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A Campanato characterization Hölder continuity holds in domains $\Omega$ such that for all $x_0 \in \Omega$ and $00.$ I'd like to know how to interpret such domains. For instance, geometrically. How can I think about a set with such a property?

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If the set is bounded and the boundary is regular the condition is satisfied. However if the domain does have cusp-like spikes in its boundary, as in this picture, the condition is not satisfied.

enter image description here

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    Thank you. English isn't my language, then i have dificulties in undertand such particular terms. Can you give me examples or references in ordem to undertand. – 2017-01-08
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    I added a picture – 2017-01-08
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    Thank you very much for your atention and effort. – 2017-01-09