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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_top

Complete declaration

Lean source

Canonical 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 :  := (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