Lecture note 37/2008

Distribution-Valued Analytic Functions - Theory and Applications

Norbert Ortner and Peter Wagner

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Submission date: 23. Jan. 2008
Pages: 133
published as:
Ortner, N. and P. Wagner: Distribution-valued analytic functions : theory and applications
   Hamburg : tredition, 2013. - 144 p.
   ISBN 978-3-8491-1968-3       

The aim of this book consists in giving a systematic and general approach to treating meromorphic distribution-valued functions of the form formula29 where the "characteristic" formula31 (with formula33 also depends meromorphically on formula35

Let us describe now the contents of the booklet more in detail. Chapter I consists of supplements to the theories of locally convex topological vector spaces and, in particular, of distribution spaces. Hereby, results from the books Schwartz [5], Robertson and Robertson [1], Horváth [4] and Treves [1] are taken for granted and are quoted only. We supplement these basic references by synopses on distributions on hypersurfaces (1.1), on convolution of measures and distributions (1.2, 1.3), on bilinear mappings defined on barrelled formula37spaces (1.4), and on holomorphic functions with values in topological vector spaces (1.5, 1.6).

In Chapter II, the quasihomogeneous distribution-valued functions formula39 are defined and their properties (analytic continuation, poles, residues, finite parts) are derived (2.1, 2.2). The structure of quasihomogeneous distributions and of its Fourier transforms is elucidated in 2.5, 2.6. In the remaining sections of Chapter II, the theory is illustrated by several concrete examples originating from the quasihomogeneous polynomials formula41 formula43 formula45 formula47 formula49 formula51 Fundamental solutions of the iterated Cauchy-Riemann operator formula53 of the iterated wave operator formula55 and, more generally, of the iterated ultrahyperbolic operator formula57 are deduced therefrom (Ex. 2.7.3, Prop. 2.4.2, Prop. 2.7.6).

In Chapter III, the convolution with the quasihomogeneous distributions arising in Chapter II is treated. For this purpose, we define weighted formula59spaces, which generalize the spaces formula61 introduced by L. Schwartz (3.1). The homogeneous distributions formula63 operate on weighted formula59spaces by convolution, and we obtain continuity properties in dependence on the regularity of the characteristic F (see 3.2 for formula69 3.3, 3.4 for formula71 3.6 for formula73 As application, we describe the convolution groups of elliptic (3.3), hyperbolic (3.5), ultrahyperbolic (3.6) and quasihyperbolic operators (3.7). Finally, the convolution groups of some singular integral operators are treated in 3.8.

18.10.2019, 02:10