Hölder condition

An inequality in which the increment of a function is expressed in terms of the increment of its argument. A function $ f $, defined in a domain $ E $ of an $ n $- dimensional Euclidean space, satisfies the Hölder condition at a point $ y \in E $ with index $ \alpha $( of order $ \alpha $), where $ 0 < \alpha \leq 1 $, and with coefficient $ A( y) $, if

$$ \tag{1 } | f ( x) - f ( y) | \leq A ( y) | x - y | ^ \alpha $$

for all $ x \in E $ sufficiently close to $ y $. One says that $ f $ satisfies the (isotropic) Hölder condition with index $ \alpha $ on a set $ E ^ \prime \subset E $ if (1) is satisfied for all $ y \in E ^ \prime $. If

$$ A = \sup _ {y \in E } A ( y) < \infty , $$

the Hölder condition is called uniform on $ E $, while $ A $ is called the Hölder coefficient of $ f $ on $ E $. Functions satisfying a Hölder condition are often referred to as Hölder continuous. The quantity

$$ | f | _ \alpha = \ | f, E | _ \alpha = \ \sup _ {x, y \in E } \ \frac{| f ( x) - f ( y) | }{| x - y | ^ \alpha } ,\ \ 0 \leq \alpha \leq 1, $$

is called the Hölder $ \alpha $- semi-norm of a bounded function $ f $ on the set $ E $. The Hölder semi-norm as a function of $ \alpha $ is logarithmically convex:

$$ | f | _ {\alpha t + \beta ( 1 - t) } \leq | f | _ \alpha ^ {t} | f | _ \beta^{1-t}. $$

The non-isotropic Hölder condition is introduced similarly to the condition (1), and has the form

$$ | f( x) - f ( y) | \leq A \sum _ {i = 1 } ^ { n } \left | \sum _ {j = 1 } ^ { n } a _ {j} ^ {i} ( x ^ {j} - y ^ {j} ) \right | ^ {\alpha _ {i} } , $$

where $ 0 < \alpha _ {i} \leq 1 $ and $ \mathop{\rm det} ( a _ {i} ^ {j} ) \neq 0 $. Functions which satisfy the non-isotropic Hölder condition are continuous and have Hölder index $ \alpha _ {i} $, $ 1 \leq i \leq n $, in the direction of the covector $ a ^ {i} = ( a _ {1} ^ {i} \dots a _ {n} ^ {i} ) $.

A condition of the form (1) was introduced by R. Lipschitz in 1864 for functions of one real variable in the context of a study of trigonometric series. In such a case the Hölder condition is often called the Lipschitz condition of order $ \alpha $ with Lipschitz constant $ A $. For functions of $ n $, $ n \geq 2 $, real variables the Hölder condition was introduced by O. Hölder in his studies of the differentiability properties of the Newton potential.

The Hölder condition can be naturally extended to the case of mappings of metric spaces. One says that a mapping $ f: X \rightarrow E $ of a metric space $ X $ into a metric space $ E $ satisfies the Hölder condition with index $ \alpha $ and coefficient $ A( x _ {0} ) $ at a point $ x _ {0} \in X $ if there exists a neighbourhood $ U ( x _ {0} ) \subset X $ of $ x _ {0} $ such that for any $ x \in U ( x _ {0} ) $ the inequality

$$ \rho _ {E} ( f ( x), f ( x _ {0} )) \leq A ( x _ {0} ) \rho _ {X} ^ \alpha ( x, x _ {0} ) $$

is valid. Here $ \rho _ {X} $ and $ \rho _ {E} $ are the metrics of the spaces $ X $ and $ E $. The Hölder condition on a set $ X ^ \prime \subset X $, the uniform Hölder condition on $ X $ and the Hölder $ \alpha $- semi-norm are introduced in a similar manner.

A vector space of functions which satisfy any Hölder condition is a Hölder space.