Uniform circular distortion gives planar quasiconformality by difference quotients

Define the locally finite Borel measure $\omega (E)=\mathrm{area}(h(E))$. This is a measure because a homeomorphism maps disjoint Borel sets to disjoint Borel sets and respects unions. It is Radon because it is the pushforward of planar area by the homeomorphism $h^{-1}$; compact and open approximation transport through that homeomorphism. The open image $h(B(x,r))$ contains $B(h(x),l(x,r))$: otherwise a segment from $h(x)$ to a point of that smaller disk outside the image would first meet the boundary $h(\mathrm{\partial }B(x,r))$, contradicting the definition of $l$. The maximum distance from $h(x)$ in $h(\overline{B(x,r)})$ occurs on its boundary, because distance from a fixed point has no maximum in a nonempty open set. Hence $\mathrm{\mid }h(y)-h(x)\mathrm{\mid }\le L(x,r)\le Hl(x,r)$ for $y\in \overline{B(x,r)}$, and $\pi l(x,r{)}^2\le \omega (B(x,r))$.

For $x\in V$ and $0<\mathrm{\mid }t\mathrm{\mid }<\mathrm{dist}(V,\mathrm{\partial }U)\mathrm{/}2$, use step 1.1 with radius $2\mathrm{\mid }t\mathrm{\mid }$ to obtain $\mathrm{\mid }h(x+t{e}_i)-h(x){\mathrm{\mid }}^2\le (H^2\mathrm{/}\pi )\omega (B(x,2\mathrm{\mid }t\mathrm{\mid }))$. These balls lie in $U$. Tonelli, applied to their incidence indicator, gives ${\int }_V\omega (B(x,2\mathrm{\mid }t\mathrm{\mid }))dx={\int }_U\mathrm{\mid }V\cap B(y,2\mathrm{\mid }t\mathrm{\mid })\mathrm{\mid }d\omega (y)\le 4\pi t^2\omega (U)$. Combining the two inequalities proves the explicit difference-quotient estimate. All measures in this calculation are locally finite Radon measures and the incidence set is open, so no absolute continuity or null-set preservation of $h$ has been presumed.

Continuity makes $h$ square integrable on every relatively compact $U$. The uniform estimate in step 2.1 and [F1], with $p=2$ and complex-valued functions, give $h\in W^{1,2}(V)$. Exhausting the plane proves local Sobolev regularity. Apply [F2] on a countable collection of rectangles. On almost every coordinate line, its ACL representative agrees with the continuous representative almost everywhere by Fubini, hence everywhere on each compact line interval because both are continuous there. Thus the actual map $h$ is ACL and has finite coordinate partial derivatives almost everywhere, agreeing with the weak derivatives.

By the general-homeomorphism interface [F3], $h$ is totally differentiable almost everywhere. At such a point $x$, write $h(x+v)=h(x)+Av+o(\mathrm{\mid }v\mathrm{\mid })$; the remainder is uniform over $\mathrm{\mid }v\mathrm{\mid }=r$ after division by $r$. If $A$ has rank one, there are unit vectors $u,v$ with $Au=0$ and $\mathrm{\mid }Av\mathrm{\mid }>0$. Then $l(x,r)=o(r)$ whereas $L(x,r)\ge r\mathrm{\mid }Av\mathrm{\mid }+o(r)$, contradicting the finite circular bound. If $A$ is nonsingular with singular values $s_1\ge s_2>0$, the same uniform expansion gives $L(x,r)\mathrm{/}r\to s_1$ and $l(x,r)\mathrm{/}r\to s_2$, so $s_1\mathrm{/}{s}_2\le H$. Its determinant is positive by [F3], since $h$ preserves orientation. The remaining possibility is $A=0$.

At almost every point the classical partials of the total derivative equal the weak derivatives by step 3.1. At the nonsingular points, [F5] and step 4.1 give $\mathrm{\mid }{h}_{\bar{z}}\mathrm{\mid }\mathrm{/}\mathrm{\mid }{h}_z\mathrm{\mid }\le (H-1)\mathrm{/}(H+1)$; at zero derivatives the asserted inequality also holds. Together with $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}$ and ACL regularity, this is exactly analytic $H$-quasiconformality. The proof uses only the locally proved total-differentiability interface [F3]; it does not invoke an infinitesimal metric-to-quasiconformal criterion.