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Cesàro Summability of
Two-dimensional Walsh-Fourier Series

Author: Ferenc Weisz
Journal: Trans. Amer. Math. Soc. 348 (1996), 2169-2181
MSC (1991): Primary 42C10, 43A75; Secondary 60G42, 42B30
MathSciNet review: 1340180
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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 $L_{\infty }$ to $L_{\infty }$, then it is also bounded from $H_{p}$ to $L_{p}$ $(0<p \leq 1)$. As an application it is shown that the maximal operator of the Cesàro means of a martingale is bounded from $H_{p}$ to $L_{p}$ $(1/2<p \leq \infty )$ 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 $f \in L_{1}$ 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 \leq \infty $.

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Additional Information

Ferenc Weisz
Affiliation: Department of Numerical Analysis, Eötvös L. University, H-1088 Budapest, Múzeum krt. 6-8 , Hungary

Keywords: Martingale Hardy spaces, p-atom, atomic decomposition, p-quasi-local operator, interpolation, Walsh functions, Cesàro summability
Received by editor(s): June 28, 1994
Additional Notes: This research was partly supported by the Hungarian Scientific Research Funds (OTKA) No. F4189.
Article copyright: © Copyright 1996 American Mathematical Society

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