Regularity of fractional Schrödinger equations and sub-Laplacian multipliers on the Heisenberg group
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| Format: | Preprint |
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2026
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| _version_ | 1866918436263690240 |
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| author | Bergfeldt, Aksel |
| author_facet | Bergfeldt, Aksel |
| contents | We define functions of the sub-Laplacian $Δ$ on the Heisenberg group $\mathbb H^d$ as Fourier multipliers. In this setting, we show that the solution $u$ of the free fractional Schrödinger equation $i\partial_tu + (-Δ)^νu = 0, u|_{t=0} = u_0$, for any $ν> 0$, satisfies the Hardy space estimate that $$ \|u(t,\cdot)\|_{H^p(\mathbb H^d)} \leq C_p (1 + t)^{Q|1/p-1/2|}\|(1-Δ)^{νQ|1/p-1/2|}u_0\|_{H^p(\mathbb H^d)}, $$ with $Q = 2d + 2$, for all $p \in (0,\infty)$, and the corresponding estimate with $p = \infty$ in $\mathrm{BMO}(\mathbb H^d)$. This is done via a general regularity result for parameter dependent sub-Laplacian Fourier multipliers. We prove also that Bessel potential spaces on the Heisenberg group correspond to Sobolev spaces in the same way as in Euclidean space, also for Hardy spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_29588 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Regularity of fractional Schrödinger equations and sub-Laplacian multipliers on the Heisenberg group Bergfeldt, Aksel Analysis of PDEs 35R03, 35B65, 43A85 We define functions of the sub-Laplacian $Δ$ on the Heisenberg group $\mathbb H^d$ as Fourier multipliers. In this setting, we show that the solution $u$ of the free fractional Schrödinger equation $i\partial_tu + (-Δ)^νu = 0, u|_{t=0} = u_0$, for any $ν> 0$, satisfies the Hardy space estimate that $$ \|u(t,\cdot)\|_{H^p(\mathbb H^d)} \leq C_p (1 + t)^{Q|1/p-1/2|}\|(1-Δ)^{νQ|1/p-1/2|}u_0\|_{H^p(\mathbb H^d)}, $$ with $Q = 2d + 2$, for all $p \in (0,\infty)$, and the corresponding estimate with $p = \infty$ in $\mathrm{BMO}(\mathbb H^d)$. This is done via a general regularity result for parameter dependent sub-Laplacian Fourier multipliers. We prove also that Bessel potential spaces on the Heisenberg group correspond to Sobolev spaces in the same way as in Euclidean space, also for Hardy spaces. |
| title | Regularity of fractional Schrödinger equations and sub-Laplacian multipliers on the Heisenberg group |
| topic | Analysis of PDEs 35R03, 35B65, 43A85 |
| url | https://arxiv.org/abs/2603.29588 |