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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

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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