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.)

Citations:

Zbl 0206.126; Zbl 0195.425; Zbl 0593.43002
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