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 .
|||C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)|
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://www.encyclopediaofmath.org/index.php?title=H%C3%B6lder_space&oldid=36411