Subcritical Fourier uncertainty principles

It is well known that if a function f satisfies f(x) eπ α |x|²ₚ + f(ξ) eπ α |ξ|²q<∞ (*) with α=1 and 1≤ p,q<∞, then f≡ 0. We prove that if f satisfies $(*)$ with some 0<α<1 and 1≤ p,q≤ ∞, then |f(y)|≤ C (1+|y|)d/p e- π α |y|², y∈ Rᵈ, with C=C(α,d,p,q) and this bound is sharp for p≠ 1. We also study a related uncertainty principle for functions satisfying \;\;f(x)|x|ᵐₚ+ f(ξ)|ξ|ⁿq <∞.