Hardy’s inequality for fractional powers of the sublaplacian on the Heisenberg group. (English) Zbl 1348.43011
The authors prove analogues of Hardy-type inequalities for fractional powers of the sublaplacian \(\mathcal{L}\) on the Heisenberg group.
Instead 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 \(\mathcal{L}^{s}\).
The authors prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous.
In 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.
In 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 M. Cowling and U. Haagerup [Invent. Math. 96, No. 3, 507–549 (1989; Zbl 0681.43012)].
The 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.
Instead 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 \(\mathcal{L}^{s}\).
The authors prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous.
In 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.
In 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 M. Cowling and U. Haagerup [Invent. Math. 96, No. 3, 507–549 (1989; Zbl 0681.43012)].
The 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.
Reviewer: Koichi Saka (Akita)
MSC:
| 43A80 | Analysis on other specific Lie groups |
| 22E30 | Analysis on real and complex Lie groups |
| 26D15 | Inequalities for sums, series and integrals |
| 35A08 | Fundamental solutions to PDEs |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
Heisenberg group; Hardy inequality; fractional order operator; sublaplacian; integral representation; uncertainty principle; heat semigroup; fundamental solutionCitations:
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