Subcritical Fourier uncertainty principles

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