Most Riesz product measures are 𝐿^{𝑝}-improving

Abstract

A Borel measure μ \mu on a compact abelian group G G is L p {L^p} -improving if, given p > 1 p > 1 , there is a q = q ( p , μ ) > p q = q(p,\mu ) > p and a K = K ( p , q , μ ) > 0 {\text {a}}\;K = K(p,q,\mu ) > 0 such that ‖ μ ∗ f ‖ q ≤ K ‖ f ‖ p {\left \| {\mu * f} \right \|_q} \leq K{\left \| f \right \|_p} for each f f in L p ( G ) {L^p}(G) . Here the L p {L^p} -improving Riesz product measures on infinite compact abelian groups are characterized by means of their Fourier transforms.