Fractional integrals and weakly singular integral equations of the first kind in the \(n\)-dimensional ball
The paper is devoted to study the solvability of the integral equation of the first kind \[ (I^\alpha_\Omega \varphi) (x) \equiv c_{n, \alpha} \int_\Omega {\varphi (y) dy \over |x - y |^{n - \alpha}} = f(x), \quad x \in \Omega,\;0 < \alpha < n,\;n \geq 1, \tag{1} \] with \(c_{n, \alpha} = 2^{- \alpha} \pi^{- n/2} \Gamma [(n - \alpha)/2]/ \Gamma (\alpha/2)\) in an \(n\)-dimensional ball \(\Omega \equiv U_c = \{t \in R^n : |t |< c\}\) \((0 < c < \infty)\), a disk when \(n = 2\). Such equations arise in applied problems in which it is important to know solvability conditions in special functional spaces and explicit formulae of solutions. The formal solution of equation (1) on a disk was obtained as a series by \textit{E. T. Copson} [Proc. Edinb. Math. Soc. 8, 14-19 (1947)] and \textit{N. A. Rostovtsev} [J. Appl. Math. Mech. 25, 238-246 (1961; Zbl 0107.194)] and via fractional differentiation operators by \textit{V. I. Fabrikant} [Izv. Vyssh. Uchebn. Zaved., Mat. 1971, No. 2(105), 102-104 (1971; Zbl 0221.45006)]. Sufficient conditions for the existence of at most one solution of equation (1) and explicit forms of such solutions were given by \textit{Ch. S. Kahane} [Integral Equations Oper. Theory 4, 548-595 (1981; Zbl 0472.45001)]\ for a disk and by \textit{Ch. S. Kahane} [ibid. 6, 67-133 (1983; Zbl 0516.45001)] and the author [Scientific works of the Jubilee seminar on boundary value questions, Collect Artic., Minsk 1985, 181-185 (1985; Zbl 0623.45013)] for a ball. In this connection we note the papers of \textit{N. K. Karapetjanc} and the author [Sov. Math., Dokl. 25, No. 6, 522-525 (1982; Zbl 0566.45020)], the author [Math. Notes 34, 751-757 (1983; Zbl 0535.45008)] and the author [Soviet Math. (Iz. VUZ) 29, No. 6, 102-106 (1985; Zbl 0601.31004)] where the invertibility of the Riesz potential \(I^\alpha_{U_c} \varphi\) with the radial \(\varphi (|x |)\)- and general \(\varphi (x)\)- density was investigated. The papers above did not contain the conditions of solvability of the equation (1) necessary and sufficient simultaneously in a concrete functional space. Such a problem is more complicated because of studying the conditions for the right-hand side \(f\) on the boundary of \(\Omega\). In this paper equation (1) in a ball \(\Omega = U_c\) is studied in the weighted space \(L_p (U_c; w)\) \((1 \leq p \leq \infty)\) of \(p\)- summable functions with the power weight \(w(x) = |x |^\lambda (c - |x |)^\mu (\lambda, \mu \in R)\), necessary and sufficient solvability conditions of (1) are proved and a new explicit form for its solution \(\varphi\) is constructed. The main results presented were earlier published by the author without proof in [Sov. Math. Dokl. 40, No. 3, 648-652 (1990; Zbl 0708.45002)]. The method applied is based on a special harmonic analysis in \(R^n\), developed by the author [Sov. Math. (Iz. VUZ) 33, No. 9, 52-59 (1989; Zbl 0723.46020) and 33, No. 10, 46-55 (1989; Zbl 0723.46021)] and adopted for operators commuting with rotations and dilations in \(R^n\) and on the properties of the so-called ball fractional integrals, being modifications of the Riesz potentials, connected with a sphere in \(R^n\). Such modifications were introduced by the author [Sov. J. Contempt. Math. Anal., Arm. Acad. Sci. 21, No. 5, 68-82 (1986; Zbl 0626.46014)], see also Section 29.2, note 25.15 in the book of \textit{S. G. Samko} the reviewer and \textit{O. I. Marichev} [Integrals and derivatives of fractional order and some of their applications (1987; Zbl 0617.26004); English edition: Fractional integrals and derivatives. Theory and applications, Gordon and Breach (1993)]. It is also indicated that the results similar to the above hold for the integral equation (1) in the exterior of a ball \(U_c\). We note that the solvability conditions and the explicit solution of the integral equation (1) with even \(\alpha\) in \(n\)-spherical shell \(U_{c,d} = \{t \in R^n : c < |t |< d\}\) \((0 \leq c < d < \infty)\) in the space of continuous functions were given by \textit{S. I. Vasilets} and the reviewer [Sov. Math. Dokl. 42, No. 3, 804-808 (1992)], see also their paper [Fukuoka Univ. Sci. Rep. 24, No. 1, 1-12 (1994)] where one may find historical notices concerning solving the simplest multidimensional integral equations with power and logarithmic kernels.
multidimensional integral equations of the first kind with power kernel
scientific article; zbMATH DE number 569416
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