Criteria for Multilinear Sobolev Inequality with Non-doubling Measure in Lorentz Spaces

In this paper necessary and sufficient conditions on a measure \(\mu \) guaranteeing the boundedness of the multilinear fractional integral operator \(T_{\gamma , \mu }^{(m)}\) (defined with respect to a measure \(\mu \)) from the product of Lorentz spaces \(\prod _{k=1}^m L^{r_k, s_k}_{\mu }\) to the Lorentz space \(L^{p,q}_{\mu }(X)\) are established. The results are new even for linear fractional integrals \(T_{\gamma , \mu }\) (i.e., for \(m=1\)). From the general results we have a criterion for the validity of Sobolev–type inequality in Lorentz spaces defined for non-doubling measures. Finally, we investigate the same problem for Morrey-Lorentz spaces. To prove the main result we use the boundedness of the multilinear modifies maximal operator \(\widetilde{\mathcal {M}}\).