Hölder condition

An inequality in which the increment of a function is expressed in terms of the increment of its argument. A function , defined in a domain of an -dimensional Euclidean space, satisfies the Hölder condition at a point with index (of order ), where , and with coefficient , if

(1)

for all sufficiently close to . One says that satisfies the (isotropic) Hölder condition with index on a set if (1) is satisfied for all . If

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

is called the Hölder -semi-norm of a bounded function on the set . The Hölder semi-norm as a function of is logarithmically convex:

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

where and . Functions which satisfy the non-isotropic Hölder condition are continuous and have Hölder index , , in the direction of the covector .

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 with Lipschitz constant . For functions of , , 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 of a metric space into a metric space satisfies the Hölder condition with index and coefficient at a point if there exists a neighbourhood of such that for any the inequality

is valid. Here and are the metrics of the spaces and . The Hölder condition on a set , the uniform Hölder condition on and the Hölder -semi-norm are introduced in a similar manner.

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