TRANSLATION-INVARIANT OPERATORS OF WEAK TYPE

Let G be a locally compact group and let m be a left measure on G. For 0 < p < ∞, let LP(G) be the usual Lebesgue space of functions f on G for which fp= ∫G(f(x)pdm(x))1/p< ∞ . If T is a linear operator which takes L1/p(G), or a subspace of L1/p(G), into measurable functionson G, then T is said to be of weak type (p, p) if there exists a positive constant C such that m(x∈G:Tf(x) ≧α) ≦ Cfpα* for f∈Lp(G), α > 0 .We are interested in the translation-invariant operators of weak type (p, p). © 1979, University of California, Berkeley. All Rights Reserved.