Cesàro Summability of Hardy Spaces on the Ring of Integers in a Local Field

Abstract

Let O be the ring of integers in a local field K. We solve an open problem due to M. H. Taibleson (1975, “Math. Notes,” Vol. 15, Princeton Univ. Press, Princeton, NJ): Suppose f∈L1(O). Does the Cesàro means of f converge to f almost everywhere if K has characteristic zero? To this end we study the (Hp,Lp) boundedness of the associated maximal operator σ* to get the corresponding interpolation result on Hardy–Lorentz spaces; in particular we obtain that σ* is of weak type (1,1). The proof mainly depends on certain estimates for the oscillatory Dirichlet kernels, which are refinements of those obtained earlier by the author (1997, J. Math. Anal. Appl.208, 528–552).