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.
Hölder condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=H%C3%B6lder_condition&oldid=16871