File size: 6,519 Bytes
2fca478 | 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 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 | import Mathlib
import OAI.Geometry.PrescribedPotential.LocalRegularity
import OAI.Geometry.PrescribedPotential.LocalizedPullback
namespace OAI
/-! Chart Transport. -/
section
noncomputable section
open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction
namespace GlobalElliptic
open SobolevChart
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
structure ChartCutoff (U : Set E) where
val : 𝓢(E, ℂ)
compact : HasCompactSupport (val : E → ℂ)
support_sub : tsupport (val : E → ℂ) ⊆ U
instance {U : Set E} : CoeFun (ChartCutoff U) (fun _ => E → ℂ) := ⟨fun κ => κ.val⟩
namespace ChartCutoff
variable {U : Set E} (κ : ChartCutoff U)
lemma product_smooth {h : E → ℂ} (hU : IsOpen U) (hh : ContDiffOn ℝ ∞ h U) :
ContDiff ℝ ∞ (fun x => κ x * h x) := by
rw [contDiff_iff_contDiffAt]
intro x
by_cases hx : x ∈ U
· exact (κ.val.smooth ⊤).contDiffAt.mul (hh.contDiffAt (hU.mem_nhds hx))
· have ht : x ∉ tsupport (κ : E → ℂ) := fun ht => hx (κ.support_sub ht)
apply (contDiffAt_const (c := (0 : ℂ))).congr_of_eventuallyEq
filter_upwards [notMem_tsupport_iff_eventuallyEq.mp ht] with y hy
simp [hy]
lemma product_compact (h : E → ℂ) : HasCompactSupport (fun x => κ x * h x) := by
apply HasCompactSupport.of_support_subset_isCompact κ.compact
intro x hx
exact subset_tsupport _ (left_ne_zero_of_mul hx)
def product {h : E → ℂ} (hU : IsOpen U) (hh : ContDiffOn ℝ ∞ h U) : 𝓢(E, ℂ) :=
(κ.product_compact h).toSchwartzMap (κ.product_smooth hU hh)
@[simp] lemma product_apply {h : E → ℂ} (hU : IsOpen U) (hh : ContDiffOn ℝ ∞ h U) (x : E) :
κ.product hU hh x = κ x * h x := rfl
lemma product_tsupport {h : E → ℂ} (hU : IsOpen U) (hh : ContDiffOn ℝ ∞ h U) :
tsupport (κ.product hU hh : E → ℂ) ⊆ tsupport (κ : E → ℂ) := by
apply closure_mono
exact fun x hx => left_ne_zero_of_mul hx
def mulOn {h : E → ℂ} (hU : IsOpen U) (hh : ContDiffOn ℝ ∞ h U) : ChartCutoff U where
val := κ.product hU hh
compact := κ.product_compact h
support_sub := (κ.product_tsupport hU hh).trans κ.support_sub
lemma deriv_tsupport (v : E) :
tsupport (⇑(∂_{v} κ.val : 𝓢(E, ℂ))) ⊆ tsupport (κ : E → ℂ) := by
intro x hx
by_contra hn
have hh := notMem_tsupport_iff_eventuallyEq.mp hn
have hd : (fun y => (∂_{v} κ.val : 𝓢(E, ℂ)) y) =ᶠ[𝓝 x] 0 := by
filter_upwards [hh.eventually_nhds] with y hy
simp only [SchwartzMap.lineDerivOp_apply_eq_fderiv]
have hy' : (κ : E → ℂ) =ᶠ[𝓝 y] 0 := hy
rw [hy'.fderiv_eq]
simp
exact (notMem_tsupport_iff_eventuallyEq.mpr hd) hx
def deriv (v : E) : ChartCutoff U where
val := ∂_{v} κ.val
compact := (κ.compact.of_isClosed_subset (isClosed_tsupport _) (κ.deriv_tsupport v))
support_sub := (κ.deriv_tsupport v).trans κ.support_sub
end ChartCutoff
variable (e : OpenPartialHomeomorph E E) (he : ContDiffOn ℝ ∞ e e.source)
def chartTransport (κ : ChartCutoff e.source) : 𝓢(E, ℂ) →ₗ[ℝ] 𝓢(E, ℂ) where
