### Abstract

The two-parameter dyadic martingale Hardy spaces H_{p} are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from H_{p} to L_{p} (1/(α + 1), 1/(β + 1) <p <∞) and is of weak type (H^{#}
_{1}, L_{1}), where the Hardy space H^{#}
_{1} is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f ∈ H^{#}
_{1} converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on H_{p} whenever 1/(α + 1), 1/(β + 1) <p <∞. Thus, in case f ∈ H_{p}, the (C, α, β) means converge to f in H_{p} norm. The same results are proved for the conjugate (C, α, β) means, too.

Original language | English |
---|---|

Pages (from-to) | 389-401 |

Number of pages | 13 |

Journal | Journal of Knot Theory and its Ramifications |

Volume | 6 |

Issue number | 4 |

Publication status | Published - 2000 |

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### Keywords

- (C, α, β) summability
- Atomic decomposition
- Interpolation
- Martingale Hardy spaces
- P-quasi-local operator
- Rectangle p-atom
- Walsh functions

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**The Maximal (C, α, β) Operator of Two-Parameter Walsh-Fourier Series.** / Weisz, F.

Research output: Contribution to journal › Article

*Journal of Knot Theory and its Ramifications*, vol. 6, no. 4, pp. 389-401.

}

TY - JOUR

T1 - The Maximal (C, α, β) Operator of Two-Parameter Walsh-Fourier Series

AU - Weisz, F.

PY - 2000

Y1 - 2000

N2 - The two-parameter dyadic martingale Hardy spaces Hp are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from Hp to Lp (1/(α + 1), 1/(β + 1) # 1, L1), where the Hardy space H# 1 is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f ∈ H# 1 converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on Hp whenever 1/(α + 1), 1/(β + 1) p, the (C, α, β) means converge to f in Hp norm. The same results are proved for the conjugate (C, α, β) means, too.

AB - The two-parameter dyadic martingale Hardy spaces Hp are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from Hp to Lp (1/(α + 1), 1/(β + 1) # 1, L1), where the Hardy space H# 1 is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f ∈ H# 1 converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on Hp whenever 1/(α + 1), 1/(β + 1) p, the (C, α, β) means converge to f in Hp norm. The same results are proved for the conjugate (C, α, β) means, too.

KW - (C, α, β) summability

KW - Atomic decomposition

KW - Interpolation

KW - Martingale Hardy spaces

KW - P-quasi-local operator

KW - Rectangle p-atom

KW - Walsh functions

UR - http://www.scopus.com/inward/record.url?scp=0345847422&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0345847422&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0345847422

VL - 6

SP - 389

EP - 401

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 4

ER -