### Abstract

A new atomic decomposition of the two-parameter dyadic martingale Hardy spaces H_{p} defined by the quadratic variation is given. We introduce H_{p}-quasi-local operators and prove that if a sublinear operator V is H_{p}-quasi-local and bounded from L_{2} to L_{2} then it is also bounded from H_{p} to L_{p} (0 < p ≤ 1). By an interpolation theorem we get that V is of weak type (H^{#}_{1}, L_{1}) where the Hardy space H^{#}_{1} is defined by the hybrid maximal function. As an application it is shown that the maximal operator of the Cesàro means of a two-parameter martingale is bounded from H_{p} to L_{p} (4/5 < p ≤ ∞) and is of weak type (H^{#}_{1}, L_{1}). So we obtain that the Cesàro means of a function f ∈ H^{#}_{1} converge a.e. to the function in question. Finally, it is verified that if the supremum is taken over all two-powers, only, then the maximal operator of the Cesàro means is bounded from H_{p} to L_{p} for every 2/3 < p ≤ ∞.

Original language | English |
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Pages (from-to) | 168-192 |

Number of pages | 25 |

Journal | Journal of Approximation Theory |

Volume | 88 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 1997 |

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Mathematics(all)
- Applied Mathematics