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In [ABB] Asmar, Berkson and Bourgain prove that for a sequence of weak type (1, 1) multipliers in and a function the weak type (1,1) constant of the maximal operator associated with is controlled by that of the maximal operator associated with . In [ABG] this theorem is extended to LCA groups with an extra hypothesis: the multipliers must be continuous. In this paper we prove a more general version of this last result without assuming the continuity of the multipliers. The proof arises...
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