A property of Ck,? functions

Abstract

Robert Dalmasso

Let f be a nonnegative function of class Ck (k ≥ 2) such that f(k) is Holder continuous with exponent α in (0,1]. If f ' (x) = ••• = f(k)(x) = 0 when f(x) = 0, we show that fμ is differentiable for μ ∈ (1/(k + α),1) and under an additional condition we show that (fμ)' is Holder continuous with exponent β = μ(1 + α) − 1 (if β ≤ 1) at x ∈ [0,T] when f(x) = 0. (fμ)' is Lipschitz continuous at x if f(x) > 0.

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