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

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L. Roncal, S. Thangavelu

Classical Analysis and ODEs · Submitted 4 Aug 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group $\mathbb{H}^n$.

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arXiv ID1508.00714 (version 2)
Pith IDZZUIHRN3, Pith's permanent record of this paper and its review
CitationL. Roncal, S. Thangavelu. "Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group." arXiv:1508.00714 (2015). https://pith.science/paper/ZZUIHRN3
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arXiv listing abstract v2. Results and the main claim are underlined; .

We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group $\mathbb{H}^n$. We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous. In the first case, the constant arising in the Hardy inequality turns out to be optimal. In order to get our results, we will use ground state representations. The key ingredients to obtain the latter are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by M. Cowling and U. Haagerup. The approach to prove the integral representations is via the language of semigroups. As a consequence of the Hardy inequalities we also obtain versions of Heisenberg uncertainty inequality for the fractional sublaplacian.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group $\mathbb{H}^n$. contribution 0.85 0.97 0.93 asserted 1.00
s1 We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous. contribution 0.69 0.94 0.16 asserted 1.00
s2 In the first case, the constant arising in the Hardy inequality turns out to be optimal. result 1.00 0.93 0.15 asserted 1.00
s3 In order to get our results, we will use ground state representations. method 1.00 0.80 0.03 asserted 0.98
s4 The key ingredients to obtain the latter are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by M. Cowling and U. Haagerup. method 0.99 0.44 0.03 asserted 0.99
s5 The approach to prove the integral representations is via the language of semigroups. method 1.00 0.75 0.02 asserted 1.00
s6 As a consequence of the Hardy inequalities we also obtain versions of Heisenberg uncertainty inequality for the fractional sublaplacian. result 0.87 0.93 0.12 asserted 1.00

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group $\mathbb{H}^n$. We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous. In the first case, the constant arising in the Hardy inequality turns out to be optimal. In order to get our results, we will use ground state representations. The key ingredients to obtain the latter are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by M. Cowling and U. Haagerup. The approach to prove the integral representations is via the language of semigroups. As a consequence of the Hardy inequalities we also obtain versions of Heisenberg uncertainty inequality for the fractional sublaplacian.

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