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.