Product domains, Multi-Cauchy transforms, and the ∂ equation

2019/04/20 by Chen, Liwei, McNeal, Jeffery D.
#32A26 #32W05 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.09401

Abstract

Solution operators for the equation ∂ u=f are constructed on general product domains in ℂⁿ. When the factors are one-dimensional, the operator is a simple integral operator: it involves specific derivatives of f integrated against iterated Cauchy kernels. For higher dimensional factors, the solution is constructed by solving sub- ∂ equations with modified data on the factors. Estimates of the operators in several norms are proved.

Related

• The Cauchy-Riemann equations on product domains
• Uniform estimates for the canonical solution to the ∂-equation on product domains
• Uniform estimates of the Cauchy-Riemann equation on product domains
• Optimal Hölder Regularity of Solution Operator to the ∂-equation on Product Domains
• ∂-Estimates on the product of bounded Lipschitz domain