On restricted weak type
Author:
K. H. Moon
Journal:
Proc. Amer. Math. Soc. 42 (1974), 148152
MSC:
Primary 47G05
MathSciNet review:
0341196
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 kth 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)^{1𝜖}, J. London
Math. Soc. 44 (1969), 643–654. MR 0240539
(39 #1886)
 [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
(38 #6296)
 [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
WalshFourier series., Ark. Mat. 7 (1969),
551–570 (1969). MR 0241885
(39 #3222)
 [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 (21 #5888)
 [1]
 Y. M. Chen, An almost everywhere divergent Fourier series of the class , London Math. Soc. J. 44 (1969), 643654. MR 39 #1886. MR 0240539 (39:1886)
 [2]
 R. 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. 235255. MR 38 #6296. MR 0238019 (38:6296)
 [3]
 K. H. Moon, Divergent Fourier series of functions in Orlicz classes (preprint in preparation).
 [4]
 P. Sjölin, An inequality of Paley and convergence a.e. of WalshFourier series, Ark. Mat. 7 (1969), 551570. MR 39 #3222. MR 0241885 (39:3222)
 [5]
 E. M. Stein and G. Weiss, An extension of a theorem of Marcinkiewicz and some of its applications, J. Math. Mech. 8 (1959), 263284. MR 21 #5888. MR 0107163 (21:5888)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
47G05
Retrieve articles in all journals
with MSC:
47G05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197403411964
PII:
S 00029939(1974)03411964
Keywords:
Weak type (p, q),
restricted weak type (p, q),
maximal operators
Article copyright:
© Copyright 1974
American Mathematical Society
