The Fefferman-Phong uncertainty principle for representations of Lie groups and applications
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909125725650944 |
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| author | Nicola, Fabio |
| author_facet | Nicola, Fabio |
| contents | We prove a new uncertainty principle for square-integrable irreducible unitary representations of connected Lie groups. The concentration of the matrix coefficients is measured in terms of weighted $L^p$ norms, with weights in the local Muckenhoupt class $A_{\infty,{\rm loc}}$ associated with a subRiemannian left-invariant metric and a relatively invariant measure. The result is reminiscent of the Fefferman-Phong uncertainty principle, and is new even for the Schrödinger representation of the reduced Heisenberg group, which corresponds to the short-time Fourier transform. As an application, we give an optimal estimate of the order of magnitude of the bottom of the spectrum and of the essential spectrum of semiclassical anti-Wick operators in $\mathbb{R}^d$ with a nonnegative symbol $a$ in the class $A_{\infty}$ (in particular, for polynomial symbols). Precisely, we show that the infimum $\inf _{(x_0,ω_0)\in{\mathbb{R}^{2d}}} -\!\!\!\!\!\int_{B((x_0,ω_0),\sqrt{h})} a(x,ω)\, dx\,dω$ represents (up to multiplicative constants) both a lower bound and an upper bound for the bottom of the spectrum, uniformly with respect to $h>0$. Similarly the quantity $$\liminf_{(x_0,ω_0)\to\infty} -\!\!\!\!\!\!\int_{B((x_0,ω_0),\sqrt{h})} a(x,ω)\, dx\,dω$$ represents both a lower bound and an upper bound for the bottom of the essential spectrum, uniformly with respect to $h>0$. Similar results are proved for semiclassical symbol classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Fefferman-Phong uncertainty principle for representations of Lie groups and applications Nicola, Fabio Classical Analysis and ODEs Mathematical Physics Functional Analysis We prove a new uncertainty principle for square-integrable irreducible unitary representations of connected Lie groups. The concentration of the matrix coefficients is measured in terms of weighted $L^p$ norms, with weights in the local Muckenhoupt class $A_{\infty,{\rm loc}}$ associated with a subRiemannian left-invariant metric and a relatively invariant measure. The result is reminiscent of the Fefferman-Phong uncertainty principle, and is new even for the Schrödinger representation of the reduced Heisenberg group, which corresponds to the short-time Fourier transform. As an application, we give an optimal estimate of the order of magnitude of the bottom of the spectrum and of the essential spectrum of semiclassical anti-Wick operators in $\mathbb{R}^d$ with a nonnegative symbol $a$ in the class $A_{\infty}$ (in particular, for polynomial symbols). Precisely, we show that the infimum $\inf _{(x_0,ω_0)\in{\mathbb{R}^{2d}}} -\!\!\!\!\!\int_{B((x_0,ω_0),\sqrt{h})} a(x,ω)\, dx\,dω$ represents (up to multiplicative constants) both a lower bound and an upper bound for the bottom of the spectrum, uniformly with respect to $h>0$. Similarly the quantity $$\liminf_{(x_0,ω_0)\to\infty} -\!\!\!\!\!\!\int_{B((x_0,ω_0),\sqrt{h})} a(x,ω)\, dx\,dω$$ represents both a lower bound and an upper bound for the bottom of the essential spectrum, uniformly with respect to $h>0$. Similar results are proved for semiclassical symbol classes. |
| title | The Fefferman-Phong uncertainty principle for representations of Lie groups and applications |
| topic | Classical Analysis and ODEs Mathematical Physics Functional Analysis |
| url | https://arxiv.org/abs/2402.03250 |