# Hölder space

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A Banach space of bounded continuous functions $f( x) = f( x ^ {1} \dots x ^ {n} )$ defined on a set $E$ of an $n$- dimensional Euclidean space and satisfying a Hölder condition on $E$.

The Hölder space $C _ {m} ( E)$, where $m \geq 0$ is an integer, consists of the functions that are $m$ times continuously differentiable on $E$( continuous for $m = 0$).

The Hölder space $C _ {m + \alpha } ( E)$, $0 < \alpha \leq 1$, where $m \geq 0$ is an integer, consists of the functions that are $m$ times continuously differentiable (continuous for $m = 0$) and whose $m$- th derivatives satisfy the Hölder condition with index $\alpha$.

For bounded $E$ a norm is introduced in $C _ {m} ( E)$ and $C _ {m + \alpha } ( E)$ as follows:

$$| f | _ {m} = \| f, E \| _ {m} = \ \sum _ {| k | = 0 } ^ { m } \sup _ {x \in E } | f ^ { ( k) } ( x) |,$$

$$| f | _ {m + \alpha } = \| f, E \| _ {m + \alpha } = | f | _ {m} + \sum _ {| k | = m } \| f ^ { ( k) } , E \| _ \alpha ,$$

where $k = ( k _ {1} \dots k _ {n} )$, $k _ {j} \geq 0$ is an integer,

$$| k | = k _ {1} + \dots + k _ {n} ,\ \ f ^ { ( k) } ( x) = \ \frac{\partial ^ {| k | } f ( x) }{\partial x _ {1} ^ {k _ {1} } \dots \partial x _ {n} ^ {k _ {n} } } .$$

The fundamental properties of Hölder spaces for a bounded connected domain ( $\overline{E}\;$ is the closure of $E$) are:

1) $C _ {m + \beta } ( \overline{E}\; )$ is imbedded in $C _ {k + \alpha } ( \overline{E}\; )$ if $0 \leq k + \alpha \leq m + \beta$, where $k$ and $m$ are integers, $0 < \alpha \leq 1$, $0 \leq \beta \leq 1$. Here $| f | _ {k + \alpha } \leq A | f | _ {m + \beta }$ and the constant $A$ is independent of $f \in C _ {m + \beta } ( \overline{E}\; )$.

2) The unit ball of $C _ {m + \beta } ( \overline{E}\; )$ is compact in $C _ {m + \alpha } ( \overline{E}\; )$ if $0 < \alpha < \beta$. Consequently, any bounded set of functions from $C _ {m + \beta } ( \overline{E}\; )$ contains a sequence of functions that converges in the metric of $C _ {m + \alpha } ( \overline{E}\; )$ to a function of $C _ {m + \alpha } ( \overline{E}\; )$.

#### References

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

If, in the above, $0 < \alpha < 1$, then $\| f , E \| _ \alpha$ is the Hölder $\alpha$- semi-norm of $f$ on $E$, i.e.
$$\| f , E \| _ \alpha = \sup _ {x,y \in E } \ \frac{| f( x) - f( y) | }{| x- y | ^ \alpha } .$$
See Hölder condition, where this norm is denoted $| f, E | _ \alpha$.