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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A maximal inequality for $ H\sp{1}$-functions on a generalized Walsh-Paley group


Author: Nobuhiko Fujii
Journal: Proc. Amer. Math. Soc. 77 (1979), 111-116
MSC: Primary 42B30; Secondary 26D10, 42C10, 43A70
MathSciNet review: 539641
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Abstract: Let $ G = \prod _{i = 0}^\infty Z({p_i})$ be the countable product of discrete cyclic groups of order $ {p_i}$. We assume that $ {\sup _{i \geqslant 0}}{p_i} < \infty $. We consider Walsh-Fourier series on G and define $ {H^1}$-functions on G by the Coifman-Weiss atoms. Let $ {K_n}(x)$ be the nth (C, 1)-kernel. We prove that $ {\smallint_G {{{\sup }_{n \geqslant 1}}\vert({K_n} \ast f)(x)\vert d\mu \leqslant C\left\Vert f \right\Vert} _{{H^1}}}$. Here $ d\mu $ is the normalized Haar measure, $ {\left\Vert \right\Vert _{{H^1}}}$ is the $ {H^1}$-norm and C is a constant independent of f.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0539641-9
PII: S 0002-9939(1979)0539641-9
Article copyright: © Copyright 1979 American Mathematical Society