Boundedness of Bilinear Bessel Potentials

In analogy with bilinear Riesz potentials, we introduce bilinear Bessel potentials and characterize their boundedness from $L^p\times L^q$ into Lebesgue and Lorentz spaces $L^{r,\alpha }\mathrm{.}$ In several cases we identify the optimal Lorentz indices by constructing explicit counterexamples.