The Martingale Hardy Type Inequalities for Dyadic Derivative and Integral

Abstract

Since the Leibniz–Newton formula for derivatives cannot be used in local fields, it is important to investigate the new concept of derivatives in Walsh–analysis, or harmonic analysis on local fields. On the basis of idea of derivatives introduced by Butzer, Schipp and Wade, Weisz has proved that the maximal operators of the one–dimensional dyadic derivative and integral are bounded from the dyadic Hardy space Hp,q to Lp,q, of weak type (L1,L1), and the corresponding maximal operators of the two–dimensional case are of weak type \( {\left( {H^{\# }_{1} ,L_{1} } \right)} \). In this paper, we show that these maximal operators are bounded both on the dyadic Hardy spaces Hp and the hybrid Hardy spaces \( \begin{array}{*{20}c} {{H^{\# }_{p} }} & {{0 < p \leqslant 1}} \\ \end{array} \).