Maximal functions and weighted norm inequalities on local fields

Weight functions are characterized so that Hardy–Littlewood maximal operator is bounded in certain spaces. The reverse weak type estimates with applications to some singular integrals and to the class L(1+log+L) of Zygmund are established. These results are also compared with the ones in Euclidean case which are obtained by K.F. Andersen and W.S. Young, thereby showing the differences between the two cases. We introduce a weak type estimate for a new class of maximal function and employ it to deduce a special result on singular operators over a local field which is obtained by K. Phillips and M. Taibleson.