On Carleman formulas for the Dolbeault cohomology (Q2756129)

Let \(D\) be a bounded domain in \(\mathbb{C}^n\) with piecewise smooth boundary and \(S\) be an open subset of \(\partial D\). The aim of the paper is to construct Carleman type formulas for \(\overline\partial\)-closed \((p,q)\)-forms on \(\overline D\): NEWLINE\[NEWLINEu(z)= \lim_{\varepsilon\to 0} \int_S u\wedge C^{(p)}_{q+ 1}(\varepsilon; z,.)+ \overline\partial h^{(p)}_q u(z),\qquad z\in U\cap D,NEWLINE\]NEWLINE where \(U\) is an open neighborhood of \(\partial D\setminus S\), \(C^{(p)}_{q+1}(\varepsilon; z,\zeta)\) is a double differential form on \((U\cap D)\times S\) and \(h^{(p)}_q\) is a \(\overline\partial\)-homotopy operator.NEWLINENEWLINENEWLINEWhen \(S\) has a piecewise smooth boundary in \(\partial D\) and \(\partial D\setminus S\) is smooth and strictly \(q\)-pseudoconcave, the authors construct a decomposition on \((U\cap D)\times (U\setminus D)\) of the Koppelman kernel: NEWLINE\[NEWLINEK^{(p)}_{q+ 1}(z, \zeta)= R^{(p)}_{q+ 1}(z, \zeta)+(- 1)^{p+ q}\overline\partial_z P^{(p)}_{q+ 1}(z, \zeta).NEWLINE\]NEWLINE Suppose now that \(R^{(p)}_{q+ 1}(z,\zeta)\) can be approximated uniformly in \(\zeta\in\partial D\setminus S\) by a family \(R^{(p)}_{q+ 1}(\varepsilon; z,\zeta)\) \((\varepsilon\to 0)\) of double differential forms on \((U\cap D)\times\overline D\), \(\overline\partial\)-closed in \(\zeta\in D\), then a Carleman type formula is obtained with \(C^{(p)}_{q+ 1}(\varepsilon; z,\zeta)= K^{(p)}_{q+ 1}(z, \zeta)- R^{(p)}_{q+ 1}(\varepsilon; z,\zeta)\).