Sobolev-BMO and fractional integrals on super-critical ranges of Lebesgue spaces

In this article, we explore the mapping and boundedness properties of linear and bilinear fractional integral operators acting on Lebesgue spaces with large indices. The prototype ν-order fractional integral operator is the Riesz potential Iν, and the standard estimates for Iν are from Lp into Lq when 1