### Abstract

We introduce p-quasi-local operators and the two-dimensional dyadic Hardy spaces H_{p} defined by the dyadic squares. It is proved that, if a sublinear operator T is p-quasi-local and bounded from L_{∞} to L_{∞}, then it is also bounded from H_{p} to L_{p} (0 < p ≤ 1). As an application it is shown that the maximal operator of the Cesaβro means of a martingale is bounded from H_{p} to L_{p} (1/2 < p ≤ ∞) and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. So we obtain the dyadic analogue of a summability result with respect to two-dimensional trigonometric Fourier series due to Marcinkievicz and Zygmund; more exactly, the Cesàro means of a function f ∈ 6 L_{1} converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Finally, it is verified that if we take the supremum in a cone, but for two-powers, only, then the maximal operator of the Cesàro means is bounded from H_{p} to L_{p} for every 0 < p ≤ ∞.

Original language | English |
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Pages (from-to) | 2169-2181 |

Number of pages | 13 |

Journal | Transactions of the American Mathematical Society |

Volume | 348 |

Issue number | 6 |

Publication status | Published - Dec 1 1996 |

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### Keywords

- Atomic decomposition
- Cesàro summability
- Interpolation
- Martingale hardy spaces
- Walsh functions
- p-Atom
- p-Quasi-local operator

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*348*(6), 2169-2181.