| import OAI.Geometry.RCD.Heat.CompactKernelFactor |
| import Mathlib.MeasureTheory.Function.LpSeminorm.Indicator |
| import Mathlib.MeasureTheory.MeasurableSpace.Embedding |
|
|
| namespace OAI |
|
|
| /-! # Zero extension of continuous weights on compact endpoint supports |
|
|
| The extension is bounded and measurable and belongs to L² for locally |
| finite measures. Continuity across the support boundary is not needed. |
| -/ |
|
|
| open Set Filter MeasureTheory |
| open scoped ENNReal NNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
|
|
| def compactFactorExtension {S : Set X} (F : C(S, ℝ)) : X → ℝ := |
| Function.extend Subtype.val F (fun _ => 0) |
|
|
| omit [MeasurableSpace X] [BorelSpace X] in |
| @[simp] theorem compactFactorExtension_coe {S : Set X} (F : C(S, ℝ)) (x : S) : |
| compactFactorExtension F x=F x := Subtype.val_injective.extend_apply _ _ x |
|
|
| omit [MeasurableSpace X] [BorelSpace X] in |
| theorem compactFactorExtension_off {S : Set X} (F : C(S, ℝ)) {x : X} (hx : x ∉ S) : |
| compactFactorExtension F x=0 := by |
| apply Function.extend_apply' |
| rintro ⟨y,hy⟩ |
| exact hx (hy ▸ y.property) |
|
|
| theorem compactFactorExtension_measurable {S : Set X} (hS : MeasurableSet S) (F : C(S, ℝ)) : |
| Measurable (compactFactorExtension F) := |
| (MeasurableEmbedding.subtype_coe hS).measurable_extend F.continuous.measurable measurable_const |
|
|
| omit [MeasurableSpace X] [BorelSpace X] in |
| theorem compactFactorExtension_bounded {S : Set X} (hS : IsCompact S) (F : C(S, ℝ)) : |
| ∃ C : ℝ≥0, ∀ x, |compactFactorExtension F x| ≤ C := by |
| let : CompactSpace S := isCompact_iff_compactSpace.mp hS |
| refine ⟨‖F‖₊,fun x => ?_⟩ |
| by_cases hx : x ∈ S |
| · have he := compactFactorExtension_coe F ⟨x,hx⟩ |
| rw [he] |
| exact F.norm_coe_le_norm ⟨x,hx⟩ |
| · rw [compactFactorExtension_off F hx,abs_zero] |
| exact NNReal.coe_nonneg _ |
|
|
| theorem compactFactorExtension_memLp {S : Set X} (hS : IsCompact S) (F : C(S, ℝ)) |
| (μ : Measure X) [IsFiniteMeasureOnCompacts μ] : MemLp (compactFactorExtension F) 2 μ := by |
| have hm := compactFactorExtension_measurable hS.isClosed.measurableSet F |
| obtain ⟨C,hC⟩ := compactFactorExtension_bounded hS F |
| have he : S.indicator (compactFactorExtension F) = compactFactorExtension F := by |
| funext x |
| by_cases hx : x ∈ S |
| · simp only [Set.indicator_of_mem hx] |
| · simp only [Set.indicator_of_notMem hx,compactFactorExtension_off F hx] |
| rw [← he,memLp_indicator_iff_restrict hS.isClosed.measurableSet] |
| let : IsFiniteMeasure (μ.restrict S) := ⟨by simpa using hS.measure_lt_top⟩ |
| exact MemLp.of_bound hm.aestronglyMeasurable (C : ℝ) |
| (Eventually.of_forall fun x => hC x) |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
