发布时间： 2018-07-17     17:33   【返回上一页】 发布人：龚向宏

We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct, for m \geq 3,  two  smooth families of smoothly bounded m-connected domains in the complex plane, and for n\geq2, two families of strictly pseudoconvex domains in C^n, that are  equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms.  Finally, we give sufficient conditions for  the smooth equivalence of two smooth families of domains. This is joint work with Herv\'e Gaussier.

We derive a new homotopy formula for a strictly pseudoconvex domain of C^2  boundary in C^n by using a method of Lieb and Range and obtain estimates in H\"older-Zygmund spaces for the homotopy operators. In particular, this yields a sharp regularity for d-bar solutions.  We apply the estimates to obtain boundary regularities of D-solutions for a domain in C^n\times R^m.