Adams inequalities for Riesz subcritical potentials

The Riesz potential $|x|^{\alpha-n}*f$ fails to be continuous from $L^p$ to $L^q$ at the critical index $p=n/\alpha$. In this case the classical Hardy-Littlewood –Sobolev inequality is replaced by an inequality for the exponential integral, the sharp form of which is due to Adams, on domains with finite measure.

In joint work with Luigi Fontana we derived versions of such "Adams inequalities" which are valid in general measure spaces, and for integral operators whose kernels behave like a Riesz kernel locally, but with slightly better decay at infinity. I will present several examples of such "Riesz subcritical potentials", and their connections with Moser-Trudinger and Poincaré inequalities.