Abstract
We show that, under the conditions of the theorems of Paley and Stein, a system with bounded norms in L∞ can be replaced by a system with bounded norms in the space of functions of bounded mean oscillation.
This is a preview of subscription content, log in via an institution to check access.
References
R. E. Paley, “Some theorems on orthogonal functions,” Stud. Math., No. 3, 226–238 (1931).
E. M. Stein, “Interpolation of linear operators,” Trans. Amer. Math. Soc., 83, No. 21, 482–492 (1956).
V. I. Kolyada, “On some generalizations of the Hardy-Littlewood-Paley theorem,” Mat. Zametki, 51, Issue 3, 24–34 (1993).
S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian] Nauka, Moscow 1978.
S. Bennet and R. Sharpley, “Weak-type inequalities for H p and BMO” Proc. Symp. Pure Math., 35, No. 1, 201–229 (1979).
J. B. Garnett, Bounded Analytic Functions [Russian translation] Mir, Moscow 1984.
H. L. Montgomery, “A note on rearrangements of Fourier coefficients,” Ann. Inst. Fourier, 26, 29–34 (1979).
L. Leindler, “Generalization of inequalities of Hardy and Littlewood,” Acta Sci. Math., 31, No 3-4, 279–285 (1971).
J. Marcinkiewicz and A. Zygmund, “Some theorems on orthogonal systems,” Fund. Math., 28, 309–335 (1937).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kirillov, S.A. A note on the theorems of paley and stein. Ukr Math J 52, 643–648 (2000). https://doi.org/10.1007/BF02515405
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02515405