Trace inequalities for fractional integrals in grand Lebesgue spaces

Criteria guaranteeing the trace inequality for integral transforms of various types with fractional order in (generalized) grand Lebesgue spaces defined, generally speaking, on quasi-metric measure spaces are established. In particular, we derive necessary and sufficient conditions on a measure nu governing the boundedness for fractional maximal and potential operators defined on quasi-metric measure spaces from L-p),L-theta(X, mu) to L-q),L-q theta/p(X, nu) (trace inequality), where 1 < p < q < infinity, theta > 0 and mu satisfies the doubling condition in X. The results are new even for Euclidean spaces. For example, from our general results D. Adams-type necessary and sufficient conditions guaranteeing the trace inequality for fractional maximal functions and potentials defined on so-called s-sets in R-n follow. Trace inequalities for one-sided potentials, strong fractional maximal functions and potentials with product kernels, fractional maximal functions and potentials defined on the half-space are also proved in terms of Adams-type criteria. Finally, we remark that a Fefferman-Stein-type inequality for Hardy-Littlewood maximal functions and Calderon-Zygmund singular integrals holds in grand Lebesgue spaces.