An Orlicz variational formula for David-type Beltrami equations

Abstract

Let $\mathcal{U} \Subset \mathbb{C}$ be fixed, $\Phi(s)=e^s-s-1$, $F(\nu)=\frac{\nu}{2+|\nu|}$, and let $f^{F(\nu)}$ denote the principal solution of the corresponding Beltrami equation. The identity $K_{F(\nu)}=1+|\nu|$ identifies compactly supported David coefficients with exponential-Orlicz parameters. We prove that, on the open subset of $L^\Phi_{\mathcal{U}}(\mathbb{C})$ where a sufficiently high finite exponential moment is available, the principal solution map is locally real $C^{1,1}$ with values in $W^{1,2}_{\mathrm{loc}}(\mathbb{C})$. The derivative in a direction $\eta \in L^\Phi_{\mathcal{U}}(\mathbb{C})$ is the principally normalized solution of $\bar{\partial} V - F(\nu) \partial_z V = DF_\nu(\eta) \partial_z f^{F(\nu)}$. The proof uses a pullback by the base principal solution. The key estimate is the pointwise cancellation $\frac{\|DF_\nu\|_{\mathrm{op}}}{1-|F(\nu)|^2} \le \frac{1}{2}$, which converts the linearized equation into a $\bar{\partial}$-equation whose source is controlled directly by the $L^\Phi$-norm of the direction. Combined with the principal degenerate $L^2$-resolvent and the optimal Jacobian regularity for exponentially integrable distortion, this yields a uniform quadratic remainder estimate. At the origin one obtains $D \mathrm{Sol}_0[\eta]=(1/2)\mathcal{C}\eta$ in $W^{1,2}_{\mathrm{loc}}$.