A real function associated with convolution of measures (Q1978841)

A measure \(\mu \) is called \(L^p\) improving, \(1<p<\infty \), if there is a \(q\in (p,\infty)\) such that \(\mu *f\in L^q\) for every \(f\in L^p\), where \(\mu *f(x)=(2\pi)^{-1}\int _0^{2\pi } f(x-s) d\mu (s)\). The index function \(I(\mu ,p)\) is given by \(I(\mu ,p)=\sup \{ q\geq p;\mu *L^p\subset L^q\}\). The paper is devoted to the study of the index function. The author states the following results: 1) the domain of \(I(\mu ,p)\) is either empty or an open interval \((1,p_0)\) for some \(p_0\in (1,\infty ]\); 2) \(I(\mu ,\cdot)\) is continuous on its domain, and 3) if \(I(\mu ^n;p_0)=\infty \) for some \(p_0\in (1,\infty)\) and \(n\in \mathbb N\), then the Fourier transform of \(\mu \), \(\widehat \mu (n)=(2\pi)^{-1}\int _0^{2\pi }e ^{-ins} d\mu (s)\), satisfies \(\widehat \mu (n)\to 0\) as \(n\to \infty \). Several examples are pointed out. The proofs are not included but a reference to a forthcoming paper is made.