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ef96466 | 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 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralActionGronwall
namespace OAI
/-! The residual correction is small relative to the same action scale when
its scaled integral is small. -/
open Set MeasureTheory
namespace DefocusingNLS
theorem spectral_action_convolution_bound
(a r A : ℝ) (har : a≤ r) (hA : 0≤ A) (X g H : ℝ → ℝ)
(hX : ContinuousOn X (Icc a r)) (hg : ContinuousOn g (Icc a r))
(hH : ContinuousOn H (Icc a r)) (hg0 : ∀ t ∈ Icc a r, 0≤ g t)
(hbound : ∀ t ∈ Icc a r, X t≤ A*Real.exp (H t+∫ s in a..t, g s)) :
(∫ t in a..r, g t*Real.exp (H r-H t)*X t)≤
A*Real.exp (H r+∫ s in a..r, g s)*(∫ t in a..r, g t) := by
have hG (t : ℝ) (ht : t ∈ Icc a r) : (∫ s in a..t, g s)≤∫ s in a..r, g s := by
have h1 : IntervalIntegrable g volume a t := ContinuousOn.intervalIntegrable_of_Icc ht.1
(hg.mono (Icc_subset_Icc le_rfl ht.2))
have h2 : IntervalIntegrable g volume t r := ContinuousOn.intervalIntegrable_of_Icc ht.2
(hg.mono (Icc_subset_Icc ht.1 le_rfl))
have ha := intervalIntegral.integral_add_adjacent_intervals h1 h2
have hn : 0≤∫ s in t..r, g s :=
intervalIntegral.integral_nonneg ht.2 (fun s hs => hg0 s ⟨ht.1.trans hs.1,hs.2⟩)
linarith
have hleft : ContinuousOn (fun t => g t*Real.exp (H r-H t)*X t) (Icc a r) :=
(hg.mul (Real.continuous_exp.comp_continuousOn (continuousOn_const.sub hH))).mul hX
have hright : ContinuousOn (fun t => (A*Real.exp (H r+∫ s in a..r, g s))*g t) (Icc a r) :=
continuousOn_const.mul hg
calc
_ ≤ ∫ t in a..r, (A*Real.exp (H r+∫ s in a..r, g s))*g t := by
apply intervalIntegral.integral_mono_on har
(ContinuousOn.intervalIntegrable_of_Icc har hleft)
(ContinuousOn.intervalIntegrable_of_Icc har hright)
intro t ht
calc
_ ≤ g t*Real.exp (H r-H t)*(A*Real.exp (H t+∫ s in a..t, g s)) :=
mul_le_mul_of_nonneg_left (hbound t ht) (mul_nonneg (hg0 t ht) (Real.exp_pos _).le)
_ = g t*A*Real.exp (H r+∫ s in a..t, g s) := by
have he : Real.exp (H r-H t)*Real.exp (H t+∫ s in a..t, g s)=
Real.exp (H r+∫ s in a..t, g s) := by
rw [← Real.exp_add]
congr 1
ring
calc
_ = g t*A*(Real.exp (H r-H t)*Real.exp (H t+∫ s in a..t, g s)) := by ring
_ = _ := by rw [he]
_ ≤ g t*A*Real.exp (H r+∫ s in a..r, g s) := by
exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr (add_le_add le_rfl (hG t ht)))
(mul_nonneg (hg0 t ht) hA)
_ = _ := by ring
_ = _ := intervalIntegral.integral_const_mul _ _
end DefocusingNLS
end OAI
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