toFun f := κ.product e.open_source ((f.smooth ⊤).comp_contDiffOn he)
map_add' f h := by
ext x
simp [mul_add]
map_smul' c f := by
ext x
simp [mul_left_comm]
@[simp] lemma chartTransport_apply (κ : ChartCutoff e.source) (f : 𝓢(E, ℂ)) (x : E) :
chartTransport e he κ f x = κ x * f (e x) := rfl
variable [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
theorem chartTransport_l2_bound (hs : ContDiffOn ℝ ∞ e.symm e.target)
(κ : ChartCutoff e.source) : ∃ C : ℝ, 0 ≤ C ∧ ∀ f : 𝓢(E, ℂ),
‖schwartzCoord 0 (chartTransport e he κ f)‖ ≤ C * ‖schwartzCoord 0 f‖ := by
let K := tsupport (κ : E → ℂ)
obtain ⟨C, hC, hmeas, hmap⟩ := chart_map_restrict_bound e he hs κ.compact κ.support_sub
let B : L2 E →L[ℝ] L2 E :=
(Lp.extendZero (μ := (volume : Measure E)) (p := 2) (F := ℂ) (isClosed_tsupport _).measurableSet).toContinuousLinearMap ∘L
Lp.pullbackBounded (K.indicator e) hmeas ENNReal.ofReal_ne_top hmap
let T : L2 E →L[ℝ] L2 E := (multiply κ.val.toBoundedContinuousFunction).restrictScalars ℝ ∘L B
refine ⟨‖T‖, norm_nonneg _, fun f => ?_⟩
have hc (u : 𝓢(E, ℂ)) : schwartzCoord 0 u = u.toLp 2 volume := by
apply realize_injective 0
simpa [realize] using realize_schwartzCoord 0 u
rw [hc, hc]
have heq : T (f.toLp 2 volume) = (chartTransport e he κ f).toLp 2 volume := by
apply Lp.ext
have hb := Lp.coeFn_pullbackBounded (K.indicator e) hmeas ENNReal.ofReal_ne_top hmap
(f.toLp 2 volume)
have hz := Lp.coeFn_extendZero (μ := (volume : Measure E))
(isClosed_tsupport (κ : E → ℂ)).measurableSet
(Lp.pullbackBounded (K.indicator e) hmeas ENNReal.ofReal_ne_top hmap (f.toLp 2 volume))
have hf := (f.memLp 2 volume).coeFn_toLp
have hfc := (ae_of_ae_map hmeas.aemeasurable
(ae_mono hmap (Measure.ae_smul_measure hf _)))
have ht := (chartTransport e he κ f).coeFn_toLp (p := 2) (μ := volume)
have hm : (fun x => multiply κ.val.toBoundedContinuousFunction (B (f.toLp 2 volume)) x) =ᵐ[volume]
fun x => κ x * B (f.toLp 2 volume) x := by
filter_upwards [(ContinuousLinearMap.mul ℂ ℂ).coeFn_holder (r := 2)
(κ.val.toBoundedContinuousFunction.memLp_top.toLp κ.val.toBoundedContinuousFunction : Lp ℂ ⊤ volume)
(B (f.toLp 2 volume)),
(κ.val.toBoundedContinuousFunction.memLp_top (μ := volume)).coeFn_toLp] with x h₁ h₂
exact h₁.trans (by simp only [ContinuousLinearMap.mul_apply', h₂]; rfl)
filter_upwards [hm, hz, (ae_restrict_iff' (isClosed_tsupport _).measurableSet).mp hb,
(ae_restrict_iff' (isClosed_tsupport _).measurableSet).mp hfc, ht]
with x h₁ h₂ h₃ h₄ h₅
change multiply κ.val.toBoundedContinuousFunction (B (f.toLp 2 volume)) x = _
rw [h₁, h₅]
change B (f.toLp 2 volume) x = _ at h₂
rw [h₂]
by_cases hx : x ∈ K
· rw [indicator_of_mem hx, h₃ hx]
have hh := h₄ hx
change (f.toLp 2 volume) (K.indicator e x) = f (K.indicator e x) at hh
rw [hh, indicator_of_mem hx]
rfl
· have hk : κ x = 0 := image_eq_zero_of_notMem_tsupport hx
simp [chartTransport_apply, hk]
rw [← heq]
exact T.le_opNorm _
end GlobalElliptic
end
end
end OAI
|