Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group (Q317301)

The authors prove analogues of Hardy-type inequalities for fractional powers of the sublaplacian \(\mathcal{L}\) on the Heisenberg group.NEWLINENEWLINEInstead of considering powers of \(\mathcal{L}\) the authors consider conformally invariant fractional powers \(\mathcal{L}_{s}\). From the inequalities for \(\mathcal{L}_{s}\) the authors deduce corresponding inequalities for the fractional powers \(\mathcal{L}^{s}\).NEWLINENEWLINEThe authors prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous.NEWLINENEWLINEIn the non-homogenuos case, the constant arising in the Hardy inequaltiy turns out to be optimal. In order to get the results, ground state representations are used.NEWLINENEWLINEIn the homogeneous case, the key ingredients to obtain the results are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by \textit{M. Cowling} and \textit{U. Haagerup} [Invent. Math. 96, No. 3, 507--549 (1989; Zbl 0681.43012)].NEWLINENEWLINEThe approach to prove the integral representations is via the language of semigroups. As a consequence of the Hardy inequalities the authors also obtain versions of the Heisenberg uncertainty inequality for the fractional sublaplacian.