EXPLICIT SOBOLEV ESTIMATES FOR THE CAUCHY-RIEMANN EQUATION ON PARAMETERS

• Published : 2008.05.31

Abstract

Let $\bar{M}$ be a smoothly bounded pseudoconvex complex manifold with a family of almost complex structures $\{L^{\tau}\}_{{\tau}{\in}I}$, $0{\in}I$, 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 $L^0|_{bM}$ in any direction where the Levi-form vanishes on bM. We obtain explicit estimates for the $\bar{\partial}$-Neumann problem in Sobolev spaces both in space and parameter variables. Also we get a similar result when $\bar{M}$ is strongly pseudoconvex.

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2. SOBOLEV ESTIMATES FOR THE LOCAL EXTENSION OF BOUNDARY HOLOMORPHIC FORMS ON REAL HYPERSURFACES IN ℂn vol.50, pp.3, 2013, https://doi.org/10.4134/JKMS.2013.50.3.479