Boundedness and Applications of Fractional Integral Operators in Nonlocal Problems with Fractional Laplacians

In this paper, we investigate the properties of the boundedness of fractional integral operators Kα defined on general measure metric spaces. We study their action in Lebesgue spaces Lp(Y), Morrey spaces Lφp(Y), and extend our analysis to fractional Sobolev spaces Wα,p(Y). Using classical dyadic decomposition and the Hardy–Littlewood maximal operator, we establish sharp bounds for Kα in terms of kernel parameters and the geometric structure of the space. A significant contribution of this work is the proof that Kα is bounded from Wα,p(Y) to Lq(Y), where thus linking our operator-theoretic framework with the theory of nonlocal and fractional partial differential equations. These results provide valuable tools for studying regularity, a priori estimates, and solution mappings in nonlocal problems involving the fractional Laplacian and related operators on irregular or non- Euclidean domains.