## Notation for space of Lipschitz continuous functions

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 2010 at 19:37
Aren't these just $C^{0,1}(D)$ and $C^{1,1}(D)$? Or am I missing something? – Willie Wong Jul 8 2010 at 19:37
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 2010 at 20:34
Both Yemon and Willie gave good answers. If either one wants to leave their comment as an answer to the question, I'll be happy to choose it to give them some points. Willie, since your reputation score is lower, you have first dibs. – Tom LaGatta Jul 9 2010 at 4:19
\operatorname ftw! – Kevin Lin Jul 10 2010 at 20:03