Jerison and Kenig give the following two conditions (alongside an exterior corkscrew condition, which is not relevant to this discussion) in the definition of an $(M,r_0)$ nontangentially accessible domain $\Omega$:
- Inner corkscrew condition (IC): if $q\in\partial\Omega$, and $r<r_0$, then there is a point $a\in\Omega$ such that $d(a,\partial\Omega)>r/M$ and $d(a,q)<r$
- Harnack chain condition (HC): if $a_1,a_2\in\Omega$ satisfy $d(a_i,\partial\Omega)>\varepsilon$ and $d(a_1,a_2)<C\varepsilon$ then there is a chain of balls $B_1,\ldots,B_n\subset\Omega$, $a_1\in B_1$, $a_2\in B_n$, such that $B_i$ intersects $B_{i+1}$, and $n$ depends only on $C$ and $\Omega$ (i.e. not on $\varepsilon$ or $a_i$). Moreover, the balls each satisfy the nontangentiality condition $r_j/M<d(B_j,\partial\Omega)<r_jM$ (where $r_j$ is the radius of $B_j$).
They also define the following condition:
- Condition $(*)$: if $a_1,a_1\in\Omega$ satisfy $d(a_i,\partial\Omega)>\varepsilon$ and $d(a_1,a_2)<2^k\varepsilon$, then there is a chain of $M$-nontangential balls as in (HC) from $a_1$ to $a_2$ of length $Mk$. Moreover, each ball in this chain has radius $r(B_j)>\min_i(d(a_i,B_j))/M$.
Jerison and Kenig then goes on to state that $(*)$ is equivalent to (IC)+(HC) combined; I don't quite see how this follows. If the $M$ in $(*)$ is supposed to be different from the $M$ in (IC), then $(*)$ just seems like a slightly improved version of (HC), in which the dependence on $C$ is logarithmic; and if the $M$ is supposed to be the same, then it seems absurdly strong. Moreover, I simply can't see any sensible way to pick the $q,a(q)$ in (IC) in a way that is suited to constructing a Harnack chain at all.
I am interested in extracting constants, so along with a proof of equivalence I would quite like to see at least an explicit dependence of the $M$ in $(*)$ to the $M$ in (IC); ideally they'd be equal, but that might be asking too much.
