# III The Bochner–Martinelli–Koppelman kernel and formula and applications

## Abstract

In this chapter we define one of the fundamental tools of integral representation theory in complex analysis, namely the Bochner–Martinelli–Koppelman kernel. This kernel generalises the Cauchy kernel on ℂ to ℂ^{n}. It enables us to prove an integral representation formula, the Bochner–Martinelli–Koppelman formula, which extends Cauchy’s formula to (p, q) differential forms in ℂ^{n}. This formula plays an important role in the study of the operator \(\overline{\partial}\): in particular, we prove using this formula our first results on the existence of solutions to the Cauchy–Riemann equation in ℂ^{n} by considering the case where the data has compact support. Hartog’s phenomenon, a special case of which was studied in Chapter I, follows from the existence of a compactly supported solution to the Cauchy–Riemann equations for n ⩾ 2 when the right-hand side of the equation is a compactly supported form of bidegree (0, 1). The links between the vanishing of compactly supported Dolbeault cohomology groups in bidegree (0, 1) and Hartog's phenomenon will be explored in greater detail in Chapter V. We will also use the Bochner–Martinelli–Koppelman formula to study the regularity of the operator \(\bar{\partial}\) by proving a Hölder hypoellipticity theorem.

## Keywords

Bounded Domain Holomorphic Function Compact Support Erential Form Real Hypersurface## Preview

Unable to display preview. Download preview PDF.