Cauchy non-integral formulas

TL;DR

This paper develops generalized Cauchy integral formulas for gradients of solutions to elliptic systems, extending classical formulas using functional calculus and duality methods for weighted spaces.

Contribution

It introduces new Cauchy formulas for elliptic system solutions with gradients in weighted L2 spaces, including endpoint cases, beyond traditional singular integral approaches.

Findings

Established Cauchy formulas for solutions with weighted gradient spaces

Extended formulas to endpoint cases using Carleson duality

Demonstrated formulas beyond classical singular integral scope

Abstract

We study certain generalized Cauchy integral formulas for gradients of solutions to second order divergence form elliptic systems, which appeared in recent work by P. Auscher and A. Ros\'en. These are constructed through functional calculus and are in general beyond the scope of singular integrals. More precisely, we establish such Cauchy formulas for solutions $u$ with gradient in weighted $L_2(\R^{1+n}_+,t^{\alpha}dtdx)$ also in the case $|\alpha|<1$. In the end point cases $\alpha= \pm 1$, we show how to apply Carleson duality results by T. Hyt\"onen and A. Ros\'en to establish such Cauchy formulas.