Boundary values of integral operators

✍ Scribed by Torsten Hefer

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 278 KB
• Volume: 261-262
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310114

✦ Synopsis

Abstract

Consider a differential form u in the image of an integral operator K on a smooth domain D in a Riemannian manifold, i.e. u(y) = Kf(y) = ∫~x∈D~ f(x) ∧ K(x, y) for y ∈ D. If the kernel K of the integral operator is sufficiently regular and defined also for y in the boundary bD of D, one may define two different kinds of boundary values of u on bD. Firstly, u may have boundary values in the sense of distributions, i.e. boundary values satisfying a Stokes' formula in a suitably weak sense. Secondly, one can simply restrict (pull back) y in the kernel K(x, y) to the boundary of D, then integrate with respect to x in the above formula. It is interesting to know under which hypotheses on f and K both types of boundary values agree, because the boundary values defined by restricting the kernel can often be estimated by direct methods, whereas the abstractly given distributional boundary values are less tractable but analytically interesting objects linked to u. We will give counter‐examples to show that even in quite regular situations the two notions do not necessarily agree. Then we study conditions implying equality. We also mention some interesting applications, e.g. generalizations of Stokes' formula and applications in the theory of integral representations in several complex variables. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)