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The Lipschitz norm of a function over a domain $D \subseteq \mathbb R^n$ is easy to define: $$\|f\|_{\mathrm{Lip}} = \sup_{x,y \in D} \frac{|f(x) - f(y)|}{|x-y|}.$$ Is there a standard notation for the space of functions whose Lipschitz norm is finite? i.e., $$X(D) = \{f \in C(D) : \|f\|_{\mathrm{Lip}} < \infty \}.$$ Even better, what about functions which are continuously-differentiable and whose derivative has finite Lipschitz norm? $$X^1(D) = \{f \in C^1(D) : \|\nabla f\|_{\mathrm{Lip}} < \infty \}$$ I am using $X$ and $X^1$ here only as placeholders, and would prefer a better notation.

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Your first space would, I think, usually be denoted by $Lip_1(D)$ or even just $Lip(D)$. There are several papers on "Lipschitz algebras" if you look on MathSciNet. –  Yemon Choi Jul 8 '10 at 19:37
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Aren't these just $C^{0,1}(D)$ and $C^{1,1}(D)$? Or am I missing something? –  Willie Wong Jul 8 '10 at 19:37
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Yeah, but if you go with Yemon's notation then for goodness' sake write it as $\operatorname{Lip}_1(D)$ or $\operatorname{Lip}(D)$. –  Harald Hanche-Olsen Jul 8 '10 at 20:34
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\operatorname ftw! –  Kevin H. Lin Jul 10 '10 at 20:03
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For a domain $D\subset \mathbb{R}^n$ I would definitely go for Willie's suggestion as the standard ones. The notations $\mathrm{Lip}(X)$ and $\mathrm{Lip}_k(X)$ are quite standard for the Lipschitz functions on the metric space $X$, resp., the k-Lipschitz functions on $X .$ –  Pietro Majer Oct 23 '10 at 9:46
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