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

Abstract

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<p<nν and 1p=1q+νn. We show that a ν-order linear fractional integral operator can be continuously extended to a bounded operator from Lp into the Sobolev-BMO space Is(BMO) when nν≤p<∞ and 0≤s<ν satisfy 1p=ν−sn. Likewise, we prove estimates for ν-order bilinear fractional integral operators from Lp1×Lp2 into Is(BMO) for various ranges of the indices p1, p2, and s satisfying 1p1+1p2=ν−sn.