Non-Archimedean Radial Calculus: Volterra Operator and Laplace Transform
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Integral Equations and Operator Theory
Non-Archimedean Radial Calculus: Volterra Operator and Laplace Transform Anatoly N. Kochubei Abstract. In an earlier paper (A. N. Kochubei, Pacif. J. Math. 269 (2014), 355–369), the author considered a restriction of Vladimirov’s fractional differentiation operator Dα , α > 0, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse I α that the appropriate change of variables reduces equations with Dα (for radial functions) to integral equations whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In the present paper, we begin an operator-theoretic investigation of the operator I α , and study a related analog of the Laplace transform. Mathematics Subject Classification. Primary 47G10, Secondary 11S80, 35S10, 43A32. Keywords. Fractional differentiation operator, Non-Archimedean local field, Radial functions, Volterra operator, Laplace transform.
1. Introduction The basic linear operator defined on real- or complex-valued functions on a non-Archimedean local field K (such as K = Qp , the field of p-adic numbers) is the Vladimirov pseudo-differential operator Dα , α > 0, of fractional differentiation [19]; for further development of this subject see [1,3,8,11,12,22]. Note also the recent publications devoted to applications in geophysical models and to the study of related nonlinear equations [9,10,17,18]. It was found in [13] that properties of Dα become much simpler on radial functions. Moreover, in this case it was found to possess a right inverse I α , which can be seen as a p-adic counterpart of the Riemann-Liouville fractional integral or, for α = 1, the classical anti-derivative. The change of an unknown function u = I α v reduces the Cauchy problem for an equation with the radial restriction of Dα to an integral equation with properties resembling those of classical Volterra equations. In other words, we found, in the 0123456789().: V,-vol
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framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In [13], we studied linear equations of this kind; nonlinear ones were investigated in [15]. Note that radial functions appear as exact solutions of the p-adic analog of the classical porous medium equation [9]. In this paper we study the operator I 1 on the ring of integers O ⊂ K as an object of operator theory. The operator I 1 on L2 (O) happens to be a sum of a bounded selfadjoint operator and a simple Volterra operator I01 with a rank two imaginary part J, such that tr J = 0. The characteristic matrix-function W (z) of I01 is such that W (z −1 ) is, in contrast to classical examples, an entire matrix function of zero order. While the theory of Volterra operators and their characteristic functions is well-developed (see [4–6,16,21]), properties of the above operator are very different from
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