Asymmetry of convolution norms on Lie groups
In the case of locally compact abelian groups if a convolution operator on \(L^p\) for some \(p>1\) is bounded, then it is also bounded on the dual space \(L^q\) and has the same norm. The present authors show the situation to be completely different for all nonabelian Lie groups except those locally isomorphic to \(R^3\) or to the semidirect product \(R^+\times R^2\) where the techniques do not apply. In particular the authors show that for all Lie groups of the above type there is a convolution operator with kernel supported on an arbitrary neighborhood of the identity and, for a given \(p>1\) and \(p\) not equal to 2, it is bounded on \(L^p\) but unbounded on \(L^q\) for every \(q\) outside the closed interval with endpoints \(p\) and 2.
scientific article; zbMATH DE number 1496046