Revistes Catalanes amb Accés Obert (RACO)

Semi-global solutions of $\bar\partial_b$ with $L^p$ $(1 \le p \le \infty)$ bounds on strongly pseudoconvex real hypersurfaces in ${\mathbf C}^n$ $(n\ge 3)$

C. H. Chang, H. P. Lee


Let $M$ be an open subset of a compact strongly pseudoconvex hypersurface $\{\rho = 0\}$ defined by $M = D \times {\mathbf C}^{n-m}\cap \{\rho = 0\}$, where $1 \le m \le n-2$, $D = \{\sigma(z_1,\ldots, z_m) < 0\} \subset {\mathbf C}^m$ is strongly pseudoconvex in ${\mathbf C}^m$. For $\bar\partial_b$ closed $(0,q)$ forms $f$ on $M$, we prove the semi-global existence theorem for $\bar {\partial}_b$ if $1 \le q \le n-m-2$, or if $q=n-m-1$ and $f$ satisfies an additional "moment condition". Most importantly, the solution operator satisfies $L^p$ estimates for $1\le p\le \infty$ with $p=1$ and $\infty$ included.

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