# 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:

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 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.

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=12054