English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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.

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