A property of $C^{k,\alpha}$ functions

TL;DR

This paper investigates the differentiability and Hölder continuity properties of fractional powers of nonnegative functions with certain smoothness and vanishing derivative conditions at zeros, extending classical regularity results.

Contribution

It establishes new regularity results for fractional powers of functions with Hölder continuous derivatives, including conditions for differentiability and Hölder continuity of the derivatives.

Findings

$(f^b)'}$ is differentiable for specific b in (1/(k+a), 1)

$(f^b)'}$ is Hölder continuous with exponent b(1+a) - 1 under additional conditions

$(f^b)'}$ is Lipschitz continuous where $f(x) > 0$

Abstract

Let $f$ be a nonnegative function of class $C^k$ ($k \geq 2$) such that $f^{(k)}$ is H\''older continuous with exponent $\alpha$ in $(0,1]$. If $f'(x) = \cdots = f^{(k)}(x) = 0$ when $f(x) = 0$, we show that $f^{\mu}$ is differentiable for $\mu \in (1/(k+\alpha), 1)$ and under an additional condition we show that $(f^\mu)'$ is H\''older continuous with exponent $\beta = \mu(1+\alpha) - 1$ (if $\beta \leq 1$) at $x \in [0,T]$ when $f(x) = 0$. $(f^\mu)'$ is Lipschitz continuous at $x$ if $f(x) > 0$.