On restricted weak type

Author:
K. H. Moon

Journal:
Proc. Amer. Math. Soc. **42** (1974), 148-152

MSC:
Primary 47G05

DOI:
https://doi.org/10.1090/S0002-9939-1974-0341196-4

MathSciNet review:
0341196

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Abstract: Let be a sequence of linear operators defined on such that for every for some , and . Then the inequality holds for characteristic functions *f* (*T* is of restricted weak type (1, 1)) if and only if it holds for all functions (*T* is of weak type (1, 1)). In particular, if is the *k*th partial sum of Fourier series of *f*, this theorem implies that the maximal operator *T* related to is not of restricted weak type (1, 1).

**[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**, https://doi.org/10.1112/jlms/s1-44.1.643**[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**, https://doi.org/10.1007/BF02590894**[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**

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DOI:
https://doi.org/10.1090/S0002-9939-1974-0341196-4

Keywords:
Weak type (*p, q*),
restricted weak type (*p, q*),
maximal operators

Article copyright:
© Copyright 1974
American Mathematical Society