Weighted norm inequalities for maximal operator of Fourier series

Let M be the maximal partial sum operator for Fourier series on the ring of integers Z of a local field K. For 1< p < infinity, we establish weighted norm inequalities for M on the weighted spaces L-P (D, w), where w is a Muckenhoupt A(p) weight. As a consequence of this result, we prove that the Fourier partial sum operators are uniformly of weak type (1, 1) on L-1(D). Further, we establish vector-valued inequalities for Fourier series on D. These results include the cases when Z is the ring of integers of the p-adic field O-p and the field F-q((X)) of formal Laurent series over a finite field F-q, and in particular, when D is the Walsh-Paley or dyadic group 2(omega).