Potentials with product kernels in grand Lebesgue spaces: One-weight criteria*

Abstract

We study the boundedness problem for fractional integral operators with product kernels and corresponding strong fractional maximal operators in unweighted and weighted grand Lebesgue spaces. Among other statements, we prove that the one-weight inequality \( {{\left\| {{T_{\alpha }}\left( {f{w^{\alpha }}} \right)} \right\|}_{{L_w^{{q),\theta q/p}}}}}\leqslant c{{\left\| f \right\|}_{{L_w^{{p),\theta }}}}} \), where q is the Hardy–Littlewood–Sobolev exponent of p, holds for potentials with product kernels Tα if and only if the weight w belongs to the Muckenhoupt class A1+q/p′ defined with respect to n-dimensional intervals with sides parallel to the coordinate axes. We also provide a motivation of choosing θq/p as the second parameter of the target space.