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Strong boundary values: independence of the defining function and spaces of test functions. (English) Zbl 1051.46026

The purpose of the paper under review is the further study of the authors’ notion of strong boundary values, introduced in [Mem. Am. Math. Soc. 725 (2001; Zbl 0988.46032)]. In contrast to the usual theory of analytic functionals and hyperfunction boundary values, where the defining function is holomorphic or harmonic, the concept relies on the boundary values at a real-analytic manifold \(\mathcal{M}\) of functions which are only continuous on \(\mathcal{M}\times(-\infty,0)\). Such a function \(U\) is said to have a strong boundary value if \[ \psi_V(t)=\int_{\mathcal{M}} U(t,x)V(t,x)\,dx \] extends to a holomorphic function on a small neighbourhood of \(0\) in \(t\), for all real-analytic test functions \(V\in\mathcal{A}(\mathcal{M}\times\{0\})\). In this case, \(\lim_{t\rightarrow 0} \psi_V(t)\) defines an analytic functional in \(\mathcal{A}'(\mathcal{M})\).
The main result of the present paper is the invariance of the notion of strong boundary value and the analytic functional defined by it, with respect to real-analytic reparametrisations of the \(t\)-level sets defining the boundary value. To achieve this, a decomposition of \(U\) into eigenfunctions of the Laplace-Beltrami operator on \(\mathcal{M}\) is used.
Further results concern a counterexample when the condition of real-analyticity of the manifold and the test functions is relaxed, smaller test function spaces, and special examples of strong boundary values of functions not satisfying any particular noncharacteristic PDE.

MSC:

46F15 Hyperfunctions, analytic functionals
32A10 Holomorphic functions of several complex variables
32A40 Boundary behavior of holomorphic functions of several complex variables
42C15 General harmonic expansions, frames

Citations:

Zbl 0988.46032
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References:

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