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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 46 47 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Sphere.SphereMeasure
namespace OAI
/-! Angular cutoffs and derivative estimates. -/
section
section
namespace Attempt4FullFar
open Set Metric
open scoped ContDiff Topology
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
lemma compact_contDiffOn_derivative_bound {f : E → F} {U K : Set E}
(hU : IsOpen U) (hK : IsCompact K) (hKU : K ⊆ U) (hf : ContDiffOn ℝ ∞ f U)
(m : ℕ) : ∃ C : ℝ, 1 ≤ C ∧ ∀ j ≤ m, ∀ x ∈ K,
‖iteratedFDerivWithin ℝ j f U x‖ ≤ C := by
have hbound (j : ℕ) : ∃ C : ℝ, ∀ x ∈ K,
‖iteratedFDerivWithin ℝ j f U x‖ ≤ C :=
hK.exists_bound_of_continuousOn
((hf.continuousOn_iteratedFDerivWithin (by exact_mod_cast (le_top : (j:ℕ∞) ≤ ⊤)) hU.uniqueDiffOn).mono hKU)
induction m with
| zero =>
obtain ⟨C,hC⟩ := hbound 0
refine ⟨max 1 C, le_max_left _ _, ?_⟩
intro j hj x hx
have : j=0 := Nat.eq_zero_of_le_zero hj
subst j
exact (hC x hx).trans (le_max_right _ _)
| succ m ih =>
obtain ⟨D,hD,hboundD⟩ := ih
obtain ⟨C,hC⟩ := hbound (m+1)
refine ⟨max D C, hD.trans (le_max_left _ _), ?_⟩
intro j hj x hx
by_cases hjm : j ≤ m
· exact (hboundD j hjm x hx).trans (le_max_left _ _)
· have : j=m+1 := by omega
subst j
exact (hC x hx).trans (le_max_right _ _)
end Attempt4FullFar
end
end
end OAI
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