  This is a contribution to the theory of Lizorkin--Triebel spaces having mixed
Lebesgue norms and quasi-homogeneous smoothness. We discuss their
characterisation in terms of general quasi-norms based on convolutions. In
particular, this covers the case of local means, in Triebel's terminology. The
main step is an extension of some crucial inequalities due to Rychkov to the
case with mixed norms.
