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 $\;\;\displaystyle\|f(x)|x|^m\|_p+ \|\widehat{ f}(\xi)|\xi|^n\|_q <\infty.$
Categories: math.CA math.FA