Assume that $0<\alpha<1/2$ and $f$ is analytic in the unit disk $D$ with $|f(z)-f(w)|\le M|z-w|^\alpha$. Can we state that in general $$\int_D |f'(z)|^2 dx dy <\infty?$$
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$\begingroup$ Is there a reason to believe or hope that it is true? $\endgroup$– Joonas IlmavirtaSep 20, 2014 at 20:05
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1$\begingroup$ There is an example of a function which satisfies the Holder condition on the boundary circle (which I suspect implies the Holder condition on the interior as well) but has Taylor coefficients $O(n^{-1})$, so does not lie in Dirichlet space $\endgroup$– Yemon ChoiSep 21, 2014 at 0:19
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$\begingroup$ @Yemon choi: Of course if, the function in Holder in the boundary, then it is Holder continuous inside. Probably you right! $\endgroup$– user57714Sep 21, 2014 at 11:44
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