The Fefferman-Phong uncertainty principle for representations of Lie groups and applications

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 $A_{\mathrm{\infty },\mathrm{l}\mathrm{o}\mathrm{c}}$ 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 $A_{\mathrm{\infty }}$ (in particular, for polynomial symbols). Precisely, we show that the infimum ${\inf }_{(x_0,{\omega }_0)\in {\mathbb{R}}^{2d}}-\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{B((x_0,{\omega }_0),\sqrt{h})}a(x,\omega )dxd\omega$ 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 $$\underset{(x_0,{\omega }_0)\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}-\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{\int }_{B((x_0,{\omega }_0),\sqrt{h})}a(x,\omega )dxd\omega$$ 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.