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7f37411 | 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 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Sphere.SphereAmbient2
namespace OAI
/-! Graph energies and pair estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer
open Set MeasureTheory
open scoped ComplexConjugate ContDiff
lemma GraphSymbolJet.conj {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
{b : V → V → ℂ} {m : ℕ} {s B : ℝ}
(h : GraphSymbolJet b m s B) : GraphSymbolJet (fun x w => conj (b x w)) m s B := by
refine ⟨Complex.continuous_conj.measurable.comp h.measurable,
fun x => Complex.conjCLE.contDiff.comp (h.smooth x), h.nonneg, ?_⟩
intro x j hj w hw
have he := Complex.conjLIE.norm_iteratedFDeriv_comp_left (b x) w j
change ‖iteratedFDeriv ℝ j (Complex.conjLIE ∘ b x) w‖ ≤ _
rw [he]
exact h.bound x j hj w hw
lemma GraphSymbolJet.mono_order {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
{b : V → V → ℂ} {m n : ℕ} {s B : ℝ}
(h : GraphSymbolJet b m s B) (hn : n ≤ m) : GraphSymbolJet b n s B :=
⟨h.measurable,h.smooth,h.nonneg,fun x j hj w hw => h.bound x j (hj.trans hn) w hw⟩
lemma GraphSymbolJet.mono_bound {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
{b : V → V → ℂ} {m : ℕ} {s B C : ℝ}
(h : GraphSymbolJet b m s B) (hs : 0 ≤ s) (hBC : B ≤ C) : GraphSymbolJet b m s C :=
⟨h.measurable,h.smooth,h.nonneg.trans hBC,fun x j hj w hw =>
(h.bound x j hj w hw).trans (mul_le_mul_of_nonneg_right hBC (pow_nonneg hs _))⟩
end CurrentFalconer
end
end
end OAI
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