A real or complex-valued function \(u\) on \(d\)-dimensional euclidean spaces is Hölder continuous, when there are real constants \(C>0\) and \(\gamma>0\), such that for all \(x\) and \(y\) in the domain of \(u\)
\begin{equation*} |u(x)-u(y)|\le C \lVert x-y \rVert^{\gamma}. \end{equation*}In this manner we may define a Hölder space. Let \(U\subset \mathbb{R}^n\), \(k\in \mathbb{N}\) and \(\gamma\in (0,1]\). Then the Hölder space \(C^{k,\gamma}(U)\) consists of all functions \(u\subset C^{k,\gamma}(U)\) for which the norm
\begin{equation*} \lVert u\rVert_{C^{k,\gamma}(U)} = \lVert u\rVert_{C^k(U)} + \sum_{\lvert \alpha\rvert=k} \sup_{\substack{x, y\in U \\ x\neq y}} \frac{\lvert D^\alpha u(x)-D^\alpha u(y)\rvert}{\lvert x-y\rvert^\gamma}. \end{equation*}is finite.
Remarks
- Every Hölder space is a Banach space .
- The space \(C^{0,1}(U)\) consists of Lipschitz continuous functions.
- \(\alpha=0\) and \(\alpha>1\) are omitted, since the former is satisfied by bounded functions and the latter is only satisfied by constant functions.