Almost everywhere convergence of sequences of multiplier operators on local fields

Let K n be the n -dimensional vector space over a local field K . Two maximal multiplier theorems on L p( K n) are proved for certain multiplier operator sequences associated with regularization and dilation respectively. Consequently the a. e. convergence of such multiplier operator sequences is obtained. This sharpens Taibleson’s main result and applies to several important singular integral operators on K n.