Abstract
We prove analogues of P. L. Ul’yanov and V. A. Andrienko results concerning embeddings of Lq Hölder spaces into Lebesgue spaces Lr or Lr Hölder spaces in the case 1 ≤ q < r ≤ 2 for functions defined on p-adic linear spaces. The conditions presented in these theorems are sharp. Also we give necessary and sufficient conditions for such embeddings in the case 1 ≤ q < r < ∞ that generalize recent results of S. S. Platonov.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. A. Andrienko, “Necessary conditions for imbedding the function classes \(H_p^\omega \),” Mat. Sb. 78, 280–300 (1969) [In Russian]; Engl. transl.: Math. USSR-Sb. 7, 273–292 (1969).
B. Golubov, A. Efimov and V. Skvortsov, Walsh Series and Transforms. Theory and Applications (Kluwer, Dordrecht, 1991).
B. I. Golubov and S. S. Volosivets, “On the integrability and uniform convergence of multiplicative Fourier transforms,” Georgian Math. J. 16, 533–546 (2009).
N. Koblitz, p-Adic Numbers, p-Adic Analysis, and Zeta Functions (Springer-Verlag, N.Y., 1984).
A. A. Konyushkov, “Best approximation by trigonometric polynomials and Fourier coefficients,” Mat. Sb. 44, 53–84 (1958) [in Russian].
L. Leindler, “Generalization of inequalities of Hardy and Littlewood,” Acta Sci. Math. (Szeged). 31, 279–285 (1970).
S. S. Platonov, “Some problems in the theory of approximation of functions on the group of p-adic numbers,” p-Adic Numbers Ultrametric Anal. Appl. 10, 118–129 (2018).
M. H. Taibleson, Fourier Analysis on Local Fields (Princeton Univ. Press, Princeton, 1975).
P. L. Ul’yanov, “On absolute and uniform convergence of Fourier series,” Mat. Sb. 72, 193–225 (1967) [In Russian]; Engl. transl.: Math. USSR-Sb. 1, 169–197 (1967).
P. L. Ul’yanov, “Embedding of some function classes \(H_p^\omega \),” Izv. AN SSSR. Ser. Mat. 32, 649–686 (1968) [In Riussian]; Engl. transl.: Math. USSR-Izv. 2, 601–637 (1968).
V. S. Vladimirov, I. V. Volovich and E.I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).
S. S. Volosivets, “Hausdorff operators on p-adic linearspaces and their properties in Hardy, BMO and Hölder spaces,” Math. Notes. 93, 382–391 (2013).
S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems (Springer-Verlag, Berlin-Heidelberg-New York, 1975).
S. S. Platonov, “Some problems in the theory of approximation of functions on locally compact Vilenkin groups,” p-Adic Numbers Ultrametric Anal. Appl. 11, 163–175 (2019).
Author information
Authors and Affiliations
Corresponding author
Additional information
The text was submitted by the author in English.
Rights and permissions
About this article
Cite this article
Volosivets, S.S. Embedding Theorems for Hölder Classes Defined on p-Adic Linear Spaces. P-Adic Num Ultrametr Anal Appl 12, 60–67 (2020). https://doi.org/10.1134/S2070046620010069
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S2070046620010069