Estimates for (\overline∂-𝜇∂)⁻¹ and Calderón’s theorem on the Cauchy integral

Abstract

One can view the Cauchy integral operator as giving the solution to a certain ∂ ¯ \overline \partial problem. If one has a quasiconformal mapping on the plane that takes the real line to the curve, then this ∂ ¯ \bar \partial problem on the curve can be pulled back to a ∂ ¯ − μ ∂ \bar \partial - \mu \partial problem on the line. In the case of Lipschitz graphs (or chordarc curves) with small constant, we show how a judicial choice of q.c. mapping and suitable estimates for ∂ ¯ − μ ∂ \bar \partial - \mu \partial gives a new approach to the boundedness of the Cauchy integral. This approach has the advantage that it is better suited to related problems concerning H ∞ {H^\infty } than the usual singular integral methods. Also, these estimates for the Beltrami equation have application to quasiconformal and conformal mappings, taken up in a companion paper.