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10) Since r(x) > σ(m), it follows from the definition of ε(m) that ||Φε(m) ∗Vm ||(Ω| ∪ {B(x, r(x)) : x ∈ E1 (R, m)}) ≥ (1/2)||Vm ||E1 (R, m). 11) together givee ||Φε(m) ∗ δVm ||(Ω) ≥ B(n)−1 exp(−2R)(1/4R)||Vm ||E1 (R, m) or, by Schwarz’ inequality, Ω, |Φε(m) ∗ δVm |2 /Φε(m) ∗ ||Vm || ||Φε(m) ∗Vm ||(Ω) ≥ [B(n)−1 exp(−2rR)(1/4R)||Vm ||E1 (R, m)]2 . Hypothesis (1) now implies that lim sup ||Vm ||E1 (R, m) ≤ B(n) exp(2R)4R(B||V ||(Ω))1/2 . 12) Now suppose that x ∈ E2 (R, m). 7) implies that (2−1/2 σ(m))−k ||Vm ||B(x, 2−1/k σ(m)) ≤ c3 .
Lemma: If ν ∈ N, 0 < µ < 1, and 0 < ζ < 1 then there is γ > 0 such that if (1) V ∈ IVk (Rn ), σ > 0, w > 0, 0 < R < σ, (1 − ζ)/2ν > 1 − exp(−4wσ), ρ/R > µ; 0 < 2ρ < σ, (2) T = e1 ∧ . . ∧ ek ∈ G(n, k); (3) Y ⊂ T ⊥ , diam Y < σ, and ν = ∑{θk (||V ||, y) : y ∈ Y }; (4) for r > 0 and ξ > 0 we define E(r, ξ) = {x ∈ Rn : |T (x)| ≤ r, dist(T ⊥ (x),Y ) < ξ} (5) if 0 < r ≤ R then E(r,2ρ) (6) if 0 < r ≤ R then then ||S − T || dV (x, S) < γ αr k ; and ∆σ,w ||V ||E(r, ρ) > −γ αRk ; ||V ||E(R, ρ) ≥ (ν − ζ)αRk .
10) Since r(x) > σ(m), it follows from the definition of ε(m) that ||Φε(m) ∗Vm ||(Ω| ∪ {B(x, r(x)) : x ∈ E1 (R, m)}) ≥ (1/2)||Vm ||E1 (R, m). 11) together givee ||Φε(m) ∗ δVm ||(Ω) ≥ B(n)−1 exp(−2R)(1/4R)||Vm ||E1 (R, m) or, by Schwarz’ inequality, Ω, |Φε(m) ∗ δVm |2 /Φε(m) ∗ ||Vm || ||Φε(m) ∗Vm ||(Ω) ≥ [B(n)−1 exp(−2rR)(1/4R)||Vm ||E1 (R, m)]2 . Hypothesis (1) now implies that lim sup ||Vm ||E1 (R, m) ≤ B(n) exp(2R)4R(B||V ||(Ω))1/2 . 12) Now suppose that x ∈ E2 (R, m). 7) implies that (2−1/2 σ(m))−k ||Vm ||B(x, 2−1/k σ(m)) ≤ c3 .