Fractional integrals with measure in grand Lebesgue and Morrey spaces

A complete characterization of a measure μ governing the boundedness of fractional integral operators defined on a quasi-metric measure space (X,d,μ) (non-homogeneous space) from one grand Lebesgue spaces Lμp),θ1(X) into another one Lμq),θ2(X) is established. As a corollary, we have a generalization of the Sobolev inequality for potentials with measure. An appropriate problem for grand Morrey spaces is studied. D. Adams trace inequality (i.e. Lμp),θ1(X)↦Lνq),θ2(X) boundedness) is also derived for these operators in grand Lebesgue spaces. In the case of Morrey spaces, we assume that the underlying sets of spaces might be of infinite measure. Under some additional conditions on a measure, we investigate the sharpness of the second parameter θ2 in the target space.