Integral Operators in Grand Mixed-Normed Function Spaces

In this chapter a new scale of weighted grand mixed-norm Lebesgue spaces (WGMNLSs shortly) is introduced and the boundedness for multiple singular operators in these spaces is established. Further, we derive necessary and sufficient conditions on weights guaranteeing the one-weight Sobolev-type inequality for strong fractional maximal and multiple fractional integral operators in grand mixed-norm Lebesgue spaces (GMNLS briefly). As a consequence, appropriate statements for fractional integral operators defined on regular curves and domains in \({\mathbb {R}}^n\) are obtained. D. Adams-type trace inequalities for multiple fractional integral operators in GMNLSs are established. Operators under consideration contain multiple fractional integrals defined on product of quasi-metric measure spaces and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously necessary and sufficient for appropriate trace inequalities. The derived results are new even for multiple Riesz potential operators defined on the product of Euclidean spaces.