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Difference between revisions of "Hölder space"

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m (moved Holder space to Hölder space over redirect: accented title)
(ball, not sphere (translation error, probably))
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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" />.
 
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" />.
  
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" />.
+
2) The unit ball 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" />.
  
 
====References====
 
====References====

Revision as of 11:46, 16 May 2015

A Banach space of bounded continuous functions defined on a set of an -dimensional Euclidean space and satisfying a Hölder condition on .

The Hölder space , where is an integer, consists of the functions that are times continuously differentiable on (continuous for ).

The Hölder space , , where is an integer, consists of the functions that are times continuously differentiable (continuous for ) and whose -th derivatives satisfy the Hölder condition with index .

For bounded a norm is introduced in and as follows:

where , is an integer,

The fundamental properties of Hölder spaces for a bounded connected domain ( is the closure of ) are:

1) is imbedded in if , where and are integers, , . Here and the constant is independent of .

2) The unit ball of is compact in if . Consequently, any bounded set of functions from contains a sequence of functions that converges in the metric of to a function of .

References

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


Comments

If, in the above, , then is the Hölder -semi-norm of on , i.e.

See Hölder condition, where this norm is denoted .

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=23330
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article