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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 \(1\leq p\leq\infty.\)

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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