# 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?

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".

• would you please introduce to me some reference where the Ahlfors condition is discussed from a complex analysis point of view?
– XIII
Feb 4, 2021 at 20:50
• @XIE: I gave a reference on the primary source. Feb 5, 2021 at 0:41
• Did you mean Ahlfors's original paper? By sure it had a version of the condition. unfortunately I can't read German. I even was not able to find Ahlfors's original term (maybe it is Regularitätsbedingung). I know some books on complex analysis, such as Nevanlinna and Hayman discussed the work of Ahlfors, but I didn't find their exposition of the "Ahlfors condition". @Alexandre Eremenko
– XIII
Feb 5, 2021 at 14:00

As I know the second inequality in the definition is due to Ahlfors, and the first one is due to Guy David. Let $$E$$ be a subset of $$\mathbb {R} ^ n$$. One says that $$E$$ is Ahlfors-David regular of order d if $$C^{-1}r^d \leq \mathcal{H}^d(E \cap B(x,r)) \leq Cr^d$$ for every $$x \in \mathbb {R} ^ n$$ and $$r>0$$ (In other word, if $$\mu = \mathcal{H}^d |_{E}$$ be an Ahlfors-David regular measure).

An important theorem of Mattila-Melnikov-Verdera (DOI: 10.2307/2118585) says that if $$E \subset \mathbb {C}$$ be an Ahlfors-David regular set of order 1, then:

1- The Cauchy integral $$\mathcal{C}_E$$ is bounded in $$L^2 (E) \iff E \subset \Gamma$$, which $$\Gamma$$ is a Ahlfors-David regular curve.

2- The analytic capacity $$\gamma (E)=0 \iff E$$ is totaly unrectifiable.

This a partial answer to the Vitushkin conjecture, and Ahlfors-David regularity plays a crucial rule in the proof.

Mattila, Pertti; Melnikov, Mark S.; Verdera, Joan, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. Math. (2) 144, No. 1, 127-136 (1996). ZBL0897.42007.

Christ, Michael, Lectures on singular integral operators. Expository lectures from the CBMS regional conference held at the University of Montana, Missoula, MT (USA) from August 28-September 1, 1989, Regional Conference Series in Mathematics. 77. Providence, RI: American Mathematical Society (AMS). ix, 132 p. (1990). ZBL0745.42008.