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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 | import Mathlib
import OAI.Geometry.PrescribedPotential.GlobalSobolev
import OAI.Geometry.PrescribedPotential.LocalPotentialABP
namespace OAI
/-! A B P Integral Bound. -/
section
noncomputable section
open Set Metric Filter Topology MeasureTheory Matrix
open scoped ContDiff InnerProductSpace
namespace PotentialABP
variable {d : ℕ}
local notation "E" => EuclideanSpace ℂ (Fin d)
local instance abpIntegralRealIP (d : ℕ) : InnerProductSpace ℝ (EuclideanSpace ℂ (Fin d)) :=
InnerProductSpace.rclikeToReal ℂ (EuclideanSpace ℂ (Fin d))
local notation "e" => EuclideanSpace.equiv (Fin d) ℂ
theorem abp_integral_minimum_bound
(μ : Measure E) [μ.IsAddHaarMeasure]
{f : (Fin d → ℂ) → ℝ} (hf : ContDiff ℝ ∞ f)
(G : E → Matrix (Fin d) (Fin d) ℂ)
{c : E} {r δ K B : ℝ} (hr : 0 < r) (hδ : 0 < δ) (hK : 0 < K)
(hmin : ∀ x ∈ closedBall c r, f (e c) ≤ f (e x))
(hG : ∀ x ∈ closedBall c r, (G x).IsHermitian)
(hGδ : ∀ x ∈ closedBall c r, ∀ v : E,
δ*‖v‖^2 ≤ inner ℝ ((G x).toEuclideanLin v) v)
(hdet : ∀ x ∈ closedBall c r,
|(G x + PotentialKaehler.potentialMatrix f (e x)).det.re| ≤ K)
{U : Set E} (hU : MeasurableSet U) (hsub : closedBall c r ⊆ U)
(hn : ∀ x ∈ U, f (e x) ≤ 0)
(hi : IntegrableOn (fun x => -f (e x)) U μ)
(hB : (∫ x in U, -f (e x) ∂μ) ≤ B) :
-f (e c) ≤ δ*r^2/2 + B*((4:ℝ)^(2*d)*K^2) /
μ.real (closedBall (0:E) (δ*r/2)) := by
let S : Set E := {x | x ∈ closedBall c r ∧ f (e x) ≤ f (e c)+δ*r^2/2}
let C : ℝ := (4:ℝ)^(2*d)*K^2
let V : ℝ := μ.real (closedBall (0:E) (δ*r/2))
have hC : 0 < C := by positivity
have hV : 0 < V := ENNReal.toReal_pos
(measure_closedBall_pos μ _ (by positivity)).ne'
(isCompact_closedBall _ _).measure_lt_top.ne
have hSc : IsClosed S := isClosed_closedBall.inter
(isClosed_le (hf.continuous.comp (EuclideanSpace.equiv (Fin d) ℂ).continuous) continuous_const)
have hSsub : S ⊆ U := fun x hx => hsub hx.1
have hSi : IntegrableOn (fun x => -f (e x)) S μ := hi.mono_set hSsub
have hSfin : μ S ≠ ⊤ :=
((measure_mono (show S ⊆ closedBall c r from fun _ hx => hx.1)).trans_lt
(isCompact_closedBall _ _).measure_lt_top).ne
have hABP := local_potential_sublevel_volume μ hf G hr hδ hK.le hmin hG hGδ hdet
have hABPr : V ≤ C * μ.real S := by
have hne : ENNReal.ofReal C * μ S ≠ ⊤ := ENNReal.mul_ne_top ENNReal.ofReal_ne_top hSfin
have hh := ENNReal.toReal_mono hne hABP
simpa only [V,C,S,measureReal_def,ENNReal.toReal_mul,ENNReal.toReal_ofReal hC.le] using hh
have hBn : 0 ≤ B := (setIntegral_nonneg hU (fun x hx => neg_nonneg.mpr (hn x hx))).trans hB
have hIS : (-f (e c)-δ*r^2/2)*μ.real S ≤ B := by
have hc : IntegrableOn (fun _ : E => -f (e c)-δ*r^2/2) S μ :=
integrableOn_const hSfin
have hm := setIntegral_mono_on hc hSi hSc.measurableSet (fun x (hx : x ∈ S) => by
have hh := hx.2
linarith)
have hm' := setIntegral_mono_set hi
(by
filter_upwards [ae_restrict_mem hU] with x hx
exact neg_nonneg.mpr (hn x hx))
(Filter.Eventually.of_forall hSsub)
have hh := hm.trans (hm'.trans hB)
simpa only [setIntegral_const, smul_eq_mul, mul_comm] using hh
by_cases hm : 0 ≤ -f (e c)-δ*r^2/2
· have hmul := mul_le_mul_of_nonneg_left hABPr hm
have hmul' := mul_le_mul_of_nonneg_left hIS hC.le
have hh : (-f (e c)-δ*r^2/2)*V ≤ B*C := by nlinarith
have hd := (le_div_iff₀ hV).mpr hh
change -f (e c) ≤ δ*r^2/2 + B*C/V
linarith
· have hdiv : 0 ≤ B*C/V := div_nonneg (mul_nonneg hBn hC.le) hV.le
change -f (e c) ≤ δ*r^2/2 + B*C/V
linarith
end PotentialABP
end
end
end OAI
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