About

Let $X,Y$ be metric spaces and denote their distance functions by $d_X$ and $d_Y$. A mapping $f: X\to Y$ is said to be Hölder continuous with exponent $\alpha \in (0,1)$ (sometimes $\alpha$-Hölder continuous), and denoted $f\in C^\alpha(X,Y)$ or $f\in C^{0,\alpha}(X,Y)$ if

$$ \sup_{x_1,x_2\in X; x_1\neq x_2} \frac{d_Y\left( f(x_1),f(x_2)\right)}{d_X\left(x_1,x_2\right)^\alpha} < \infty$$

The value of the supremum is sometimes denoted $[f]_\alpha$ and is called the Hölder coefficient.

The case $\alpha = 1$ (which is always denoted $C^{0,1}$ and not $C^1$ so as not to be confused with the space of continuously differentiable functions) corresponds to Lipschitz continuity.

The notion of Hölder continuity is used to quantify how continuous (and how close to differentiable) a function is. It can be extended also to higher derivatives: letting $X$ and $Y$ be subsets of Euclidean spaces, we can define the space $C^{k,\alpha}(X,Y)$ to be subspace of $k$-times continuously differentiable functions all of whose $k$th partial derivatives are $\alpha$-Hölder continuous. This can be made in to a Banach space with the norm

$$ \|f\|_{k,\alpha} = \|f\|_{C^k} + \sum_{|\gamma| = k} [D^\gamma f]_\alpha $$

The various Hölder spaces are frequently used to study quantitative estimates of differentiability in harmonic analysis and analysis of elliptic and parabolic partial differential equations.

history | show excerpt | excerpt history