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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On restricted weak type $ (1,\,1)$

Author: K. H. Moon
Journal: Proc. Amer. Math. Soc. 42 (1974), 148-152
MSC: Primary 47G05
MathSciNet review: 0341196
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Abstract: Let $ {\{ {S_k}\} _{k \geqq 1}}$ be a sequence of linear operators defined on $ {L^1}({R^n})$ such that for every $ f \in {L^1}({R^n}),{S_k}f = f \ast {g_k}$ for some $ {g_k} \in {L^1}({R^n}),k = 1,2, \cdots $, and $ Tf(x) = {\sup _{k \geqq 1}}\vert{S_k}f(x)\vert$. Then the inequality $ m\{ x \in {R^n};Tf(x) > y\} \leqq C{y^{ - 1}}\smallint_{{R^n}} {\vert f(t)\vert dt} $ holds for characteristic functions f (T is of restricted weak type (1, 1)) if and only if it holds for all functions $ f \in {L^1}({R^n})$ (T is of weak type (1, 1)). In particular, if $ {S_k}f$ is the kth partial sum of Fourier series of f, this theorem implies that the maximal operator T related to $ {S_k}$ is not of restricted weak type (1, 1).

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Keywords: Weak type (p, q), restricted weak type (p, q), maximal operators
Article copyright: © Copyright 1974 American Mathematical Society