Asmar, Nakhlé; Berkson, Earl; Gillespie, T. A. Transference of strong type maximal inequalities by separation-preserving representations. (English) Zbl 0729.43003 Am. J. Math. 113, No. 1, 47-74 (1991). Let G be a locally compact abelian group and R: \(G\to L^ p(\mu)\) a uniformly bounded strongly continuous representation. If \(R_ u\) are separation preserving (fg\(\equiv 0\Rightarrow\) \(R_ uf\cdot R_ ug\equiv 0)\), \(u\in G\), the authors show that for any sequence of convolution operators on \(L^ p(G)\) for which the corresponding maximal operator is bounded on \(L^ p(G)\), the maximal operator on \(L^ p(\mu)\) corresponding to the “transferred” multipliers by R is also bounded. It is shown however that the corresponding fact for weak type estimates is not true. Reviewer: V.V.Peller (St.Petersburg) Cited in 1 ReviewCited in 14 Documents MSC: 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc. Keywords:locally compact abelian group; strongly continuous representation; convolution operators; maximal operator; multipliers; weak type estimates PDFBibTeX XMLCite \textit{N. Asmar} et al., Am. J. Math. 113, No. 1, 47--74 (1991; Zbl 0729.43003) Full Text: DOI