The Framed Beltrami-Vekua Normal Form and its Pseudo-Analytic Mass

We normalize a first-order real planar elliptic system, by pointwise algebra, to a framed Beltrami-Vekua equation $\mathrm{\Phi }(w_{\bar{z}}-\mu w_z)+\mathrm{\Psi }(\overline{w_z}-\mu \overline{w_{\stackrel{ˉ}{z}}})+\mathfrak{a}w+\mathfrak{b}\bar{w}=\mathfrak{f}$, with $\mathrm{\mid }\mu \mathrm{\mid }<1$ and $\mathrm{\mid }\mathrm{\Phi }\mathrm{\mid }>\mathrm{\mid }\mathrm{\Psi }\mathrm{\mid }$, and compute the closed transformation laws of its data under the recombination of unknowns $w\mapsto \phi w+\psi \bar{w}$ and under orientation-preserving $C^1$ changes of variables. The 2-form $\mathrm{\Theta }=\frac{\mid \mathrm{\Phi }\mathfrak{b}-\mathrm{\Psi }\mathfrak{a}-(\mathrm{\Phi }L\mathrm{\Psi }-\mathrm{\Psi }L\mathrm{\Phi }){\mid }^2}{(\mathrm{\mid }\mathrm{\Phi }{\mathrm{\mid }}^2-\mathrm{\mid }\mathrm{\Psi }{\mathrm{\mid }}^2{)}^2(1-\mathrm{\mid }\mu {\mathrm{\mid }}^2)}dxdy$, with $L=\bar{\mathrm{\partial }}-\mu \mathrm{\partial }$, is invariant under the recombination and covariant under the changes of variables. The total mass $\mathcal{M}={\int }_{\mathrm{\Omega }}\mathrm{\Theta }$ is therefore an invariant of the equivalence class. One recombination and one scaling carry any framed equation, in closed form, onto the trivial-frame slice - a Beltrami-Vekua equation over the same $\mu$ - there identifying $\mathrm{\Theta }$ with the pseudo-analytic mass density of the unframed equation. We then show all of this persists at measurable regularity: it suffices that $\mu$ be measurable and locally elliptic and that the frame lie in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}$, the changes of variables then being quasiconformal homeomorphisms. In that class every equation with $\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}<1$ is quasiconformally equivalent, of equal mass, to one over $\mu =0$.