fpvandoorn/carleson
Source indexedtheorem · leanprover/lean4:v4.32.0
MeasureTheory.BoundedCompactSupport.adjointCarleson
Carleson.Operators · Carleson/Operators.lean:289 to 313
Mathematical statement
Exact Lean statement
@[fun_prop]
theorem BoundedCompactSupport.adjointCarleson (hf : BoundedCompactSupport f) :
BoundedCompactSupport (adjointCarleson p f) where
memLp_topComplete declaration
Lean source
Full Lean sourceLean 4
@[fun_prop]theorem BoundedCompactSupport.adjointCarleson (hf : BoundedCompactSupport f) : BoundedCompactSupport (adjointCarleson p f) where memLp_top := by obtain ⟨C, hC⟩ := hf.bddAbove_norm_adjointCarleson p simp only [mem_upperBounds, mem_range, forall_exists_index, forall_apply_eq_imp_iff] at hC apply MeasureTheory.memLp_top_of_bound (by fun_prop) C (.of_forall hC) hasCompactSupport := by obtain x₀ : X := Classical.choice (by infer_instance) obtain ⟨r₀, h⟩ := hf.isBoundedSupport.subset_ball x₀ let C : ℝ := (D ^ 𝔰 p / 2) + r₀ apply HasCompactSupport.of_support_subset_closedBall (x := x₀) (r := C) intro x hx apply mem_support.mp at hx have : ∃ y, conj (Ks (𝔰 p) y x) * exp (.I * (Q y y - Q y x)) * f y ≠ 0 := by by_contra! hc exact hx <| setIntegral_eq_zero_of_forall_eq_zero fun x _ ↦ hc x simp only [ne_eq, mul_eq_zero, map_eq_zero, exp_ne_zero, or_false, not_or] at this obtain ⟨y, hKy, hfy⟩ := this rw [mem_closedBall] apply le_trans <| dist_triangle _ y _ unfold C gcongr · rw [dist_comm]; exact dist_mem_Icc_of_Ks_ne_zero hKy |>.2 · exact le_of_lt <| h hfy