Difference between revisions of "Hölder space"
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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 | + | 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.
Hölder space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=H%C3%B6lder_space&oldid=36411