Weak solutions of the Beltrami equation

On each relatively compact $K⋐\mathrm{\Omega }$, $\mathrm{\parallel }\mu f_z{\mathrm{\parallel }}_{L^2(K)}\le \mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{f}_z{\mathrm{\parallel }}_{L^2(K)}$, so $h:=f_{\bar{z}}-\mu f_z\in L^2(K)\subseteq L^1(K)$ by [F5]. If $h=0$ almost everywhere, its regular distribution is zero; conversely, if its regular distribution is zero, [F4] gives $h=0$ almost everywhere. Thus the almost-everywhere and distributional equations in (b) are equivalent. Applying the signed weak-derivative identity to the real partials and combining them as in [F8] gives $⟨f_{\bar{z}},\eta ⟩=-\int f{\eta }_{\bar{z}}dA$, which yields the test-function form in (b).

Replacing $f$, $\mu$, or either weak derivative by an almost-everywhere equal representative changes the equation only on the finite union of the corresponding null sets. The weak derivative classes are representative-independent by [F2], and the coefficient class is representative-independent by [F1]. Therefore the plane weak-solution condition is well-defined on these classes.

Let $\psi :{\mathrm{\Omega }}^{\mathrm{\prime }}\to \mathrm{\Omega }$ be biholomorphic. For each relatively compact $U_0⋐{\mathrm{\Omega }}^{\mathrm{\prime }}$, choose $V_0⋐\mathrm{\Omega }$ containing $\psi ({\bar{U}}_0)$; the derivatives of $\psi$ and ${\psi }^{-1}$ are bounded on these compact patches. Applying [F7] with $k=1,p=2$ and using the real chain rule [F10], then rewriting the real derivative matrix by [F8], gives the displayed weak chain-rule formulas on $U_0$. Since ${\psi }^{-1}$ maps null sets to null sets by [F6], the almost-everywhere equation for $f$ remains valid after composition.

Substitute $f_{\bar{z}}=\mu f_z$ into the second identity of step 1.3 and use the pullback formula from [F1]: $(f\circ \psi {)}_{\bar{\zeta }}=(\mu \circ \psi )(f_z\circ \psi )\overline{{\psi }^{\mathrm{\prime }}}=({\psi }^{\ast }\mu )(f_z\circ \psi ){\psi }^{\mathrm{\prime }}=({\psi }^{\ast }\mu )(f\circ \psi {)}_{\zeta }$ almost everywhere on $U_0$. The patches cover ${\mathrm{\Omega }}^{\mathrm{\prime }}$, so $f\circ \psi$ is a weak solution for ${\psi }^{\ast }\mu$. Applying the same argument to ${\psi }^{-1}$ proves the converse.

For the sphere clause, continuity of $f$ ensures that near any source point its image lies in a target chart, so the local coordinate maps in (d) are defined on open plane domains. The chart transitions are biholomorphic by [F9] and the sphere atlas is given by [F11]. On overlaps, source-chart changes preserve the equation by steps 1.3 and 2.1. A target-chart change is a local biholomorphism $\tau$; after shrinking the source neighborhood so its compact image lies in the overlap, [F12] gives $(\tau \circ F{)}_z=({\tau }^{\mathrm{\prime }}\circ F)F_z$ and $(\tau \circ F{)}_{\bar{z}}=({\tau }^{\mathrm{\prime }}\circ F)F_{\bar{z}}$. Multiplication by ${\tau }^{\mathrm{\prime }}\circ F$ proves preservation without division. Thus the local definition is independent of both chart choices and agrees with (a) in the finite chart.