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

In: Holomorphic Function Theory in Several Variables

Author

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  • Christine Laurent-Thiébaut

    (Université Joseph Fourier, Institut Fourier)

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 ℂn. 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 ℂn. 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 ℂn 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.

Suggested Citation

  • Christine Laurent-Thiébaut, 2011. "III The Bochner–Martinelli–Koppelman kernel and formula and applications," Springer Books, in: Holomorphic Function Theory in Several Variables, pages 57-73, Springer.
    • Handle: RePEc:spr:sprchp:978-0-85729-030-4_3
    DOI: 10.1007/978-0-85729-030-4_3
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