Integral convolution operators (Q1112291)

Let \(E_ 1,E_ 2\) and \(E_ 3\) be symmetric function spaces on an interval [0,a]. The following problem is under investigation: under which conditions is the convolution operator continuous from \(E_ 1\times E_ 2\) into \(E_ 3?\) The author gives some sufficient conditions for the continuity in terms of the Boyd indices \(\alpha_ E\) and \(\beta_ E\) of a space E. For example, if \(\beta_{E_ 3}+\beta_{E_ 1}<1+\alpha_{E_ 2}\) and \(\alpha_{E_ 1}+\beta_{E_ 3}>1\), then the convolution operator maps \(E_ 1\times E_ 2\) continuously into \(E_ 3\).