| import Mathlib |
| import OAI.Geometry.IntegralFillings.Charts.Differentiability |
| import OAI.Geometry.IntegralFillings.Differentiation.HilbertCharts |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open Filter Set |
| open scoped Topology NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
|
|
| namespace SharpIntegralFillings |
| open Filter MeasureTheory Set Metric |
| open scoped Topology NNReal |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (μ : Measure E) [Measure.IsAddHaarMeasure μ] |
| lemma seminorm_le_of_centered_bound {s : Set E} {x : E} |
| {r : E → ℝ} {p q : Seminorm ℝ E} {B : ℝ} |
| (hx : Tendsto (fun t => μ (s ∩ closedBall x t) / μ (closedBall x t)) |
| (𝓝[>] 0) (𝓝 1)) |
| (hr : (fun y => r y-p (y-x)) =o[𝓝[s] x] (fun y => y-x)) |
| (hbound : ∀ y ∈ s, r y ≤ B*q (y-x)) (v : E) : p v ≤ B*q v := by |
| have hv : v ∈ tangentConeAt ℝ s x := by |
| rw [tangentConeAt_eq_univ_of_density_one μ s x hx] |
| exact mem_univ v |
| obtain ⟨c,d,hc,hd,hcd⟩ := mem_tangentConeAt_iff_exists_seq_norm_tendsto_atTop.mp hv |
| have hd₀ := tangentConeAt.lim_zero atTop hc hcd |
| have ht : Tendsto (fun j => x+d j) atTop (𝓝[s] x) := |
| tendsto_nhdsWithin_iff.2 ⟨by simpa using tendsto_const_nhds.add hd₀,hd⟩ |
| have hsmall := (hr.comp_tendsto ht).norm_right.mul_isBigO |
| (Asymptotics.isBigO_refl (fun j => ‖c j‖) atTop) |
| have hnorm : Tendsto (fun j => ‖(x+d j)-x‖*‖c j‖) atTop (𝓝 ‖v‖) := by |
| simpa only [norm_smul,add_sub_cancel_left,mul_comm] using hcd.norm |
| have hzero := hsmall.tendsto_zero_of_tendsto hnorm |
| have hp := (seminorm_continuous_finiteDimensional p).tendsto v |>.comp hcd |
| have hlim : Tendsto (fun j => ‖c j‖*r (x+d j)) atTop (𝓝 (p v)) := by |
| convert hzero.add hp using 1 |
| · ext j |
| simp only [Function.comp_apply,add_sub_cancel_left,map_smul_eq_mul] |
| ring |
| · simp |
| have hq : Tendsto (fun j => B*(‖c j‖*q (d j))) atTop (𝓝 (B*q v)) := by |
| simpa only [Function.comp_apply,map_smul_eq_mul] using |
| (((seminorm_continuous_finiteDimensional q).tendsto v).comp hcd).const_mul B |
| apply le_of_tendsto_of_tendsto hlim hq |
| filter_upwards [hd] with j hj |
| have hh := mul_le_mul_of_nonneg_left (hbound (x+d j) hj) (norm_nonneg (c j)) |
| simpa only [add_sub_cancel_left,mul_left_comm] using hh |
|
|
| end SharpIntegralFillings |
|
|
| namespace SharpIntegralFillings.MetricDifferentiation |
| open Filter MeasureTheory Set Metric |
| open scoped Topology NNReal |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (μ : Measure E) [Measure.IsAddHaarMeasure μ] |
| {X : Type*} [MetricSpace X] |
|
|
| lemma HasCenteredMetricDifferentialWithin.seminorm_le_on_piece |
| {s t : Set E} (hts : t ⊆ s) {f : s → X} {p q : Seminorm ℝ E} {x : s} |
| (h : HasCenteredMetricDifferentialWithin s f p x) |
| (hx : Tendsto (fun r => μ (t ∩ closedBall (x : E) r) / μ (closedBall (x : E) r)) |
| (𝓝[>] 0) (𝓝 1)) {B : ℝ} |
| (hb : ∀ y (hy : y ∈ t), dist (f ⟨y,hts hy⟩) (f x) ≤ B*q (y-(x : E))) : |
| ∀ v : E, p v ≤ B*q v := by |
| classical |
| let r : E → ℝ := fun z => if hz : z ∈ s then dist (f ⟨z,hz⟩) (f x) else 0 |
| have hr : (fun y => r y-p (y-(x : E))) =o[𝓝[s] (x : E)] (fun y => y-(x : E)) := by |
| rw [nhdsWithin_eq_map_subtype_coe x.property,Asymptotics.isLittleO_map] |
| simpa only [HasCenteredMetricDifferentialWithin,Function.comp_def,r, |
| dite_eq_left (Subtype.coe_prop _),Subtype.coe_eta] using h |
| apply seminorm_le_of_centered_bound μ hx (hr.mono (nhdsWithin_mono _ hts)) |
| intro y hy |
| simpa only [r,dite_eq_left (hts hy)] using hb y hy |
|
|
| end SharpIntegralFillings.MetricDifferentiation |
|
|
| end |
|
|
| end OAI |
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|
