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Is Is $u:B_1(0)\rightarrow \mathbb{R}, \beta$-Hölder continuous given by $u(x) =|x|^\beta $? i.e \begin{equation} \sup \left \{ \dfrac{||x|^\beta-|y|^{\beta}|}{|x-y|^{\beta}} : \begin{matrix}|x|<1 \\ |y|<1 \end{matrix}\right \} \le C <\infty \end{equation} Assume $0< \beta <1$.

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Do you really think that it was necessary to create a new tag regularity for this question? And if you think that this tag would be useful, you should add description of the tag in the tag-wiki and tag-excerpt; since the word regularity is used in many different meanings in mathematics. (I don't think that such tag is needed.) – Martin Sleziak Jul 2 '12 at 4:21
If you don't know what did is talking about in his comment, you can read more here: How do I accept an answer? and Why should we accept answers?. – Martin Sleziak Jul 2 '12 at 10:04
@Martin: I agree. Re-tagged as holder-spaces. (Sorry, but tags don't support umlauts...) – Willie Wong Jul 2 '12 at 11:24
up vote 4 down vote accepted

Since we know that $|a+b|^\beta\le |a|^\beta + |b|^\beta$ (see e.g. here), we get for $a=x-y$, $b=y$ $$|x|^\beta \le |x-y|^\beta + |y|^\beta$$ which is equivalent to $$|x|^\beta - |y|^\beta \le |x-y|^\beta.$$ By symmetry we also have $|y|^\beta - |x|^\beta \le |x-y|^\beta$, which together gives $$||x|^\beta - |y|^\beta| \le |x-y|^\beta.$$

This is basically the same trick as in the proof of this form of triangle inequality.

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