Principal Beltrami solutions and uniform inverse-gradient integrability

Take a solution from [F1] fixing infinity. Outside $D_R$ its coefficient is zero, so it is conformal there. In the infinity chart it is an injective holomorphic map fixing zero with nonzero derivative. Its Taylor expansion there gives $az+b+O(1\mathrm{/}z)$ in the original chart, with $a\mathrm{\ne }0$. Postcomposition by $(w-b)\mathrm{/}a$ gives the principal normalization. Any two principal solutions differ by a Möbius map by [F1]; since both fix infinity this map is affine, and their leading and constant Laurent coefficients force it to be the identity. This proves existence and uniqueness of $\mathrm{\Phi }$.

Set $h=\bar{\mathrm{\partial }}\mathrm{\Phi }$, which is in $L^2$ and has support in ${\bar{D}}_R$ by local quasiconformal Sobolev regularity and the zero coefficient outside. The Cauchy integral for compact $L^2$ data is locally $L^2$: on a fixed target disk the kernel restricted to all differences with $D_R$ is in $L^1$, so [F4] applies; outside a larger disk its bound is $C\mathrm{\parallel }h{\mathrm{\parallel }}_1\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$. Approximating $h$ by smooth functions in a fixed support disk yields distributionally $\bar{\mathrm{\partial }}\mathcal{C}h=h$ and $\mathrm{\partial }\mathcal{C}h=\mathcal{B}h$, the latter by Beurling $L^2$ boundedness. Thus $\bar{\mathrm{\partial }}(\mathrm{\Phi }-z-\mathcal{C}h)=0$. Applying $\mathrm{\partial }$ and [F4] makes this distribution smooth harmonic; the original Cauchy–Riemann equation makes its representative holomorphic. It tends to zero at infinity by the principal normalization and the exterior Cauchy bound. Liouville gives $\mathrm{\Phi }-z=\mathcal{C}h$. Therefore $\mathrm{\partial }\mathrm{\Phi }=1+\mathcal{B}h$, and the Beltrami equation says $(I-\mu \mathcal{B})h=\mu$.

Let $h_r=(I-\mu \mathcal{B}{)}^{-1}\mu$ in $L^r$. Its norm is at most $C_{\kappa }\mathrm{\parallel }\mu {\mathrm{\parallel }}_r\le C_{\kappa ,R,r}$ by [F3]. The equation gives $h_r=0$ outside $D_R$, hence $h_r\in L^2$ by Hölder on that disk. Compatibility of the $L^2$ and $L^r$ Beurling actions and uniqueness of the $L^2$ inverse imply $h_r=h$ from step 2.1. Apply the Cauchy bounds in [F3] to $\mathrm{\Phi }-z=\mathcal{C}h$. This proves its $W^{1,r}$, supremum and Hölder bounds. In particular $\mathrm{\Phi }(D_S)$ lies in a disk of radius $S+C_{\kappa ,R,r}$, uniformly in $\mu$.

By [F2], $\mathrm{\Psi }={\mathrm{\Phi }}^{-1}$ has coefficient bounded by $\kappa$, supported in $\mathrm{\Phi }({\bar{D}}_R)$, and has the same principal expansion at infinity. Step 3.1 puts that support inside a fixed disk; applying step 3.1 to $\mathrm{\Psi }$ therefore gives a uniform $L^r$ bound for $\mathrm{\partial }\mathrm{\Psi }$ on every fixed disk. At common differentiability points, the derivative inverse formula gives $\mathrm{\mid }\mathrm{\partial }\mathrm{\Psi }(\mathrm{\Phi }(z))\mathrm{\mid }=1\mathrm{/}[(1-\mathrm{\mid }\mu (z){\mathrm{\mid }}^2)\mathrm{\mid }\mathrm{\partial }\mathrm{\Phi }(z)\mathrm{\mid }]$. Thus $\mathrm{\mid }\mathrm{\partial }\mathrm{\Phi }(z){\mathrm{\mid }}^{-\sigma }\le \mathrm{\mid }\mathrm{\partial }\mathrm{\Psi }(\mathrm{\Phi }(z)){\mathrm{\mid }}^{\sigma }$. Change variables with $z=\mathrm{\Psi }(w)$ using [F2]; the bound $J_{\mathrm{\Psi }}\le \mathrm{\mid }\mathrm{\partial }\mathrm{\Psi }{\mathrm{\mid }}^2$ gives ${\int }_{D_S}\mathrm{\mid }\mathrm{\partial }\mathrm{\Phi }{\mathrm{\mid }}^{-\sigma }\le {\int }_{\mathrm{\Phi }(D_S)}\mathrm{\mid }\mathrm{\partial }\mathrm{\Psi }(w){\mathrm{\mid }}^{\sigma +2}dA(w)$. Since $\sigma +2=r$, step 3.1 and the uniform disk containing $\mathrm{\Phi }(D_S)$ bound the right side. Null-set preservation in [F2] justifies all inverse identities and negative powers on common full-measure sets.

On $D_S$, $\mathrm{\mid }\mathrm{\Phi }\mathrm{\mid }\le S+C_{\kappa ,R,r}$. For $\mathrm{\mid }k\mathrm{\mid }=1$, the exponential and its derivative are bounded on this range. To justify the Sobolev chain rule, set $h=\bar{\mathrm{\partial }}\mathrm{\Phi }$ and approximate it in $L^r$ by smooth functions $h_n$ supported in one fixed larger disk; the density and Cauchy bounds in [F3] give ${\mathrm{\Phi }}_n=z+\mathcal{C}{h}_n\to \mathrm{\Phi }$ in $W^{1,r}(D_{S+1})$ and uniformly on $D_S$. The Cauchy integrals of these smooth data are smooth, and the uniform convergence keeps their ranges on $D_S$ in a fixed bounded set. The classical chain rule for $f_n=e^{ik{\mathrm{\Phi }}_n}$ gives $\mathrm{\partial }{f}_n=ik{e}^{ik{\mathrm{\Phi }}_n}\mathrm{\partial }{\mathrm{\Phi }}_n$ and the corresponding $\bar{\mathrm{\partial }}$ formula. Uniform convergence of the exponential factors and $L^r$ convergence of the derivatives pass these identities to $f=e^{ik\mathrm{\Phi }}$ in $W^{1,r}(D_S)$. Step 3.1 supplies the uniform $W^{1,r}$ bound, while $\mathrm{\mid }\mathrm{\partial }f{\mathrm{\mid }}^{-\sigma }\le e^{\sigma (S+C)}\mathrm{\mid }\mathrm{\partial }\mathrm{\Phi }{\mathrm{\mid }}^{-\sigma }$ and step 4.1 supply the negative-power bound. This proves the final assertion, including the case $\sigma <1$ with its stated integral convention.