Positive power translation moduli after principal quasiconformal composition

Choose a fixed nonnegative smooth unit-mass mollifier supported in the unit disk and let $v_h={\rho }_h\ast v$, $0<h\le 1$. The probability form of Hölder, followed by integration, gives $\mathrm{\parallel }v-v_h{\mathrm{\parallel }}_p\le {\omega }_p(v,h)\le A{h}^{\theta }$. Also $\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_{\mathrm{\infty }}\le B$, $\mathrm{\parallel }\mathrm{\nabla }{v}_h{\mathrm{\parallel }}_{\mathrm{\infty }}\le C_{\rho }B\mathrm{/}h$, and its support is contained in ${\bar{D}}_{R+1}$. The latter derivative bound follows by differentiating the smooth convolution kernel and taking its derivative's $L^1$ norm. All supports involved in $v_h\circ \mathrm{\Psi }$ and $v\circ \mathrm{\Psi }$ lie in $\mathrm{\Phi }({\bar{D}}_{R+1})$, hence inside one fixed disk by [F1]. Null-set preservation of $\mathrm{\Phi }$ makes composition with $\mathrm{\Psi }$ independent of the representative of $v$.

Apply the weighted area formula with $w=\mathrm{\Phi }(z)$ and then Hölder. Since $J_{\mathrm{\Phi }}\le \mathrm{\mid }\mathrm{\partial }\mathrm{\Phi }{\mathrm{\mid }}^2$, its $L^a$ norm on $D_{R+1}$ is uniformly bounded by [F1]. Thus $\mathrm{\parallel }(v-v_h)\circ \mathrm{\Psi }{\mathrm{\parallel }}_q^q=\int \mathrm{\mid }v-v_h{\mathrm{\mid }}^q{J}_{\mathrm{\Phi }}dA\le \mathrm{\parallel }v-v_h{\mathrm{\parallel }}_{q{a}^{\mathrm{\prime }}}^q\mathrm{\parallel }{J}_{\mathrm{\Phi }}{\mathrm{\parallel }}_a\le C{A}^q{h}^{q\theta }$, using $q{a}^{\mathrm{\prime }}=p$. Therefore the translation difference of this error has $L^q$ norm at most $2CA{h}^{\theta }$, by translation invariance of Lebesgue measure.

For $\mathrm{\mid }y\mathrm{\mid }\le t\le 1$, the inverse deformation estimate gives $\mathrm{\mid }\mathrm{\Psi }(w-y)-\mathrm{\Psi }(w)\mathrm{\mid }\le \mathrm{\mid }y\mathrm{\mid }+C\mathrm{\mid }y{\mathrm{\mid }}^{{\theta }_0}\le C{t}^{{\theta }_0}$. The Lipschitz bound for $v_h$ therefore bounds $\mathrm{\mid }{v}_h(\mathrm{\Psi }(w-y))-v_h(\mathrm{\Psi }(w))\mathrm{\mid }$ by $CB{h}^{-1}{t}^{{\theta }_0}$. The support of this difference lies in the fixed disk from step 1.1 enlarged by one, whose area is uniformly bounded. Its $L^q$ norm is consequently at most $CB{h}^{-1}{t}^{{\theta }_0}$. Combine with step 2.1 and take $h=t^{{\theta }_0\mathrm{/}(1+\theta )}$, which lies in $(0,1]$. The two powers $h^{\theta }$ and $h^{-1}{t}^{{\theta }_0}$ coincide with $t^{\theta {\theta }_0\mathrm{/}(1+\theta )}$. Taking the supremum over the shifts proves the estimate.