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Transference of strong type maximal inequalities by separation-preserving representations. (English) Zbl 0729.43003

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.

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.
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