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A characterization of \(L^ p\)-improving measures. (English) Zbl 0664.43001

A Borel measure \(\mu\) on a compact abelian group G is \(L^ p\)-improving if \(\mu\) takes \(L^ p\) to \(L^{p+\epsilon}\), for some \(\epsilon >0\), by convolution. If T is the dual group of G and \(\mu\) any Borel measure on G, define for \(\epsilon >0\), the set \(E(\epsilon)=\{\gamma \in T:\) \(| {\hat \mu}(\gamma)| \geq \epsilon \}\). It is proved that a necessary and sufficient condition for \(\mu\) to be \(L^ p\)-improving is that the sets E(\(\epsilon)\) are \(\Lambda\) (p)-sets for all \(2<p<\infty\). In [Ann. Inst. Fourier 20, No.2, 335-402 (1970; Zbl 0195.425)] A. Bonami showed that certain Riesz product measures on the circle are \(L^ p\)- improving and satisfy the above condition. In [Proc. Am. Math. Soc. 97, 291-295 (1986; Zbl 0593.43002)] D. Ritter gave a characterization of \(L^ p\)-improving Riesz product measures in terms of their Fourier transforms.
Reviewer: S.Poornima

MSC:

43A05 Measures on groups and 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.)
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