File size: 2,759 Bytes
b37e936 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | 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
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