Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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).

References [Enhancements On Off] (What's this?)

  • [1] Y. M. Chen, An almost everywhere divergent Fourier series of the class 𝐿(log⁺log⁺𝐿)^{1-𝜖}, J. London Math. Soc. 44 (1969), 643–654. MR 0240539
  • [2] Richard A. Hunt, On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967) Southern Illinois Univ. Press, Carbondale, Ill., 1968, pp. 235–255. MR 0238019
  • [3] K. H. Moon, Divergent Fourier series of functions in Orlicz classes (preprint in preparation).
  • [4] Per Sjölin, An inequality of Paley and convergence a.e. of Walsh-Fourier series., Ark. Mat. 7 (1969), 551–570. MR 0241885
  • [5] E. M. Stein and Guido Weiss, An extension of a theorem of Marcinkiewicz and some of its applications, J. Math. Mech. 8 (1959), 263–284. MR 0107163

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47G05

Retrieve articles in all journals with MSC: 47G05

Additional Information

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