Hölder space

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)

Comments

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 $.

Hölder spaces play a role in partial differential equations, potential theory, complex analysis, functional analysis (cf. Imbedding theorems), etc.