## Most Riesz product measures are $$L^ p$$-improving.(English)Zbl 0593.43002

Let G be an infinite compact abelian group with dual $$\Gamma$$. A Borel measure $$\mu$$ defined on G is called $$L^ p$$-improving if, given $$p>1$$, there is a $$q=q(p,\mu)>p$$ and a $$K=K(p,q,\mu)>0$$ such that $$\mu$$ satisfies $$\| \mu *f\|_ q\leq K \| f\|_ p$$ for each $$f\in L^ p(G)$$. For the notion of Riesz product measure, we refer to C. C. Graham and O. C. McGehee [Essays in commutative harmonic analysis (1979; Zbl 0439.43001), Ch. 7]. The author characterizes the $$L^ p$$-improving Riesz product measures on G by means of their Fourier transforms, as follows: A Riesz product measure $$\mu$$ on G is $$L^ p$$- improving iff there is a positive number $$\delta <1$$ such that the Fourier transform of $$\mu$$ satisfies $$| {\hat \mu}(\gamma)| \leq \delta$$ for each nonzero $$\gamma$$ in $$\Gamma$$.
Reviewer: G.Crombez

### MSC:

 43A05 Measures on groups and semigroups, etc. 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups 43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)

Zbl 0439.43001
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