Commutators associated with Schrödinger operators on the nilpotent Lie group
Abstract Assume that G is a nilpotent Lie group. Denote by L = − Δ + W $L=-\Delta +W $ the Schrödinger operator on G, where Δ is the sub-Laplacian, the nonnegative potential W belongs to the reverse Hölder class B q 1 $B_{q_{1}}$ for some q 1 ≥ D 2 $q_{1} \geq \frac{D}{2}$ and D is the dimension at...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-12-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-017-1584-8 |
| _version_ | 1818995245549879296 |
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| author | Tianzhen Ni Yu Liu |
| author_facet | Tianzhen Ni Yu Liu |
| author_sort | Tianzhen Ni |
| collection | DOAJ |
| description | Abstract Assume that G is a nilpotent Lie group. Denote by L = − Δ + W $L=-\Delta +W $ the Schrödinger operator on G, where Δ is the sub-Laplacian, the nonnegative potential W belongs to the reverse Hölder class B q 1 $B_{q_{1}}$ for some q 1 ≥ D 2 $q_{1} \geq \frac{D}{2}$ and D is the dimension at infinity of G. Let R = ∇ ( − Δ + W ) − 1 2 $\mathcal{R}=\nabla (-\Delta +W)^{-\frac{1}{2}}$ be the Riesz transform associated with L. In this paper we obtain some estimates for the commutator [ h , R ] $[h,\mathcal{R}]$ for h ∈ Lip ν θ $h\in \operatorname{Lip}^{\theta }_{\nu }$ , where Lip ν θ $\operatorname{Lip}^{\theta }_{\nu }$ is a function space which is larger than the classical Lipschitz space. |
| first_indexed | 2024-12-20T21:10:47Z |
| format | Article |
| id | doaj.art-4774d7e9b6f24c97adb2fd72800bf68a |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-20T21:10:47Z |
| publishDate | 2017-12-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-4774d7e9b6f24c97adb2fd72800bf68a2022-12-21T19:26:32ZengSpringerOpenJournal of Inequalities and Applications1029-242X2017-12-012017111710.1186/s13660-017-1584-8Commutators associated with Schrödinger operators on the nilpotent Lie groupTianzhen Ni0Yu Liu1School of Mathematics and Physics, University of Science and Technology BeijingSchool of Mathematics and Physics, University of Science and Technology BeijingAbstract Assume that G is a nilpotent Lie group. Denote by L = − Δ + W $L=-\Delta +W $ the Schrödinger operator on G, where Δ is the sub-Laplacian, the nonnegative potential W belongs to the reverse Hölder class B q 1 $B_{q_{1}}$ for some q 1 ≥ D 2 $q_{1} \geq \frac{D}{2}$ and D is the dimension at infinity of G. Let R = ∇ ( − Δ + W ) − 1 2 $\mathcal{R}=\nabla (-\Delta +W)^{-\frac{1}{2}}$ be the Riesz transform associated with L. In this paper we obtain some estimates for the commutator [ h , R ] $[h,\mathcal{R}]$ for h ∈ Lip ν θ $h\in \operatorname{Lip}^{\theta }_{\nu }$ , where Lip ν θ $\operatorname{Lip}^{\theta }_{\nu }$ is a function space which is larger than the classical Lipschitz space.http://link.springer.com/article/10.1186/s13660-017-1584-8commutatorLipschitz spacenilpotent Lie groupsreverse Hölder inequalityRiesz transformSchrödinger operator |
| spellingShingle | Tianzhen Ni Yu Liu Commutators associated with Schrödinger operators on the nilpotent Lie group Journal of Inequalities and Applications commutator Lipschitz space nilpotent Lie groups reverse Hölder inequality Riesz transform Schrödinger operator |
| title | Commutators associated with Schrödinger operators on the nilpotent Lie group |
| title_full | Commutators associated with Schrödinger operators on the nilpotent Lie group |
| title_fullStr | Commutators associated with Schrödinger operators on the nilpotent Lie group |
| title_full_unstemmed | Commutators associated with Schrödinger operators on the nilpotent Lie group |
| title_short | Commutators associated with Schrödinger operators on the nilpotent Lie group |
| title_sort | commutators associated with schrodinger operators on the nilpotent lie group |
| topic | commutator Lipschitz space nilpotent Lie groups reverse Hölder inequality Riesz transform Schrödinger operator |
| url | http://link.springer.com/article/10.1186/s13660-017-1584-8 |
| work_keys_str_mv | AT tianzhenni commutatorsassociatedwithschrodingeroperatorsonthenilpotentliegroup AT yuliu commutatorsassociatedwithschrodingeroperatorsonthenilpotentliegroup |