Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices
We consider operators acting on a UMD Banach lattice $X$ that have the same algebraic structure as the position and momentum operators associated with the harmonic oscillator $-\frac{1}{2}\Delta + \frac{1}{2}|x|^{2} $ acting on $L^{2}(\mathbb{R}^{d})$. More precisely, we consider abstract harmonic o...
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Académie des sciences
2023-07-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.370/ |
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| author | van Neerven, Jan Portal, Pierre Sharma, Himani |
| author_facet | van Neerven, Jan Portal, Pierre Sharma, Himani |
| author_sort | van Neerven, Jan |
| collection | DOAJ |
| description | We consider operators acting on a UMD Banach lattice $X$ that have the same algebraic structure as the position and momentum operators associated with the harmonic oscillator $-\frac{1}{2}\Delta + \frac{1}{2}|x|^{2} $ acting on $L^{2}(\mathbb{R}^{d})$. More precisely, we consider abstract harmonic oscillators of the form $\frac{1}{2} \sum _{j=1} ^{d}(A_{j}^{2}+B_{j}^{2})$ for tuples of operators $A=(A_{j})_{j=1} ^{d}$ and $B=(B_{k})_{k=1} ^{d}$, where $iA_j$ and $iB_k$ are assumed to generate $C_{0}$ groups and to satisfy the canonical commutator relations. We prove functional calculus results for these abstract harmonic oscillators that match classical Hörmander spectral multiplier estimates for the harmonic oscillator $-\frac{1}{2}\Delta + \frac{1}{2}|x|^{2}$ on $L^{p}(\mathbb{R}^{d})$. This covers situations where the underlying metric measure space is not doubling and the use of function spaces that are not particularly well suited to extrapolation arguments. For instance, as an application we treat the harmonic oscillator on mixed norm Bargmann–Fock spaces. Our approach is based on a transference principle for the Schrödinger representation of the Heisenberg group that allows us to reduce the problem to the study of the twisted Laplacian on the Bochner spaces $L^{2}(\mathbb{R}^{2d};X)$. This can be seen as a generalisation of the Stone–von Neumann theorem to UMD lattices $X$ that are not Hilbert spaces. |
| format | Article |
| id | doaj-art-77bdeb21f1734f0db659ed6498e6adce |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-07-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-77bdeb21f1734f0db659ed6498e6adce2025-02-07T11:08:07ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-07-01361G583584610.5802/crmath.37010.5802/crmath.370Spectral multiplier theorems for abstract harmonic oscillators on UMD latticesvan Neerven, Jan0Portal, Pierre1Sharma, Himani2Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The NetherlandsThe Australian National University, Mathematical Sciences Institute, Hanna Neumann Building, Ngunnawal and Ngambri Country, Canberra ACT 2601, AustraliaThe Australian National University, Mathematical Sciences Institute, Hanna Neumann Building, Ngunnawal and Ngambri Country, Canberra ACT 2601, Australia.We consider operators acting on a UMD Banach lattice $X$ that have the same algebraic structure as the position and momentum operators associated with the harmonic oscillator $-\frac{1}{2}\Delta + \frac{1}{2}|x|^{2} $ acting on $L^{2}(\mathbb{R}^{d})$. More precisely, we consider abstract harmonic oscillators of the form $\frac{1}{2} \sum _{j=1} ^{d}(A_{j}^{2}+B_{j}^{2})$ for tuples of operators $A=(A_{j})_{j=1} ^{d}$ and $B=(B_{k})_{k=1} ^{d}$, where $iA_j$ and $iB_k$ are assumed to generate $C_{0}$ groups and to satisfy the canonical commutator relations. We prove functional calculus results for these abstract harmonic oscillators that match classical Hörmander spectral multiplier estimates for the harmonic oscillator $-\frac{1}{2}\Delta + \frac{1}{2}|x|^{2}$ on $L^{p}(\mathbb{R}^{d})$. This covers situations where the underlying metric measure space is not doubling and the use of function spaces that are not particularly well suited to extrapolation arguments. For instance, as an application we treat the harmonic oscillator on mixed norm Bargmann–Fock spaces. Our approach is based on a transference principle for the Schrödinger representation of the Heisenberg group that allows us to reduce the problem to the study of the twisted Laplacian on the Bochner spaces $L^{2}(\mathbb{R}^{2d};X)$. This can be seen as a generalisation of the Stone–von Neumann theorem to UMD lattices $X$ that are not Hilbert spaces.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.370/spectral multipliersharmonic oscillatortwisted convolutionscanonical commutation relationsWeyl pseudo-differential calculusUMD spacestransference$H^\infty $-calculusHörmander calculus |
| spellingShingle | van Neerven, Jan Portal, Pierre Sharma, Himani Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices Comptes Rendus. Mathématique spectral multipliers harmonic oscillator twisted convolutions canonical commutation relations Weyl pseudo-differential calculus UMD spaces transference $H^\infty $-calculus Hörmander calculus |
| title | Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices |
| title_full | Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices |
| title_fullStr | Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices |
| title_full_unstemmed | Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices |
| title_short | Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices |
| title_sort | spectral multiplier theorems for abstract harmonic oscillators on umd lattices |
| topic | spectral multipliers harmonic oscillator twisted convolutions canonical commutation relations Weyl pseudo-differential calculus UMD spaces transference $H^\infty $-calculus Hörmander calculus |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.370/ |
| work_keys_str_mv | AT vanneervenjan spectralmultipliertheoremsforabstractharmonicoscillatorsonumdlattices AT portalpierre spectralmultipliertheoremsforabstractharmonicoscillatorsonumdlattices AT sharmahimani spectralmultipliertheoremsforabstractharmonicoscillatorsonumdlattices |