Reverse Hardy-Littlewood-Sobolev inequalities (Q2338120)

This paper deals with a new family of reverse Hardy-Littlewood-Sobolev inequalities which involve a power law kernel with positive exponent. The range of the admissible parameters as well as the properties of optimal functions are also studied. A type of concentration is investigated, and the relationship with free energy functionals and nonlinear diffusion equations involving mean field drifts is also analyzed. The main arguments are based on a non-conformally invariant family of interpolation inequalities, relaxed variational methods, and lower semicontinuous extension of the corresponding free energy.