Subcritical Fourier uncertainty principles

Abstract

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

arXiv comment30 pages; Remark 1.4 (vii), Acknowledgements, and Proposition 5.5 added