File size: 6,524 Bytes
67aee1d | 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 | import Mathlib
namespace OAI
/-! Automatic boundedness and integral kernels on Lp spaces. -/
open MeasureTheory
open scoped NNReal ENNReal ContDiff
noncomputable section
open Filter
open scoped Topology ContDiff
open MeasureTheory Set
open scoped ContDiff FourierTransform InnerProductSpace ENNReal
open MeasureTheory Set Filter Metric
open scoped ContDiff Topology
open Set Filter
open scoped ENNReal InnerProductSpace
open scoped FourierTransform SchwartzMap ENNReal
open scoped ENNReal FourierTransform InnerProductSpace
open MeasureTheory Set Filter
open scoped ContDiff Topology ENNReal
open scoped ENNReal NNReal
open Filter
namespace DiagonalExtension.AutomaticBound
variable {α V : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup V] [NormedSpace ℂ V] [CompleteSpace V]
{p : ℝ≥0∞} [Fact (1 ≤ p)]
def pointwiseLpMap (T : α → V →L[ℂ] ℂ)
(hT : ∀ v, MemLp (fun x => T x v) p μ) : V →ₗ[ℂ] Lp ℂ p μ where
toFun v := (hT v).toLp (fun x => T x v)
map_add' v w := by
apply Lp.ext
filter_upwards [(hT (v+w)).coeFn_toLp,(hT v).coeFn_toLp,(hT w).coeFn_toLp,
Lp.coeFn_add ((hT v).toLp _) ((hT w).toLp _)] with x h₁ h₂ h₃ h₄
exact (h₁.trans ((T x).map_add v w)).trans (by rw [h₄,Pi.add_apply,h₂,h₃])
map_smul' c v := by
apply Lp.ext
filter_upwards [(hT (c • v)).coeFn_toLp,(hT v).coeFn_toLp,
Lp.coeFn_smul c ((hT v).toLp _)] with x h₁ h₂ h₃
exact (h₁.trans ((T x).map_smul c v)).trans (by rw [RingHom.id_apply,h₃,Pi.smul_apply,h₂])
theorem continuous_pointwiseLpMap (T : α → V →L[ℂ] ℂ)
(hT : ∀ v, MemLp (fun x => T x v) p μ) :
Continuous (pointwiseLpMap T hT) := by
apply LinearMap.continuous_of_seq_closed_graph
intro u v w hu hw
obtain ⟨ns,hmono,hns⟩ := (tendstoInMeasure_of_tendsto_Lp hw).exists_seq_tendsto_ae
apply Lp.ext
have hall : ∀ᵐ x ∂μ, ∀ n : ℕ,
pointwiseLpMap T hT (u (ns n)) x = T x (u (ns n)) := by
exact ae_all_iff.mpr (fun n => (hT (u (ns n))).coeFn_toLp)
filter_upwards [hns,hall,(hT v).coeFn_toLp] with x hx hxe hxv
change w x = (hT v).toLp (fun x => T x v) x
rw [hxv]
have he : Tendsto (fun n => T x (u (ns n))) atTop (𝓝 (T x v)) :=
((T x).continuous.tendsto v).comp (hu.comp hmono.tendsto_atTop)
have hw' : Tendsto (fun n => T x (u (ns n))) atTop (𝓝 (w x)) := by
convert hx using 1
ext n
exact (hxe n).symm
exact tendsto_nhds_unique hw' he
theorem exists_bound (T : α → V →L[ℂ] ℂ)
(hT : ∀ v, MemLp (fun x => T x v) p μ) :
∃ C : ℝ≥0, ∀ v, eLpNorm (fun x => T x v) p μ ≤ (C : ℝ≥0∞) * ‖v‖ₑ := by
let L : V →L[ℂ] Lp ℂ p μ :=
⟨pointwiseLpMap T hT, continuous_pointwiseLpMap T hT⟩
refine ⟨‖L‖₊,fun v => ?_⟩
have hb := L.le_opNorm v
have he := ENNReal.ofReal_le_ofReal hb
have hv : ‖L v‖ₑ = eLpNorm (fun x => T x v) p μ :=
Lp.enorm_toLp (hT v)
rw [← hv, ENNReal.coe_nnreal_eq]
change ‖L v‖ₑ ≤ ENNReal.ofReal ‖L‖ * ‖v‖ₑ
simpa only [ENNReal.ofReal_mul (norm_nonneg _),ofReal_norm] using he
