| import Mathlib |
| import OAI.Analysis.FourierExtension.RadialMeasure |
| import OAI.Analysis.FourierExtension.L2Fourier |
|
|
| namespace OAI |
|
|
| /-! Convex graph and affine Fourier bounds. -/ |
|
|
| 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 |
|
|
| namespace DiagonalExtension.ConvexRadial |
| open MeasureTheory Set |
| open scoped ENNReal ContDiff |
|
|
| abbrev GraphSpace := WithLp 2 (E2 × ℝ) |
|
|
| def graphEmbedding (φ : E2 → ℝ) (ξ : E2) : GraphSpace := |
| WithLp.toLp 2 (ξ, φ ξ) |
|
|
| lemma graphEmbedding_continuous {φ : E2 → ℝ} (hφ : Continuous φ) : |
| Continuous (graphEmbedding φ) := |
| (WithLp.prod_continuous_toLp 2 E2 ℝ).comp (continuous_id.prodMk hφ) |
|
|
| lemma graphEmbedding_pairMap (φ : E2 → ℝ) : |
| GraphFourier.pairMap (graphEmbedding φ) = WithLp.toLp 2 ∘ pairGraph φ := by |
| ext point |
| rfl |
|
|
| lemma graphEmbedding_pairMap_le {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) |
| {c : ℝ} (hc : 0 < c) (hell : ∀ x v : E2, |
| c * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v) : |
| Measure.map (GraphFourier.pairMap (graphEmbedding φ)) (volume.prod volume) ≤ |
| ENNReal.ofReal (Real.pi / c) • (volume : Measure GraphSpace) := by |
| rw [graphEmbedding_pairMap, ← Measure.map_map |
| (WithLp.prod_continuous_toLp 2 E2 ℝ).measurable |
| (pairGraph_continuous hφ.continuous).measurable] |
| have H := Measure.map_mono (pairGraph_map_le hφ hc hell) |
| (WithLp.prod_continuous_toLp 2 E2 ℝ).measurable |
| simpa only [Measure.map_smul _ (WithLp.prod_continuous_toLp 2 E2 ℝ).measurable.aemeasurable, |
| (WithLp.volume_preserving_toLp E2 ℝ).map_eq] using H |
|
|
| theorem convex_graph_L4 {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) |
| {c : ℝ} (hc : 0 < c) (hell : ∀ x v : E2, |
| c * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v) |
| {g : E2 → ℂ} (hg : MemLp g 2 volume) (hgc : HasCompactSupport g) : |
| MemLp (GraphFourier.transform (graphEmbedding φ) g) 4 volume ∧ |
| eLpNorm (GraphFourier.transform (graphEmbedding φ) g) 4 volume ≤ |
| ENNReal.ofReal (Real.pi / c) ^ (1 / 4 : ℝ) * eLpNorm g 2 volume := by |
| exact GraphFourier.memLp_four_and_bound (graphEmbedding_continuous hφ.continuous) |
| ENNReal.ofReal_ne_top (graphEmbedding_pairMap_le hφ hc hell) hg hgc |
|
|
| end DiagonalExtension.ConvexRadial |
|
|
| namespace DiagonalExtension.AffineGraph |
| open MeasureTheory Set ConvexRadial |
| open scoped ENNReal NNReal |
| variable {Y : Type*} [NormedAddCommGroup Y] [InnerProductSpace ℝ Y] |
| [FiniteDimensional ℝ Y] [MeasurableSpace Y] [BorelSpace Y] |
|
|
| def haarFactor (A : GraphSpace ≃L[ℝ] Y) : ℝ≥0 := |
| Measure.addHaarScalarFactor (Measure.map A (volume : Measure GraphSpace)) (volume : Measure Y) |
|
|
| lemma map_affine (A : GraphSpace ≃L[ℝ] Y) (a : Y) : |
| Measure.map (fun z => a + A z) (volume : Measure GraphSpace) = |
| (haarFactor A : ℝ≥0∞) • (volume : Measure Y) := by |
| have : Measure.IsAddHaarMeasure (Measure.map A (volume : Measure GraphSpace)) := |
| A.isAddHaarMeasure_map volume |
| have hA := Measure.isAddLeftInvariant_eq_smul |
| (Measure.map A (volume : Measure GraphSpace)) (volume : Measure Y) |
| have he : (fun z => a + A z) = (fun y : Y => a + y) ∘ A := rfl |
| have hm : Measurable (fun y : Y => a + y) := by fun_prop |
| rw [he, ← Measure.map_map hm A.continuous.measurable, |
| hA, Measure.map_smul _ hm.aemeasurable, Measure.IsAddLeftInvariant.map_add_left_eq_self] |
| rfl |
|
|
| lemma pairMap_le (A : GraphSpace ≃L[ℝ] Y) (a : Y) |
| {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) {c : ℝ} (hc : 0 < c) |
| (hell : ∀ x v : E2, c * ‖v‖^2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v) : |
| Measure.map (GraphFourier.pairMap (fun ξ => a + A (graphEmbedding φ ξ))) |
| (volume.prod volume) ≤ |
| (ENNReal.ofReal (Real.pi / c) * (haarFactor A : ℝ≥0∞)) • volume := by |
| have he : GraphFourier.pairMap (fun ξ => a + A (graphEmbedding φ ξ)) = |
| (fun z => (a+a) + A z) ∘ GraphFourier.pairMap (graphEmbedding φ) := by |
| ext z |
| simp only [GraphFourier.pairMap, Function.comp_apply, map_add] |
| abel |
| have hm : Measurable (fun z : GraphSpace => (a+a) + A z) := by fun_prop |
| rw [he, ← Measure.map_map hm |
| (GraphFourier.pairMap_continuous (graphEmbedding_continuous hφ.continuous)).measurable] |
| have H := Measure.map_mono (graphEmbedding_pairMap_le hφ hc hell) |
| hm |
| simpa only [Measure.map_smul _ hm.aemeasurable, map_affine, smul_smul] using H |
|
|
| theorem L4 (A : GraphSpace ≃L[ℝ] Y) (a : Y) |
| {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) {c : ℝ} (hc : 0 < c) |
| (hell : ∀ x v : E2, c * ‖v‖^2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v) |
| {g : E2 → ℂ} (hg : MemLp g 2 volume) (hgc : HasCompactSupport g) : |
| MemLp (GraphFourier.transform (fun ξ => a + A (graphEmbedding φ ξ)) g) 4 volume ∧ |
| eLpNorm (GraphFourier.transform (fun ξ => a + A (graphEmbedding φ ξ)) g) 4 volume ≤ |
| (ENNReal.ofReal (Real.pi / c) * (haarFactor A : ℝ≥0∞)) ^ (1/4 : ℝ) * |
| eLpNorm g 2 volume := by |
| apply GraphFourier.memLp_four_and_bound |
| (continuous_const.add (A.continuous.comp (graphEmbedding_continuous hφ.continuous))) |
| (by finiteness) (pairMap_le A a hφ hc hell) hg hgc |
|
|
| end DiagonalExtension.AffineGraph |
|
|
| end |
|
|
| end OAI |
|
|
