| import OAI.Analysis.CircleDomains.Topology.PlanarMaximalInput | |
| namespace OAI | |
| /-! A maximal-function majorant pays the mass on a balloon by its mass | |
| on any measurable subset occupying a fixed fraction of the balloon. -/ | |
| open Set Function Filter MeasureTheory Metric | |
| open scoped Topology ENNReal | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| theorem ball_subset_double_ball {c x : ℂ} {d : ℝ} (hx : x ∈ ball c d) : | |
| ball c d ⊆ ball x (2*d) := by | |
| intro y hy | |
| rw [mem_ball] at hx hy ⊢ | |
| calc | |
| dist y x ≤ dist y c+dist c x := dist_triangle _ _ _ | |
| _ < d+d := add_lt_add hy (by simpa only [dist_comm] using hx) | |
| _ = 2*d := by ring | |
| theorem maximal_ball_payment {g m : ℂ → ℝ} | |
| (hg : MemLp g 2 volume) (hm : MemLp m 2 volume) (_hm0 : ∀ z, 0 ≤ m z) | |
| (hmax : ∀ r : ℝ, 0 < r → ∀ᵐ x ∂volume, | |
| (∫ y in ball x r, ‖g y‖) ≤ volume.real (ball x r)*m x) | |
| {c : ℂ} {d k : ℝ} (hd : 0 < d) (_hk : 0 < k) {E : Set ℂ} | |
| (hE : MeasurableSet E) (hEB : E ⊆ ball c d) | |
| (hsize : k*d^2 ≤ volume.real E) : | |
| k*(∫ y in ball c d, ‖g y‖) ≤ 4*Real.pi*(∫ x in E, m x) := by | |
| have hEf : volume E ≠ ∞ := measure_ne_top_of_subset hEB measure_ball_lt_top.ne | |
| let : IsFiniteMeasure (volume.restrict E) := isFiniteMeasure_restrict.mpr hEf | |
| have hmE : IntegrableOn m E := (hm.restrict E).integrable (by norm_num) | |
| have hpoint : ∀ᵐ x ∂volume.restrict E, | |
| (∫ y in ball c d, ‖g y‖) ≤ (4*Real.pi*d^2)*m x := by | |
| filter_upwards [ae_restrict_mem hE,ae_restrict_of_ae (hmax (2*d) (by positivity))] with x hx hxmax | |
| have hbig : IntegrableOn (fun y => ‖g y‖) (ball x (2*d)) := | |
| ((hg.norm.locallyIntegrable (by norm_num)).integrableOn_isCompact | |
| (isCompact_closedBall x (2*d))).mono_set ball_subset_closedBall | |
| calc | |
| _ ≤ ∫ y in ball x (2*d), ‖g y‖ := | |
| setIntegral_mono_set hbig (Eventually.of_forall (fun _ => norm_nonneg _)) | |
| (Filter.Eventually.of_forall (ball_subset_double_ball (hEB hx))) | |
| _ ≤ volume.real (ball x (2*d))*m x := hxmax | |
| _ = _ := by rw [show volume.real (ball x (2*d)) = Real.pi*(2*d)^2 from | |
| real_volume_ball x (by positivity)]; ring | |
| have hi := integral_mono_ae (integrable_const (∫ y in ball c d, ‖g y‖)) | |
| (hmE.const_mul (4*Real.pi*d^2)) hpoint | |
| rw [integral_const,integral_const_mul,smul_eq_mul,Measure.real,Measure.restrict_apply_univ] at hi | |
| have hI : 0 ≤ ∫ y in ball c d, ‖g y‖ := integral_nonneg (fun _ => norm_nonneg _) | |
| have hl := (mul_le_mul_of_nonneg_right hsize hI).trans hi | |
| apply (mul_le_mul_iff_right₀ (sq_pos_of_pos hd)).mp | |
| nlinarith only [hl] | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |