Abstract
For a wide class of functional spaces we obtain a necessary and sufficient condition on a space that guarantees a Hardy-Littlewood type of assertion about whether the sum of a cosine series with monotonic coefficients belongs to a functional space, e.g., Lp (p > 1). As examples we consider Lorentz spaces, Marcinkiewicz spaces, Orlicz spaces, and Lp spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
E. M. Semenov, “Imbedding theorems for Banach spaces of measurable functions,” Dokl. Akad. Nauk SSSR,156, No. 6, 1292–1295 (1964).
T. Shimogaki, “On the complete continuity of operators in an interpolation theorem,” J. Fac. Sci. Hokkaido Univ., Ser. 1,20, 104–114 (1968).
A. Zygmund, Trigonometric Series, Vols. 1 and 2 (rev. ed.), Cambridge University Press, New York (1968).
G. G. Lorentz, “On the theory of spaces A,” Pacific J. Math.,1, 411–423 (1951).
M. A. Krasnosel'skii and Ya. B. Rutitskii, Convex Functions and Orlicz Spaces [in Russian], Fizmatgiz, Moscow (1953).
T. Shimogaki, “Hardy-Littlewood majorants in functions spaces,” J. Math. Soc. Japan,17, 365–373 (1965).
V. A. Rodin, “On the convergence of the partial sums of a trigonometric cosine series with monotonically decreasing coefficients,” Matern. Issled.,8, No. 3, 46–55 (1973).
Y. Sagher, “An application of interpolation theory to Fourier series,” Studia Math.,41, No. 2, 169–181 (1972).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 241–246, August, 1976.
The author thanks E. M. Semenov for stating the problem and for his attention to this paper.
Rights and permissions
About this article
Cite this article
Rodin, V.A. The Hardy-Littlewood theorem for the cosine series in a symmetric space. Mathematical Notes of the Academy of Sciences of the USSR 20, 693–696 (1976). https://doi.org/10.1007/BF01155876
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01155876