Most Riesz product measures are $L^ p$-improving

Abstract:

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