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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.)

Citations:

Zbl 0439.43001
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