Acta Mathematica Sinica

, Volume 21, Issue 6, pp 1465–1474 | Cite as

The Martingale Hardy Type Inequalities for Dyadic Derivative and Integral

ORIGINAL ARTICLES
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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 H p,q to L p,q , of weak type (L 1 ,L 1 ), 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 H p and the hybrid Hardy spaces \( \begin{array}{*{20}c} {{H^{\# }_{p} }} & {{0 < p \leqslant 1}} \\ \end{array} \).

Keywords

martingale Hardy space dyadic derivative dyadic integral Walsh–Fejer kernels p–atom Quasi–local operator 

MR (2000) Subject Classification

42A56 42C10 43A70 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institute of Near Sensing Technique with Millimeter Wave & Optical WaveNanjing University of Science & TechnologyNanjing 210094P. R. China
  2. 2.School of Mathematics & Computer ScienceFuzhou UniversityFuzhou, 350002P. R. China
  3. 3.Institute of Near Sensing Technique with Millimeter Wave & Optical WaveNanjing University of Science & TechnologyNanjing 210094P. R. China
  4. 4.Institute of Near Sensing Technique with Millimeter Wave & Optical WaveNanjing University of Science & TechnologyNanjing 210094P. R. China

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