Let is a domain in , is function on ,

if

if

And for

,

Then we can define a norm

So we have generalized holder space . In arbitrary domain , we don’t have the inclusion when . But for domain, we have

Suppose , where , and . Let be a domain and . Then for any and some constant , we have

From this interpolation result, we can prove the compactness between holder space

Suppose with . as before. The inclusion is compact

GT p137

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