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Riesz potentials and amalgams. (English) Zbl 0938.47027

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.

MSC:

47B38 Linear operators on function spaces (general)
22E30 Analysis on real and complex Lie groups
31C12 Potential theory on Riemannian manifolds and other spaces
43A80 Analysis on other specific Lie groups
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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