Holomorphy property and integral representation of certain Whitney fields (Q5952931)

Suppose \(K\subset{\mathbb C}^n\) is compact and \(U\supset K\) is a bounded neighbourhood. Suppose \(\delta:U\to[0,\infty)\) is a Lipschitz function so that \(K=\delta^{-1}\{0\}\) and \(\delta\) decays faster than any power of distance to \(K\) as we approach \(K\). Then one can introduce the algebra \(C_\delta^1(U)\) of complex-valued continuous bounded functions \(h\) on \(U\) so that \(\overline\partial h\) is measurable and \((\overline\partial h)/\delta\) is integrable on \(U\). It is a Banach algebra under \(\|h\|_\delta=\sup_U|h|+\int_U|\overline\partial h|/\delta\). NEWLINENEWLINENEWLINEThe authors establish a Bochner-Martinelli formula for such functions recovering \(h|_K\) from \(h|_{U\setminus K}\). They use this to define a Whitney field on \(K\) associated to \(h\) (a list of what would be the partial derivatives of \(h\) if \(h\) were actually differentiable near \(K\)). In particular, \(h\) is holomorphic on the interior of \(K\), should this be non-empty.