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A [[Banach space|Banach space]] of bounded continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h0475161.png" /> defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h0475162.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h0475163.png" />-dimensional Euclidean space and satisfying a [[Hölder condition|Hölder condition]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h0475164.png" />.
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The Hölder space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h0475165.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h0475166.png" /> is an integer, consists of the functions that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h0475167.png" /> times continuously differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h0475168.png" /> (continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h0475169.png" />).
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{{TEX|done}}
  
The Hölder space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751610.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751611.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751612.png" /> is an integer, consists of the functions that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751613.png" /> times continuously differentiable (continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751614.png" />) and whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751615.png" />-th derivatives satisfy the [[Hölder condition|Hölder condition]] with index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751616.png" />.
+
A [[Banach space|Banach space]] of bounded continuous functions $  f( x) = f( x  ^ {1} \dots x  ^ {n} ) $
 +
defined on a set  $  E $
 +
of an  $  n $-
 +
dimensional Euclidean space and satisfying a [[Hölder condition|Hölder condition]] on  $  E $.
  
For bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751617.png" /> a norm is introduced in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751618.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751619.png" /> as follows:
+
The Hölder space  $  C _ {m} ( E) $,
 +
where  $  m \geq  0 $
 +
is an integer, consists of the functions that are  $  m $
 +
times continuously differentiable on  $  E $(
 +
continuous for  $  m = 0 $).
  
<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/h047516/h04751620.png" /></td> </tr></table>
+
The Hölder space  $  C _ {m + \alpha }  ( E) $,
 +
$  0 < \alpha \leq  1 $,
 +
where  $  m \geq  0 $
 +
is an integer, consists of the functions that are  $  m $
 +
times continuously differentiable (continuous for  $  m = 0 $)
 +
and whose  $  m $-
 +
th derivatives satisfy the [[Hölder condition|Hölder condition]] with index  $  \alpha $.
  
<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/h047516/h04751621.png" /></td> </tr></table>
+
For bounded  $  E $
 +
a norm is introduced in  $  C _ {m} ( E) $
 +
and  $  C _ {m + \alpha }  ( E) $
 +
as follows:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751622.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751623.png" /> is an integer,
+
$$
 +
| f | _ {m}  = \| f, E \| _ {m}  = \
 +
\sum _ {| k | = 0 } ^ { m }
 +
\sup _ {x \in E }  | f ^ { ( k) } ( x) |,
 +
$$
  
<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/h047516/h04751624.png" /></td> </tr></table>
+
$$
 +
| f | _ {m + \alpha }  = \| f, E \| _ {m
 +
+ \alpha }  = | f | _ {m} + \sum _ {| k |
 +
= m } \| f ^ { ( k) } , E \| _  \alpha  ,
 +
$$
  
The fundamental properties of Hölder spaces for a bounded connected domain (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751625.png" /> is the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751626.png" />) are:
+
where  $  k = ( k _ {1} \dots k _ {n} ) $,
 +
$  k _ {j} \geq  0 $
 +
is an integer,
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751627.png" /> is imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751628.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751629.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751630.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751631.png" /> are integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751632.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751633.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751634.png" /> and the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751635.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751636.png" />.
+
$$
 +
| k |  =  k _ {1} + \dots + k _ {n} ,\ \
 +
f ^ { ( k) } ( x)  = \
  
2) The unit sphere of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751637.png" /> is compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751638.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751639.png" />. Consequently, any bounded set of functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751640.png" /> contains a sequence of functions that converges in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751641.png" /> to a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751642.png" />.
+
\frac{\partial  ^ {| k | } f ( x) }{\partial  x _ {1} ^ {k _ {1} } \dots \partial  x _ {n} ^ {k _ {n} } }
 +
.
 +
$$
 +
 
 +
The fundamental properties of Hölder spaces for a bounded connected domain ( $  \overline{E}\; $
 +
is the closure of $  E $)
 +
are:
 +
 
 +
1)  $  C _ {m + \beta }  ( \overline{E}\; ) $
 +
is imbedded in  $  C _ {k + \alpha }  ( \overline{E}\; ) $
 +
if  $  0 \leq  k + \alpha \leq  m + \beta $,
 +
where  $  k $
 +
and  $  m $
 +
are integers,  $  0 < \alpha \leq  1 $,
 +
0 \leq  \beta \leq  1 $.  
 +
Here  $  | f | _ {k + \alpha }  \leq  A  | f | _ {m + \beta }  $
 +
and the constant  $  A $
 +
is independent of  $  f \in C _ {m + \beta }  ( \overline{E}\; ) $.
 +
 
 +
2) The unit ball of  $  C _ {m + \beta }  ( \overline{E}\; ) $
 +
is compact in $  C _ {m + \alpha }  ( \overline{E}\; ) $
 +
if  $  0 < \alpha < \beta $.  
 +
Consequently, any bounded set of functions from $  C _ {m + \beta }  ( \overline{E}\; ) $
 +
contains a sequence of functions that converges in the metric of $  C _ {m + \alpha }  ( \overline{E}\; ) $
 +
to a function of $  C _ {m + \alpha }  ( \overline{E}\; ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Miranda,  "Partial differential equations of elliptic type" , Springer  (1970)  (Translated from Italian)</TD></TR></table>
  
 +
====Comments====
 +
If, in the above,  $  0 < \alpha < 1 $,
 +
then  $  \| f , E \| _  \alpha  $
 +
is the Hölder  $  \alpha $-
 +
semi-norm of  $  f $
 +
on  $  E $,
 +
i.e.
  
