Subcritical Fourier uncertainty principles

It is well known that if a function f satisfies f (x) e^ |x|²ₚ + f () e^{ ||²}q< (*) with =1 and 1 p, q<, then f 0. We prove that if f satisfies (*) with some 0<<1 and 1 p, q, then |f (y) | C (1+|y|) ^d{p} e^- |y|², y Rᵈ, with C=C (, d, p, q) and this bound is sharp for p 1. We also study a related uncertainty principle for functions satisfying \;\; (x) |x|ᵐₚ+ f () ||ⁿq <.