Lorentz-improving measures (Q1328070)

The Lorentz spaces, \(L(p,q)\), are function spaces, intermediate to the \(L^ p\)-spaces. Motivated by the definition of an \(L^ p\)-improving measure, we call a measure \(\mu\) (defined on a compact abelian group) Lorentz-improving if there exist \(1<p<\infty\), \(1\leq q<r\leq\infty\), such that \(\mu* L(p,r)\subseteq L(p,q)\). We show that the class of Lorentz-improving measures properly contains the class of \(L^ p\)- improving measures, and we characterize Lorentz-improving measures in terms of the size of their Fourier transform. This requires the introduction of a new type of thin set which generalizes the notion of a \(\Lambda(p)\) set. Other properties of their size are investigated. We also study random Cantor measures and characterize, almost surely, those which are \(L^ p\)-improving.