe] using! | |
| ((continuousMultilinearCurryFin0 ℝ E F).symm.toContinuousLinearEquiv.contDiff.comp hf) | |
| | succ n ih => | |
| have hh : ContDiff ℝ ∞ (fun z : (A × E) × E => iteratedFDeriv ℝ n (f z.1.1) z.2) := | |
| ih.comp (contDiff_fst.fst.prodMk contDiff_snd) | |
| have hd : ContDiff ℝ ∞ (fun z : A × E => fderiv ℝ (iteratedFDeriv ℝ n (f z.1)) z.2) := | |
| hh.fderiv contDiff_snd (by simp) | |
| simpa only [iteratedFDeriv_succ_eq_comp_left, Function.comp_apply, ContinuousLinearEquiv.coe_coe] using! | |
| ((continuousMultilinearCurryLeftEquiv ℝ (fun _ : Fin (n + 1) => E) F).symm.toContinuousLinearEquiv.contDiff.comp hd) | |
| end SmoothFamily | |
| section CompactFamily | |
| variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| theorem tendsto_schwartz_of_common_support | |
| (f : ℝ → 𝓢(E, F)) (hf : ContDiff ℝ ∞ (fun p : ℝ × E => f p.1 p.2)) | |
| (K : Set E) (hK : IsCompact K) (hsupp : ∀ r, tsupport (f r) ⊆ K) | |
| (hzero : f 0 = 0) : Tendsto f (𝓝 0) (𝓝 0) := by | |
| let : CompactSpace K := isCompact_iff_compactSpace.mp hK | |
| rw [(schwartz_withSeminorms ℝ E F).tendsto_nhds] | |
| rintro ⟨k, n⟩ ε hε | |
| have hc : Continuous (fun p : ℝ × K => ‖(p.2 : E)‖ ^ k * | |
| ‖iteratedFDeriv ℝ n (f p.1) (p.2 : E)‖) := by | |
| exact ((continuous_subtype_val.comp continuous_snd).norm.pow k).mul | |
| (((contDiff_partial_iteratedFDeriv (fun r x => f r x) hf n).continuous.comp | |
| (continuous_fst.prodMk (continuous_subtype_val.comp continuous_snd))).norm) | |
| have hu := Continuous.tendstoUniformly | |
| (fun r (x : K) => ‖(x : E)‖ ^ k * ‖iteratedFDeriv ℝ n (f r) (x : E)‖) hc 0 | |
| have he : ∀ᶠ r in 𝓝 (0 : ℝ), ∀ x : K, | |
| ‖(x : E)‖ ^ k * ‖iteratedFDeriv ℝ n (f r) (x : E)‖ < ε / 2 := by | |
| have hu' := (Metric.tendstoUniformly_iff.mp hu) (ε / 2) (by positivity) | |
| filter_upwards [hu'] with r hr | |
| intro x | |
| simpa [hzero, dist_comm, Real.dist_eq, abs_of_nonneg (by positivity : | |
| 0 ≤ ‖(x : E)‖ ^ k * ‖iteratedFDeriv ℝ n (f r) (x : E)‖)] using hr x | |
| filter_upwards [he] with r hr | |
| simp only [sub_zero, SchwartzMap.schwartzSeminormFamily_apply] | |
| apply lt_of_le_of_lt (SchwartzMap.seminorm_le_bound ℝ k n (f r) (by positivity) ?_) | |
| (half_lt_self hε) | |
| intro x | |
| by_cases hx : x ∈ K | |
| · exact (hr ⟨x, hx⟩).le | |
| · have hz : iteratedFDeriv ℝ n (f r) x = 0 := by | |
| apply image_eq_zero_of_notMem_tsupport | |
| exact fun h => hx (hsupp r (tsupport_iteratedFDeriv_subset n h)) | |
| simp only [hz, norm_zero, mul_zero] | |
| positivity | |
| end CompactFamily | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| def rescaledCoefficient (f χ : E → ℂ) (hf : ContDiff ℝ ∞ f) | |
| (hχ : ContDiff ℝ ∞ χ) (hχc : HasCompactSupport χ) (p : E) (r : ℝ) : 𝓢(E, ℂ) := | |
| (hχc.mul_right (f' := fun x => f (p + r • x) - f p)).toSchwartzMap | |
| (hχ.mul ((hf.comp (contDiff_const.add (contDiff_const.smul contDiff_id))).sub contDiff_const)) | |
| lemma rescaledCoefficient_tendsto (f χ : E → ℂ) (hf : ContDiff ℝ ∞ f) | |
| (hχ : ContDiff ℝ ∞ χ) (hχc : HasCompactSupport χ) (p : E) : | |
| Tendsto (rescaledCoefficient f χ hf hχ hχc p) (𝓝 0) (𝓝 0) := by | |
| apply tendsto_schwartz_of_common_support _ ?_ (tsupport χ) hχc ?_ ?_ | |
| · change ContDiff ℝ ∞ (fun z : ℝ × E => χ z.2 * (f (p + z.1 • z.2) - f p)) | |
| exact (hχ.comp contDiff_snd).mul | |
| ((hf.comp (contDiff_const.add (contDiff_fst.smul contDiff_snd))).sub contDiff_const) | |
| · intro r | |
| exact tsupport_mul_subset_left | |
| · ext x | |
| simp [rescaledCoefficient] | |
| end YauCounterexamples | |
| end | |
| end OAI | |