Local Holder Continuity

The local Hölder continuity is a concept of mathematics that generalizes the Hölder continuity and thus also the Lipschitz continuity . It is named after Otto Hölder and is used, for example, in probability theory when formulating the Kolmogorow-Tschenzow theorem . This provides criteria as to when modifications of a stochastic process exist that are locally wood-continuous.

definition

Let two metric spaces and be given . An illustration ${\displaystyle (E_{1},d_{1})}$ ${\displaystyle (E_{2},d_{2})}$

${\displaystyle \varphi \colon E_{1}\to E_{2}}$

is called locally Hölder-continuous of order γ or locally Hölder-γ-continuous for short , if for each there is a really positive and a really positive number , so that for all with and the inequality ${\displaystyle t\in E_{1}}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle C=C(t,\varepsilon )>0}$ ${\displaystyle r,s\in E_{1}}$ ${\displaystyle d_{1}(r,t)<\varepsilon }$ ${\displaystyle d_{1}(s,t)<\varepsilon }$

${\displaystyle d_{2}(\varphi (r),\varphi (s))\leq C\cdot d_{1}(r,s)^{\gamma }}$

applies.

Examples

• Every Lipschitz continuous function with Lipschitz constant is locally Hölder continuous with exponent and ${\displaystyle L}$ ${\displaystyle \gamma =1}$ ${\displaystyle C=L}$
• Every Hölder-continuous mapping with constant and exponent is also locally Hölder-continuous with constant and exponent . ${\displaystyle C}$ ${\displaystyle \gamma }$ ${\displaystyle C}$ ${\displaystyle \gamma }$

properties

• If a real-valued function of a real variable is locally Hölder-continuous with an exponent , then also locally Hölder-continuous for every exponent with . ${\displaystyle f}$ ${\displaystyle \gamma \in (0,1]}$ ${\displaystyle f}$ ${\displaystyle \gamma ^{*}}$ ${\displaystyle 0\leq \gamma ^{*}\leq \gamma }$
• If the domain of definition is compact, then Hölder continuity follows from local Hölder continuity. In general, this conclusion is wrong. ${\displaystyle f}$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 468-469 , doi : 10.1007 / 978-3-642-36018-3 .