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Some singular measures on the circle which improve $$L^ p$$ spaces. (English) Zbl 0637.43002
Let $$1\leq p<\infty$$. A positive Borel measure $$\mu$$ is said to be $$L^ p$$-improving if $$\mu$$ defines a convolution operator from $$L^ p$$ to $$L^ q$$, for $$q>p$$. In [Colloq. Math. 47, 113-117 (1982; Zbl 0501.42007)] D. Oberlin proved that the classical Cantor-Lebesgue measure on the circle group T is $$L^ p$$-improving for $$p>1$$. In this paper it is shown that the result is true for Cantor Lebesgue measures on T with a rational ratio of dissection. The result is extended to the real line $${\mathbb{R}}$$. An $$L^ p$$-improving result for $$p<2$$ is given on a finite cyclic group G. These results have been independently obtained by W. Becker, S. Janson, D. Jerison and M. Christ.
Reviewer: S.Poornima

MSC:
 43A05 Measures on groups and semigroups, etc. 42A85 Convolution, factorization for one variable harmonic analysis 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 42A45 Multipliers in one variable harmonic analysis
Zbl 0501.42007
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