end DiagonalExtension.AutomaticBound
open MeasureTheory Filter
open scoped NNReal ENNReal Topology
namespace DiagonalExtension.KernelLp
variable {α : Type*} [MeasurableSpace α] {μ : Measure α} [IsFiniteMeasure μ]
{p : ℝ≥0∞} [Fact (1 ≤ p)]
lemma integral_norm_le_lp (hp : p ≠ (∞ : ℝ≥0∞)) (f : Lp ℂ p μ) :
(∫ x, ‖f x‖ ∂μ) ≤ (μ Set.univ ^ (1-1/p.toReal)).toReal * ‖f‖ := by
have hpr : 1 ≤ p.toReal := by
simpa using ENNReal.toReal_mono hp (show (1:ℝ≥0∞) ≤ p from Fact.out)
have hpow : 0 ≤ 1-1/p.toReal := by
have ht : 0 < p.toReal := by linarith
have hh : 1/p.toReal ≤ 1 := (div_le_one ht).mpr hpr
linarith
have hM : μ Set.univ ^ (1-1/p.toReal) ≠ (∞ : ℝ≥0∞) :=
ENNReal.rpow_ne_top_of_nonneg hpow (measure_ne_top μ _)
rw [integral_norm_eq_lintegral_enorm (Lp.aestronglyMeasurable f),
← eLpNorm_one_eq_lintegral_enorm (Lp.aestronglyMeasurable f)]
have hh := eLpNorm_le_eLpNorm_mul_rpow_measure_univ
(show (1:ℝ≥0∞) ≤ p from Fact.out) (Lp.aestronglyMeasurable f)
simp only [ENNReal.toReal_one,div_one] at hh
have hr := ENNReal.toReal_mono (ENNReal.mul_ne_top (Lp.memLp f).eLpNorm_ne_top hM) hh
simpa only [ENNReal.toReal_mul,← Lp.norm_def,mul_comm] using hr
lemma integrable_kernel {k : α → ℂ} (hk : AEStronglyMeasurable k μ)
(hb : ∀ᵐ x ∂μ, ‖k x‖ ≤ 1) (f : Lp ℂ p μ) :
Integrable (fun x => f x * k x) μ :=
((Lp.memLp f).integrable (show (1:ℝ≥0∞) ≤ p from Fact.out)).mul_bdd hk hb
def evalLinear {k : α → ℂ} (hk : AEStronglyMeasurable k μ)
(hb : ∀ᵐ x ∂μ, ‖k x‖ ≤ 1) : Lp ℂ p μ →ₗ[ℂ] ℂ where
toFun f := ∫ x, f x * k x ∂μ
map_add' f g := by
calc
_ = ∫ x, f x * k x + g x * k x ∂μ := by
apply integral_congr_ae
filter_upwards [Lp.coeFn_add f g] with x hx
rw [hx,Pi.add_apply,add_mul]
_ = _ := integral_add (integrable_kernel hk hb f) (integrable_kernel hk hb g)
map_smul' c f := by
simp only [RingHom.id_apply]
calc
_ = ∫ x, c • (f x * k x) ∂μ := by
apply integral_congr_ae
filter_upwards [Lp.coeFn_smul c f] with x hx
rw [hx,Pi.smul_apply,smul_mul_assoc]
_ = _ := integral_smul _ _
lemma norm_evalLinear_le (hp : p ≠ (∞ : ℝ≥0∞)) {k : α → ℂ}
(hk : AEStronglyMeasurable k μ) (hb : ∀ᵐ x ∂μ, ‖k x‖ ≤ 1)
(f : Lp ℂ p μ) :
‖evalLinear hk hb f‖ ≤ (μ Set.univ ^ (1-1/p.toReal)).toReal * ‖f‖ := by
have hi := ((Lp.memLp f).integrable (show (1:ℝ≥0∞) ≤ p from Fact.out)).norm
apply le_trans (norm_integral_le_of_norm_le hi ?_) (integral_norm_le_lp hp f)
filter_upwards [hb] with x hx
exact (norm_mul _ _).le.trans (by simpa using mul_le_mul_of_nonneg_left hx (norm_nonneg (f x)))
def eval (hp : p ≠ (∞ : ℝ≥0∞)) {k : α → ℂ} (hk : AEStronglyMeasurable k μ)
(hb : ∀ᵐ x ∂μ, ‖k x‖ ≤ 1) : Lp ℂ p μ →L[ℂ] ℂ :=
(evalLinear hk hb).mkContinuous (μ Set.univ ^ (1-1/p.toReal)).toReal
(norm_evalLinear_le hp hk hb)
@[simp] lemma eval_apply (hp : p ≠ (∞ : ℝ≥0∞)) {k : α → ℂ} (hk : AEStronglyMeasurable k μ)
(hb : ∀ᵐ x ∂μ, ‖k x‖ ≤ 1) (f : Lp ℂ p μ) :
eval hp hk hb f = ∫ x, f x * k x ∂μ := rfl
end DiagonalExtension.KernelLp
end
end OAI
|