Explicit Inversion and Error Analysis of Saigo-Type Fractional Integral Operators in Weighted Lµp Spaces

Abstract

This paper develops a rigorous theory for the inversion of standard Saigo-type fractional integral operators on various function spaces. We present and prove explicit inversion formulas showing that the Saigo operator Sα,β,γ 0+ and its inverse, represented by a corresponding fractional derivative, are mutual inverses under natural parameter constraints. The results are established for weighted L1
μ and Lp μ spaces, as well as for locally integrable, Sobolev-type, compactly supported continuous, H¨older continuous, and essentially bounded weighted function spaces. The theoretical framework is supported by detailed proofs and is illustrated through concrete numerical examples, including tabulated results for specific parameter choices and error analysis. The sharpness of parameter conditions for invertibility and the applicability of the inversion formulas across various function spaces are highlighted. Our findings provide
robust tools for exact reconstruction and analysis in fractional calculus, enhancing the theoretical foundation as well as practical computation for applications involving Saigo-type operators.