 +
$$
 +
\| f , E \| _  \alpha  =  \sup _ {x,y \in E } \
  
====Comments====
+
\frac{| f( x) - f( y) | }{| x- y |  ^  \alpha  }
If, in the above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751643.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751644.png" /> is the Hölder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751645.png" />-semi-norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751646.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751647.png" />, i.e.
+
.
 
+
$$
<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/h047516/h04751648.png" /></td> </tr></table>
 
  
See [[Hölder condition|Hölder condition]], where this norm is denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047516/h04751649.png" />.
+
See [[Hölder condition|Hölder condition]], where this norm is denoted $  | f, E | _  \alpha  $.
  
 
Hölder spaces play a role in partial differential equations, potential theory, complex analysis, functional analysis (cf. [[Imbedding theorems|Imbedding theorems]]), etc.
 
Hölder spaces play a role in partial differential equations, potential theory, complex analysis, functional analysis (cf. [[Imbedding theorems|Imbedding theorems]]), etc.

Latest revision as of 22:11, 5 June 2020


A Banach space of bounded continuous functions $ f( x) = f( x ^ {1} \dots x ^ {n} ) $ defined on a set $ E $ of an $ n $- dimensional Euclidean space and satisfying a Hölder condition on $ E $.

The Hölder space $ C _ {m} ( E) $, where $ m \geq 0 $ is an integer, consists of the functions that are $ m $ times continuously differentiable on $ E $( continuous for $ m = 0 $).

The Hölder space $ C _ {m + \alpha } ( E) $, $ 0 < \alpha \leq 1 $, where $ m \geq 0 $ is an integer, consists of the functions that are $ m $ times continuously differentiable (continuous for $ m = 0 $) and whose $ m $- th derivatives satisfy the Hölder condition with index $ \alpha $.

For bounded $ E $ a norm is introduced in $ C _ {m} ( E) $ and $ C _ {m + \alpha } ( E) $ as follows:

$$ | f | _ {m} = \| f, E \| _ {m} = \ \sum _ {| k | = 0 } ^ { m } \sup _ {x \in E } | f ^ { ( k) } ( x) |, $$

$$ | f | _ {m + \alpha } = \| f, E \| _ {m + \alpha } = | f | _ {m} + \sum _ {| k | = m } \| f ^ { ( k) } , E \| _ \alpha , $$

where $ k = ( k _ {1} \dots k _ {n} ) $, $ k _ {j} \geq 0 $ is an integer,

$$ | k | = k _ {1} + \dots + k _ {n} ,\ \ f ^ { ( k) } ( x) = \ \frac{\partial ^ {| k | } f ( x) }{\partial x _ {1} ^ {k _ {1} } \dots \partial x _ {n} ^ {k _ {n} } } . $$

The fundamental properties of Hölder spaces for a bounded connected domain ( $ \overline{E}\; $ is the closure of $ E $) are:

1) $ C _ {m + \beta } ( \overline{E}\; ) $ is imbedded in $ C _ {k + \alpha } ( \overline{E}\; ) $ if $ 0 \leq k + \alpha \leq m + \beta $, where $ k $ and $ m $ are integers, $ 0 < \alpha \leq 1 $, $ 0 \leq \beta \leq 1 $. Here $ | f | _ {k + \alpha } \leq A | f | _ {m + \beta } $ and the constant $ A $ is independent of $ f \in C _ {m + \beta } ( \overline{E}\; ) $.

2) The unit ball of $ C _ {m + \beta } ( \overline{E}\; ) $ is compact in $ C _ {m + \alpha } ( \overline{E}\; ) $ if $ 0 < \alpha < \beta $. Consequently, any bounded set of functions from $ C _ {m + \beta } ( \overline{E}\; ) $ contains a sequence of functions that converges in the metric of $ C _ {m + \alpha } ( \overline{E}\; ) $ to a function of $ C _ {m + \alpha } ( \overline{E}\; ) $.

References

[1] C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)

Comments

If, in the above, $ 0 < \alpha < 1 $, then $ \| f , E \| _ \alpha $ is the Hölder $ \alpha $- semi-norm of $ f $ on $ E $, i.e.

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

See Hölder condition, where this norm is denoted $ | f, E | _ \alpha $.

Hölder spaces play a role in partial differential equations, potential theory, complex analysis, functional analysis (cf. Imbedding theorems), etc.

How to Cite This Entry:
Hölder space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=H%C3%B6lder_space&oldid=12054
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article