Subcritical Fourier uncertainty principles

Abstract

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