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Lecture Note

Distribution-Valued Analytic Functions - Theory and Applications

Norbert Ortner and Peter Wagner


The aim of this book consists in giving a systematic and general approach to treating meromorphic distribution-valued functions of the form $\lambda\longmapsto F_{\lambda}\cdot \rho^{\lambda},$ where the "characteristic" $F_{\lambda}\in\mathcal{D}'(\Gamma)$ (with $\Gamma=\{x\in\mathbf{R}^n;\, \rho(x)=1\})$ also depends meromorphically on $\lambda.$

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 {\smc Schwartz} [5], {\smc Robertson} and {\smc Robertson} [1], {\smc Horv\'ath} [4] and {\smc 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 $\mathcal{D}\mathcal{F}-$spaces (1.4), and on holomorphic functions with values in topological vector spaces (1.5, 1.6).

InChapter II, the quasihomogeneous distribution-valued functions $\lambda\longmapsto F_{\lambda}\cdot \rho^{\lambda}$ 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 $x_1^{m_1}+\dots+x_n^{m_n},$ $x_1+\text ix_2,$ $\text ix_1+R(x'),$ $x_1^2-|x'|^2,$ $|x|^2,$ $|x'|^2-|x''|^2.$ Fundamental solutions of the iterated Cauchy-Riemann operator $(\partial_1+\text i\partial_2)^l,$ of the iterated wave operator $(\partial_1^2-\Delta_{n-1})^l,$ and, more generally, of the iterated ultrahyperbolic operator $(\partial_1^2+\dots+\partial_m^2-\partial_{m+1}^2-\dots-\partial_n^2)^l$ 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 $\mathcal{D}'_{L^p}-$spaces, which generalize the spaces $\mathcal{D}'_{L^p}$ introduced by L. Schwartz (3.1). The homogeneous distributions $F\cdot \rho^{\lambda}$ operate on weighted $\mathcal{D}'_{L^p}-$spaces by convolution, and we obtain continuity properties in dependence on the regularity of the characteristic $F$ (see 3.2 for $F\in L^p(\mathbf{S}^{n-1}),$ 3.3, 3.4 for $F\in \mathcal{C}^\infty(\mathbf{S}^{n-1}),$ 3.6 for $F\in L^p_s(\mathbf{S}^{n-1})).$ 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.