A note on multidimensional modified fractional calculus operators
For functions \(f(x)\), \(x\in\mathbb{R}^n_+\), there exist many definitions of what is fractional integration or differentiation in case of many variables. Apart from obvious definitions of partial or mixed fractional derivatives and integrals, there exist essentially multidimensional versions, such as fractional differentiation in a given direction, fractional differentiation of elliptic, hyperbolic or parabolic type and others [see the book by the reviewer and \textit{A. A. Kilbas} and \textit{O. I. Marichev}, ``Fractional integrals and derivatives. Theory and applications'' ((Russian) 1987; Zbl 0617.26004, (English translation) 1993; Zbl 0818.26003), Sections 24-29]. In \textit{Yu. A. Brychkov}, \textit{H.-J. Glaeske}, \textit{A. P. Prudnikov} and \textit{V. K. Tuan}: ``Multidimensional integral transformations'' (1992; Zbl 0752.44004) there was suggested a definition of the fractional integration in the form \[ X^\alpha_+f(x)= {1\over\Gamma(\alpha+ 1)} \sum^n_{k=1} {\partial\over\partial x_k} x^{1- n}_k \int^{x_k}_0 (x_k- t)^{\alpha}t^{n-\alpha- 1}f\Biggl({t\over x_k} x\Biggr) dt,\quad x\in\mathbb{R}^n_+, \] and a similar form \(X^\alpha_-f\) with the integration over \((x_k,+\infty)\), where it is evidently assumed that \(0<\text{Re }\alpha<n\). The author derives some formulas for calculation of such integrals \(X^\alpha_{\pm}f\) when \(f\) is expressed in a certain way in terms of the \(H\)-function of Fox, the final formulas being too complicated to be reproduced here. The action of the operators of the type \(x^\beta X^\alpha_{\pm}x^{-\beta}\) is also studied in some special spaces \(M_\gamma(\mathbb{R}^n_+)\) defined via analyticity properties of its functions \(f(x)\) after the substitution \(x\to e^x\). Reviewer's remark: By easy calculations \(X^\alpha_+f(x)\) is reduced to \[ X^\alpha_+f(x)= {1\over\Gamma(\alpha)} \int^1_0 t^{n-\alpha-1}(1- t)^{\alpha-1}f(tx)dt \] and similarly for \(X^\alpha_-f(x)\), which should simplify calculations. We remark also that \(X^\alpha_{\pm}f(x)\) is a kind of one-dimensional fractional integration applicable to functions of many variables (compare with the fractional integration in a given direction); it resembles the Hadamard approach (1892) who suggested to consider fractional integrals in the complex plane in the general form \(\int^1_0 V(t)f(tz)dt\), \(z\in\mathbb{C}^1\), with prescribed properties of \(V(t)\) as \(t\to 1\), the realization of this idea with a rich content being given by M. M. Dzhrbashyan (1966), see references in the above-mentioned book by the reviewer, A. A. Kilbas and O. I. Marichev, p. 435.
scientific article; zbMATH DE number 955218
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