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An inequality in which the increment of a function is expressed in terms of the increment of its argument. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h0475121.png" />, defined in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h0475122.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h0475123.png" />-dimensional Euclidean space, satisfies the Hölder condition at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h0475124.png" /> with index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h0475126.png" /> (of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h0475128.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h0475129.png" />, and with coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751210.png" />, if
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751211.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|done}}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751212.png" /> sufficiently close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751213.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751214.png" /> satisfies the (isotropic) Hölder condition with index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751215.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751216.png" /> if (1) is satisfied for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751217.png" />. If
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751218.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
| f ( x) - f ( y) |
 +
\leq  A ( y)  | x - y |  ^  \alpha
 +
$$
  
the Hölder condition is called uniform on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751219.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751220.png" /> is called the Hölder coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751221.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751222.png" />. Functions satisfying a Hölder condition are often referred to as Hölder continuous. The quantity
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751223.png" /></td> </tr></table>
+
$$
 +
= \sup _ {y \in E }  A ( y)  < \infty ,
 +
$$
  
is called the Hölder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751225.png" />-semi-norm of a bounded function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751226.png" /> on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751227.png" />. The Hölder semi-norm as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751228.png" /> is logarithmically convex:
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751229.png" /></td> </tr></table>
+
$$
 +
| 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
 
The non-isotropic Hölder condition is introduced similarly to the condition (1), and has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751230.png" /></td> </tr></table>
+
$$
 +
| f( x) - f ( y) |
 +
\leq  A
 +
\sum _ {i = 1 } ^ { n }  \left |
 +
\sum _ {j = 1 } ^ { n }
 +
a _ {j}  ^ {i} ( x  ^ {j} - y  ^ {j} ) \right | ^ {\alpha _ {i} } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751231.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751232.png" />. Functions which satisfy the non-isotropic Hölder condition are continuous and have Hölder index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751233.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751234.png" />, in the direction of the covector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751235.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751238.png" /> with Lipschitz constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751239.png" />. For functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751240.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751241.png" />, real variables the Hölder condition was introduced by O. Hölder in his studies of the differentiability properties of the Newton potential.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751242.png" /> of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751243.png" /> into a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751244.png" /> satisfies the Hölder condition with index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751245.png" /> and coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751246.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751247.png" /> if there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751248.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751249.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751250.png" /> the inequality
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751251.png" /></td> </tr></table>
+
$$
 +
\rho _ {E} ( f ( x), f ( x _ {0} ))
 +
\leq  A ( x _ {0} )
 +
\rho _ {X}  ^  \alpha  ( x, x _ {0} )
 +
$$
  
is valid. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751252.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751253.png" /> are the metrics of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751254.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751255.png" />. The Hölder condition on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751256.png" />, the uniform Hölder condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751257.png" /> and the Hölder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047512/h04751258.png" />-semi-norm are introduced in a similar manner.
+
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|Hölder space]].
 
A vector space of functions which satisfy any Hölder condition is a [[Hölder space|Hölder space]].

Latest revision as of 22:11, 5 June 2020


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.

How to Cite This Entry:
Hölder condition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=H%C3%B6lder_condition&oldid=47305
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article