Explicit Sobolev estimates for the Cauchy-Riemann equation on parameters

Bull. Korean Math. Soc. 2008 Vol. 45, No. 2, 321-338 Printed June 1, 2008

Sanghyun Cho and Jaeseo Choi Sogang University

Abstract : Let $\overline{M} $ be a smoothly bounded pseudoconvex complex manifold with a family of almost complex structures $\{ {\mathcal L}^\tau \} _{\tau \in I}$, $0\in I$, which extend smoothly up to $bM$, the boundary of $M$, and assume that there is $\lambda \in C^\infty (bM)$ which is strictly subharmonic with respect to the structure ${\mathcal L}^0|_{bM}$ in any direction where the Levi-form vanishes on $bM$. We obtain explicit estimates for the $\overline{\partial}$-Neumann problem in Sobolev spaces both in space and parameter variables. Also we get a similar result when $\overline{M}$ is strongly pseudoconvex.