An Orlicz variational formula for David-type Beltrami equations

Abstract

Let $\mathcal{U}⋐\mathbb{C}$ 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 $L_{\mathcal{U}}^{\Phi }(\mathbb{C})$ where a sufficiently high finite exponential moment is available, the principal solution map is locally real $C^{1,1}$ with values in $W_{\mathrm{loc}}^{1,2}(\mathbb{C})$. The derivative in a direction $\eta \in L_{\mathcal{U}}^{\Phi }(\mathbb{C})$ is the principally normalized solution of $\bar{\partial }V-F(\nu ){\partial }_{z}V=D{F}_{\nu }(\eta ){\partial }_z{f}^{F(\nu )}$. The proof uses a pullback by the base principal solution. The key estimate is the pointwise cancellation $\frac{\Vert D{F}_{\nu }{\Vert }_{\mathrm{op}}}{1-|F(\nu ){|}^2}\le \frac{1}{2}$, 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 $W_{\mathrm{loc}}^{1,2}$.

20 pages