Abstract
The Plancherel Theorem asserts the equality of the {L^2}-norms (with respect to Haar measure) of a function f on a locally compact abelian group G and of its Fourier transform \hat f. The Hausdorff-Young inequality gives conditions on p and q under which {\left// {\hat f} \
ight//_q} ≤ {\left// f \
ight//_p}. We consider a different variant: we place a measure μ on \hat G, a measure w on G, and examine
{∫_{\hat G} {\left| {\hat f} \
ight|} ^2}dμ ≤ {∫_G {\left| f \
ight|} ^2}dw
Our main results show that it is enough to consider the case in which w is equivalent to Haar measure, and we give a condition on w which is necessary and sufficient for the inequality to hold for every μ ≥ 0 with \left// μ \
ight// ≤ 1.
