6.6. Proof of Theorem 6.3 107andwhere•Proof of (6.37)Let β, γ ∈ B such that γ < β.{ }The event ˆβani = γ satisfieslim sup R 4,n (β) ≤ (M 2 (β)) p , (6.38)n→∞{R 3,n (β) = sup E f ‖ ˜f}ani − f‖ p ∞ψn−p (β)I { ˆβani
108 Sharp adaptive estimation in sup-norm for d-dimensional Hölder classesis upper bounded bywhereD 12 (log n) −pω(β,γ)− 12β ′ +1 n−p(µ(β ′ )−ω(β,γ))ω(β, γ) =If B satisfies Condition (P ), thenβ2β + 1 − min β i γi=1,...,d γ i (2γ + 1) .ω(β, γ) γ and µ(β ′ ) > 0, this implies thatlimsupn→∞ f∈Σ(β,L)β2β + 1 − γ2γ + 1 .ψ −pn (β)‖b γ,2 (·, f)‖ p ∞P f (A 2,n (β ′ )) = 0. (6.40)If B do not satisfy Condition (P ), there exists i ∈ {1, . . . , d} such that β i /γ i ≤ 1, thenThis implies thatµ(β ′ ) ≥ ω(β, γ) ≥β2β + 1 − γ2γ + 1 > 0.limsupn→∞ f∈Σ(β,L)ψ −pn (β)‖b γ,2 (·, f)‖ p ∞P f (A 2,n (β ′ )) = 0. (6.41)Reasoning as in the proof of (6.21), using the decomposition (6.25), Lemma 6.5, Lemma 6.8and thatβ2β + 1 − γ2γ + 1 ≤ µ(β′ ),we deduce thatlimsupn→∞ f∈Σ(β,L)From (6.41), (6.40) and (6.42), we obtain (6.37).E f{ψ−pn (β)‖Z γ,2 ‖ p ∞I A2,n (β ′ )}= 0. (6.42)•Proof of (6.38)(1 1/4.The proof will be similar to the proof in the proof of (6.22). We fix δ n =log n)WehaveR 4,n (β) ≤ (1 + δ n ) p (M 2 (β)) p +∑sup Q 4,n (β, γ, f),whereandγ∈B,γ≥βf∈Σ(β,L)Q 4,n (β, γ, f) = E f{‖ ˆf γ,2 − f‖ p ∞ψ −pn (β)I { ˆβani =γ}∩C 1,n},C 1,n = {‖ ˆf γ,2 − f‖ ∞ ψ −1n (β) > M 2 (β)(1 + δ n )}.