Rosay, Jean-Pierre; Stout, Edgar Lee Strong boundary values: independence of the defining function and spaces of test functions. (English) Zbl 1051.46026 Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 1, No. 1, 13-31 (2002). 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. Reviewer: Andreas U. Schmidt (Frankfurt am Main) Cited in 1 ReviewCited in 1 Document 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 Keywords:strong boundary values Citations:Zbl 0988.46032 PDFBibTeX XMLCite \textit{J.-P. Rosay} and \textit{E. L. Stout}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 1, No. 1, 13--31 (2002; Zbl 1051.46026) Full Text: EuDML References: [1] N. Aronszajn - T. M. Creese - L. J. Lipkin, “Polyharmonic Functions”, The Clarendon Press Oxford University Press, New York, 1983. Notes taken by Eberhard Gerlach, Oxford Science Publications. Zbl0514.31001 MR745128 · Zbl 0514.31001 [2] I. Chavel, “Eigenvalues in Riemannian Geometry”, Academic Press Inc., Orlando, 1984, including a chapter by Burton Randol, with an appendix by Jozef Dodziuk. Zbl0551.53001 MR768584 · Zbl 0551.53001 [3] G. B. Folland, “Introduction to Partial Differential Equations, 2nd. ed.”, Princeton University Press, Princeton, 1995. Zbl0841.35001 MR1357411 · Zbl 0841.35001 [4] L. V. Hörmander, “Linear Partial Differential Operators”, volume 116 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1964. [5] L. V. Hörmander, “The Analysis of Linear Partial Differential Operators”, volume 275 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1985. Zbl0612.35001 · Zbl 0612.35001 [6] J.-P. Rosay - E. L. Stout, “Strong Boundary Values, Analytic Functionals, and Nonlinear Paley-Wiener Theory”, Memoirs of the American Mathematical Society, American Mathematical Society, Providence, Vol. 153, n. 725, 2001. Zbl0988.46032 MR1846591 · Zbl 0988.46032 [7] W. Rudin, “Functional Analysis”, McGraw-Hill, New York, 1991. Zbl0867.46001 MR1157815 · Zbl 0867.46001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.