Hölder space

Summary

The Hölder space $C^{k,\alpha}$ consists of the functions whose derivatives up to order $k$ are bounded. In addition, the derivatives of order $k$ must be Hölder continuous with exponent $\alpha$, where $0 < \alpha \le 1$. With the natural norm it is a Banach space. Hölder spaces are the setting of the classical regularity theory of elliptic and parabolic equations, as in the Schauder estimates. They fill the gaps between the spaces $C^k$ of $k$ times continuously differentiable functions.