I am struggling a little in the proof of Proposition 3.1 of Hörmander's "Hypoelliptic second order differential equations". In particular, the following does not sound true to me.
The function $\xi\in \mathbb{R}^n\mapsto(1+|\xi|^2)^{-1}\in \mathbb{L}_{loc}^p(\mathbb{R}^n)$ for all $p\geq 1$ and decay at infinity. Ok, but then? Maybe I am misunderstanding the statement or missing something. May someone help?
Thanks in advance
The statement is about the inverse Fourier transform of $(1+\lvert\xi\rvert^2)^{-1}$, which is (possibly up to a constant) given by the Bessel potential $$ G_2(x) = \frac1{4\pi} \int_0^{\infty} \exp\left(-\frac{\pi \lvert x\rvert^2}{y} - \frac{y}{4\pi}\right)\, y^{-\frac{d}2}\,\mathrm{d}y.$$ This can be verified to decrease exponentially in $x$; note that this also follows from general properties of Fourier transforms, as the kernel $\hat K = (1+\lvert\xi\rvert^2)^{-1}$ is smooth.