Vol. 85, No. 1, 1979

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ISSN: 0030-8730
Translation-invariant operators of weak type

Daniel M. Oberlin

Vol. 85 (1979), No. 1, 155–164
Abstract

Let G be a locally compact group and let m be a left Haar measure on G. For 0 < p < , let Lp(G) be the usual Lebesgue space of functions f on G for which

        ∫
∥f∥ =  (  |f (x)|pdm (x))1∕p < ∞.
p     G

If T is a linear operator which takes Lp(G), or a subspace of Lp(G), into measurable functions on G, then T is said to be of weak type (p,p) if there exists a positive constant C such that

                           p   p         p
m {x ∈ G : |Tf (x)| ≧ α} ≦ C∥f ∥p∕α for f ∈ L (G), α > 0.

We are interested in the translation-invariant operators of weak type (p,p).

Mathematical Subject Classification 2000
Primary: 43A22
Milestones
Received: 16 October 1978
Published: 1 November 1979
Authors
Daniel M. Oberlin