# Origin of term Ahlfors-David regular

Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $$q>0$$. A metric space $$(X,d)$$ is Ahlfors(-David) $$q$$-regular if there exists $$C\geq 1$$ such that $$C^{-1}r^q \leq \mathcal{H}^q(B(x,r)) \leq Cr^q$$ for all $$x \in X$$ and $$r \in (0, \text{diam } X)$$. Here, $$\mathcal{H}^q$$ denotes the $$q$$-dimensional Hausdorff measure. The term is so ubiquitous in the literature in this area that the origin seems impossible to trace down. I believe David refers to Guy David.

Can anyone fill me in on the origin of the terms and what exactly Ahlfors and David did using this type of condition?

## 2 Answers

To answer the last question, Calderón's problem was a question regarding mapping properties of the Cauchy integral

$$C_{\Gamma}f(z)=\frac{1}{2\pi i} \int\limits_{\Gamma} \frac{f(\xi)}{\xi-z} d\xi$$

namely, to determine the rectifiable Jordan curves $$\Gamma$$ for which $$C_{\Gamma}$$ gives rise to a bounded operator on $$L^2(\Gamma)$$. This was solved by Guy David in 1984 who showed that $$C_{\Gamma}$$ is bounded on $$L^2(\Gamma)$$ precisely when $$\Gamma$$ satisfies $$\mathcal{H}(\Gamma \cap B(z_{0},r)) \leq Cr$$ for every $$z_{0}\in\mathbb{C}$$, $$r>0$$ and some constant $$C$$. This opened up a large study of (what was called then) Ahlfors regularity by David and Semmes. Some of the results of that study are collected in their monograph from the 90's, Analysis of and on Uniformly Rectifiable Sets.

When $$q=1$$ this condition appears in Ahlfors' famous paper Zur Theorie der Uberlagerungsflache, Acta Math., 65 (1935) for which he was awarded one of the two very first Fields medals. For long time it was known to complex analysts just as the "Ahlfors condition".