Subcritical Fourier uncertainty principles

It is well known that if a function $f$ satisfies $$\mathrm{\parallel }f(x)e^{\pi \alpha \mathrm{\mid }x{\mathrm{\mid }}^2}{\mathrm{\parallel }}_p+\mathrm{\parallel }\widehat{f}(\xi )e^{\pi \alpha \mathrm{\mid }\xi {\mathrm{\mid }}^2}{\mathrm{\parallel }}_q<\mathrm{\infty }(\ast )$$ with $\alpha =1$ and $1\le p,q<\mathrm{\infty }$, then $f\equiv 0.$ We prove that if $f$ satisfies $(\ast )$ with some $0<\alpha <1$ and $1\le p,q\le \mathrm{\infty }$, then $$\mathrm{\mid }f(y)\mathrm{\mid }\le C(1+\mathrm{\mid }y\mathrm{\mid }{)}^{\frac{d}{p}}{e}^{-\pi \alpha \mathrm{\mid }y{\mathrm{\mid }}^2},y\in {\mathbb{R}}^d,$$ with $C=C(\alpha ,d,p,q)$ and this bound is sharp for $p\mathrm{\ne }1$. We also study a related uncertainty principle for functions satisfying ${\displaystyle \mathrm{\parallel }f(x)\mathrm{\mid }x{\mathrm{\mid }}^m{\mathrm{\parallel }}_p+\mathrm{\parallel }\widehat{f}(\xi )\mathrm{\mid }\xi {\mathrm{\mid }}^n{\mathrm{\parallel }}_q<\mathrm{\infty }\mathrm{.}}$