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Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem , t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) , t ∈ ℝ, where and p(·) is a vector-valued polynomial of the form for some elements . We describe the spectral properties of the operator A through the existence of slowly growing solutions of the (AVE). The main tool is the notion of Carleman spectrum of a vector-valued function. Moreover, an extension of a theorem...
This paper is an extended version of an invited talk presented during the Orlicz Centenary Conference (Poznań, 2003). It contains a brief survey of applications to classical problems of analysis of the theory of the so-called PLS-spaces (in particular, spaces of distributions and real analytic functions). Sequential representations of the spaces and the theory of the functor Proj¹ are applied to questions like solvability of linear partial differential equations, existence of a solution depending...
The paper is devoted to an integral equation algorithm for studying the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with...
For an analytic functional on , we study the homogeneous convolution equation S * f = 0 with the holomorphic function f defined on an open set in . We determine the directions in which every solution can be continued analytically, by using the characteristic set.
On donne un critère très simple de continuité des opérateurs définis par des intégrales singulières sur les espaces de Besov homogènes pour . Quelques exemples, utilisant notamment l’opérateur de paraproduit, illustrent ensuite l’emploi de ce critère.
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