| import OAI.Geometry.RCD.Heat.BoundedHeatRepresentative | |
| namespace OAI | |
| /-! # Pointwise order and linearity of the actual bounded heat flow -/ | |
| open Set Filter MeasureTheory Metric TopologicalSpace | |
| open scoped NNReal Topology | |
| noncomputable section | |
| namespace RegularPointCoordinates | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] | |
| {n : ℕ} | |
| namespace HasBoundedAmplitude | |
| variable {f g : Lp ℝ 2 (hausdorff n X)} | |
| theorem add (hf : HasBoundedAmplitude f) (hg : HasBoundedAmplitude g) : | |
| HasBoundedAmplitude (f + g) := by | |
| obtain ⟨C, hC⟩ := hf | |
| obtain ⟨D, hD⟩ := hg | |
| refine ⟨C + D, ?_⟩ | |
| filter_upwards [hC, hD, Lp.coeFn_add f g] with x hx hy he | |
| rw [he] | |
| exact (abs_add_le _ _).trans (add_le_add hx hy) | |
| theorem smul (hf : HasBoundedAmplitude f) (c : ℝ) : | |
| HasBoundedAmplitude (c • f) := by | |
| obtain ⟨C, hC⟩ := hf | |
| refine ⟨⟨|c| * C, mul_nonneg (abs_nonneg _) C.coe_nonneg⟩, ?_⟩ | |
| filter_upwards [hC, Lp.coeFn_smul c f] with x hx he | |
| rw [he] | |
| change |c * f x| ≤ |c| * C | |
| simpa only [abs_mul] using | |
| mul_le_mul_of_nonneg_left hx (abs_nonneg c) | |
| end HasBoundedAmplitude | |
| omit [BorelSpace X] in | |
| private theorem continuous_le_of_ae {μ : Measure X} [μ.IsOpenPosMeasure] | |
| {f g : X → ℝ} (hf : Continuous f) (hg : Continuous g) | |
| (h : f ≤ᵐ[μ] g) : ∀ x, f x ≤ g x := by | |
| have he : (fun x => min (f x) (g x)) = f := | |
| Measure.eq_of_ae_eq (by filter_upwards [h] with x hx; exact min_eq_left hx) | |
| (hf.min hg) hf | |
| intro x | |
| exact min_eq_left_iff.mp (congrFun he x) | |
| namespace NoncollapsedRCD | |
| variable {K : ℝ} (hX : NoncollapsedRCD K n X) (hn : 2 ≤ n) (p : X) | |
| [ProperSpace X] | |
| [(hausdorff n X).Regular] | |
| theorem boundedHeat_add (f g : Lp ℝ 2 (hausdorff n X)) | |
| (hf : HasBoundedAmplitude f) (hg : HasBoundedAmplitude g) | |
| (t : ℝ) (ht : 0 < t) : | |
| hX.boundedHeat hn p (f + g) (hf.add hg) t ht = | |
| hX.boundedHeat hn p f hf t ht + hX.boundedHeat hn p g hg t ht := by | |
| apply hX.boundedHeat_unique hn p | |
| · exact (hX.boundedHeat_continuous hn p f hf t ht).add | |
| (hX.boundedHeat_continuous hn p g hg t ht) | |
| · rw [map_add] | |
| filter_upwards [hX.boundedHeat_represents hn p f hf t ht, | |
| hX.boundedHeat_represents hn p g hg t ht, | |
| Lp.coeFn_add (heatAt (hausdorff n X) hX.quadratic t f) | |
| (heatAt (hausdorff n X) hX.quadratic t g)] with x hx hy he | |
| simp only [Pi.add_apply, he, hx, hy] | |
| theorem boundedHeat_smul (f : Lp ℝ 2 (hausdorff n X)) | |
| (hf : HasBoundedAmplitude f) (c t : ℝ) (ht : 0 < t) : | |
| hX.boundedHeat hn p (c • f) (hf.smul c) t ht = | |
| c • hX.boundedHeat hn p f hf t ht := by | |
| apply hX.boundedHeat_unique hn p | |
| · exact (hX.boundedHeat_continuous hn p f hf t ht).const_smul c | |
| · rw [map_smul] | |
| filter_upwards [hX.boundedHeat_represents hn p f hf t ht, | |
| Lp.coeFn_smul c (heatAt (hausdorff n X) hX.quadratic t f)] with x hx he | |
| simp only [Pi.smul_apply, he, hx] | |
| theorem boundedHeat_mono (f g : Lp ℝ 2 (hausdorff n X)) | |
| (hf : HasBoundedAmplitude f) (hg : HasBoundedAmplitude g) | |
| (t : ℝ) (ht : 0 < t) (hfg : f ≤ g) : | |
| ∀ x, hX.boundedHeat hn p f hf t ht x ≤ hX.boundedHeat hn p g hg t ht x := by | |
| let := hX.openPos K n | |
| apply continuous_le_of_ae (μ := hausdorff n X) (hX.boundedHeat_continuous hn p f hf t ht) | |
| (hX.boundedHeat_continuous hn p g hg t ht) | |
| filter_upwards [hX.boundedHeat_represents hn p f hf t ht, | |
| hX.boundedHeat_represents hn p g hg t ht, | |
| (Lp.coeFn_le _ _).mpr (heatAt_mono (hausdorff n X) hX.quadratic t hfg)] with x hx hy he | |
| simpa only [← hx, ← hy] using he | |
| theorem boundedHeat_nonneg (f : Lp ℝ 2 (hausdorff n X)) | |
| (hf : HasBoundedAmplitude f) (t : ℝ) (ht : 0 < t) (hf0 : 0 ≤ f) : | |
| ∀ x, 0 ≤ hX.boundedHeat hn p f hf t ht x := by | |
| let := hX.openPos K n | |
| apply continuous_le_of_ae (μ := hausdorff n X) continuous_const (hX.boundedHeat_continuous hn p f hf t ht) | |
| filter_upwards [hX.boundedHeat_represents hn p f hf t ht, | |
| (Lp.coeFn_nonneg _).mpr (heatAt_nonneg (hausdorff n X) hX.quadratic t f hf0)] with x hx he | |
| simpa only [← hx, Pi.zero_apply] using he | |
| theorem boundedHeat_abs_le (f : Lp ℝ 2 (hausdorff n X)) | |
| (hf : HasBoundedAmplitude f) (t : ℝ) (ht : 0 < t) | |
| (C : ℝ≥0) (hC : ∀ᵐ x ∂hausdorff n X, |f x| ≤ C) : | |
| ∀ x, |hX.boundedHeat hn p f hf t ht x| ≤ C := by | |
| let := hX.openPos K n | |
| apply continuous_le_of_ae (μ := hausdorff n X) (hX.boundedHeat_continuous hn p f hf t ht).abs continuous_const | |
| filter_upwards [hX.boundedHeat_represents hn p f hf t ht, | |
| heatAt_abs_le_const (hausdorff n X) hX.quadratic t C f hC] with x hx he | |
| simpa only [← hx, Pi.zero_apply] using he | |
| end NoncollapsedRCD | |
| end RegularPointCoordinates | |
| end | |
| end OAI | |