Ces`aro Summability of Hardy Spaces on the Ring of Integers in a Local Field

Let O be the ring of integers in a local field K. We solve an open problem due to M. H. Taibleson Ž1975, ‘‘Math. Notes,’’ Vol. 15, Princeton Univ. Press, Prince- ton, NJ.: Suppose f L1ŽO.. Does the Ces`aro means of f converge to f almost everywhere if K has characteristic zero? To this end we study the ŽHp, Lp. boundedness of the associated maximal operator to get the corresponding interpolation result on Hardy Lorentz spaces; in particular we obtain that is of weak type Ž1, 1.. The proof mainly depends on certain estimates for the oscillatory Dirichlet kernels, which are refinements of those obtained earlier by the authorŽ1997, J. Math. Anal. Appl. 208, 528 552..