Two sharp uniqueness principles arising from refined Hartley transform

We establish two sharp uniqueness principles for the refined Hartley transform on the real line. The first result is a Hardy-type theorem. We show that if pair (f,Hf) satisfy |f(x)|≤C e−αx2 and |Hf(y)|≤C e−βy2 for some α,β>0, then f≡0 whenever αβ>14. In critical case αβ=14, then f is necessarily a Gaussian of the form f(x)=A e−αx2 (with β=14α), while in subcritical regime αβ<14 we construct nontrivial examples, thereby establishing sharpness. We further show that any decay strictly faster than Gaussian forces f≡0, independently of the product αβ. The second result is a Beurling-type theorem, if ∬R2|f(x)| |Hf(y)| e|xy| dx dy<∞, then f vanishes identically. The proofs rely on a reduction to the case of classical Fourier transform through an explicit invertible algebraic transfer. This reduction, which may be of independent interest, enables us to invoke the classical results of Hardy and Hörmander.