Riesz potentials and amalgams (Q1294747)

Le \(M\) be the infinite cylinder and let \(L\) be the Laplace-Beltrami operator of \(M\). Using the notion of amalgams the authors show that the Hardy-Littlewood-Sobolev regularity theorem does not generalize to Riesz potential operators \(L^{-\alpha/2}\). Specifically, they show that such operators do not map \(L^p(M)\) into \(L^q(M)\) where \(1<p<q<\infty\) and \(1/p- 1/q= \alpha/n\). The authors then investigate the smoothing properties of these operators in terms of amalgams.