A geometric characterization: complex ellipsoids and the Bochner-Martinelli kernel
The title of the paper is given by an analogy with the characterizing of real ellipsoids in Theorem~I, but it also comes from the final conclusion: The Leray-Aizenberg kernel is self-adjoint only for complex ellipsoids and it is only defined using the Fefferman measure. The paper analyses the problem of determining geometric conditions when a kernel is substracted from its adjoint. Variants of choosing the kernel and also a measure of boundary are found. The results about the Bochner-Martinelli kernel are extended to the case of a weighted measure. Three theorems are stated in some propositions which have as purpose to determine the conditions under which the Bochner-Martinelli kernel and the Leray-Aizenberg kernel are self adjoint (with respect to the weighted measure). The paper considers the case in which the Leray-Aizenberg kernel is Möbius invariant (with respect to the Fefferman measure) for a strictly convex hypersurface. The paper ends with an example in which the Bochner-Martinelli transform is self-adjoint with respect to the weighted measure (which is a constant multiple of the Fefferman measure). The local version of Theorem 1 is established in the case of dimension one. The paper makes several connections with other papers in which the Bochner-Martinelli kernel and the Leray-Aizenberg kernel are self-adjoint. The author is also interested and wishes to solve other extensions, for example the extension of Theorem 1 to the case of Lipschitz domains. He is also interested in finding a differential-geometric proof along the lines of the proof where he characterizes the hypersurfaces for which the skew-Hermitian part of the Bochner-Martinelli kernel is less singular than usual (the author includes a proof for the case of dimension one in Section 6). The stated theorems and propositions are correctly demonstrated.
scientific article; zbMATH DE number 5015532