| 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 |
|
|
