Uniform power phase decay for Beltrami CGO solutions

Write $\mathrm{\Phi }={\mathrm{\Phi }}_{\mu }(\cdot ,k)$ and $\mathrm{\Psi }={\mathrm{\Phi }}^{-1}$. The equation $\bar{\mathrm{\partial }}f=\mu \overline{\mathrm{\partial }f}$ with $f=e^{ik\mathrm{\Phi }}$ gives $\bar{\mathrm{\partial }}\mathrm{\Phi }=-(\bar{k}\mathrm{/}k)\mu e^{-i(k\mathrm{\Phi }+\bar{k}\bar{\mathrm{\Phi }})}\overline{\mathrm{\partial }\mathrm{\Phi }}$. The nonvanishing exponential permits division, and all derivatives agree almost everywhere by [F1]. Apply the inverse derivative identity: its two derivative factors cancel the quotient $\overline{\mathrm{\partial }\mathrm{\Phi }}\mathrm{/}\mathrm{\partial }\mathrm{\Phi }$, leaving $\bar{\mathrm{\partial }}\mathrm{\Psi }(w)=(\bar{k}\mathrm{/}k)e^{-i(kw+\bar{k}\bar{w})}\mu (\mathrm{\Psi }(w))\mathrm{\partial }\mathrm{\Psi }(w)$. Thus this is a complex-linear Beltrami equation with oscillation vector of length $2\mathrm{\mid }k\mathrm{\mid }$ and coefficient $\tilde{\mu }=\mu \circ \mathrm{\Psi }$. Its support lies in the fixed disk containing $\mathrm{\Phi }(D_1)$ by [F1].

By [F2], $\tilde{\mu }$ has uniform positive power $L^4$ translation modulus; its supremum bound remains $\kappa$, with almost-everywhere composition justified by null-set preservation in [F1]. It is legitimate that $\tilde{\mu }$ depends on $k$, because [F3] is uniform over the entire coefficient family with these support and modulus bounds. Apply [F3] to the equation in step 1.1 to obtain $\mathrm{\parallel }\mathrm{\Psi }-w{\mathrm{\parallel }}_{\mathrm{\infty }}\le C\mathrm{\mid }k{\mathrm{\mid }}^{-b}$. Since $\mathrm{\Phi }$ is onto, $\mathrm{\mid }\mathrm{\Phi }(z)-z\mathrm{\mid }=\mathrm{\mid }\mathrm{\Phi }(z)-\mathrm{\Psi }(\mathrm{\Phi }(z))\mathrm{\mid }\le C\mathrm{\mid }k{\mathrm{\mid }}^{-b}$. Combine this with the spatial decay in [F1] to obtain the stated minimum bound. Multiplication of $\mu$ by a unimodular constant preserves every support, supremum and translation-modulus hypothesis; the constants therefore work for all $\lambda \mu$.