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A convolution inequality concerning Cantor-Lebesgue measures. (English) Zbl 0644.42011

Denoting with \(\mu_{\lambda}\) the totally singular probability measure associated to the Cantor set \(E_{\lambda}\) of constant ratio of dissection \(\lambda\) on the circle group, the author proves that for any \(p\in (1,\infty)\) and for any real \(\lambda >2\), there exists \(q(p,\lambda)>p\) such that \(\| f^*\mu_{\lambda}\|_ q\leq \| f\|_ q\) for all \(f\in L^ p\). In this way the results of D. L. Ritter and W. Beckner, S. Janson, and D. Jerison are extended.
Reviewer: L.Goras

MSC:

42A85 Convolution, factorization for one variable harmonic analysis
28A25 Integration with respect to measures and other set functions
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