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Semi-global solutions of $$\overline{\partial}_b$$ with $$L^p$$ $$(1\leq p\leq \infty)$$ bounds on strongly pseudoconvex real hypersurfaces in $$\mathbb{C}^n$$ $$(n\geq 3)$$. (English) Zbl 0954.32027
Let $$M$$ be an open subset of a compact strongly pseudoconvex hypersurface $$\{\rho=0\}$$ in $$\mathbb{C}^n$$ defined by $$M=D\times\mathbb{C}^{n-m}\cap\{\rho=0\},$$ where $$1\leq m\leq n-2,$$ $$D=\{\sigma(z_1,\dots,z_m)<0\}\subset\mathbb{C}^m$$ is strongly pseudoconvex in $$\mathbb{C}^m.$$
For $$\overline{\partial}_b$$ closed $$(0,q)$$ forms $$f$$ with coefficients in $$C^1(\overline M),$$ the authors prove the semi-global existence theorem for the tangential Cauchy-Riemann equation $$\overline{\partial}_bu=f$$ if $$1\leq q\leq n-m-2,$$ or if $$q=n-m-1$$ and $$f$$ satisfies an additional “moment condition”. The solution operator $$\mathcal L$$ consists of two integral operators: one is an integral over $$M$$ defined by Henkin’s kernel $$\Omega(\mathfrak r,\mathfrak r^\star)$$ for $$\overline{\partial}_b$$ on $$\{\rho=0\}$$; the other one is a boundary integral defined by the kernel $$\Omega(\mathfrak r,\mathfrak r^\star,\mathfrak s)$$ which involves not only the Leray sections $$\mathfrak r,\mathfrak r^\star$$ of $$\{\rho=0\}$$ but also the Leray section $$\mathfrak s$$ for the lower dimensional strongly pseudoconvex domain $$\{\sigma<0\}\subset\mathbb{C}^m.$$ Most importantly, the operator $$\mathcal L$$ satisfies $$L^p$$ estimates for $$1\leq p\leq\infty.$$

##### MSC:
 32W10 $$\overline\partial_b$$ and $$\overline\partial_b$$-Neumann operators 32V20 Analysis on CR manifolds 32V40 Real submanifolds in complex manifolds
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