Journal of Operator Theory

Volume 39, Issue 1, Winter 1998 pp. 139-149.

Note on norm convergence in the space of weak type multipliers

Authors: Nakhle Asmar (1), Earl Berkson (2), and T.A. Gillespie (3)
Author institution: (1) Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211, U.S.A.
(2) Department of Mathematics, University of Illinois, 1409 West Green St., Illinois 61801, U.S.A.
(3) Department of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3JZ, Scotland

Summary: Suppose that $1\leq p<\infty$, and $G$ is a locally compact abelian group with dual group $\Gamma$. Denote by $\mpw$ the space of weak type $(p,p)$ multipliers for $L^p(G)$. We show that the injection mapping of $\mpw$ into $L^{\infty}(\Gamma)$ is bounded. This affords a short proof that $\mpw$ is complete with respect to the weak type $(p,p)$ multiplier norm. When $1<p<\infty$, the completeness of $\mpw$ is further demonstrated by characterizing the transforms of the weak type $(p,p)$ multipliers as the translation-invariant continuous linear mappings of $L^p(G)$ into the weak $L^p$ space of $G$. This result permits $\mpw$ to be supplied with a Banach space structure when $1<p<\infty$.

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