Hölder space
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:
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where ,
is an integer,
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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 sphere 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.
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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=12054