IJPAM: Volume 112, No. 3 (2017)

Title

NOTES ON ZYGMUND FUNCTIONS

Authors

H. Akhadkulov$^1$, M.S. Noorani$^2$, A.B. Saaban$^3$, S. Akhatkulov$^4$
$^{1,3}$School of Quantitative Sciences
University Utara Malaysia
CAS 06010, UUM Sintok, Kedah Darul Aman, MALAYSIA
$^{2,4}$School of Mathematical Sciences
University Kebangsaan
43600 UKM Bangi, Selangor DE, MALAYSIA

Abstract

In this paper we study a class of continuous functions satisfying a certain Zygmund condition dependent on a parameter $\gamma>0.$ It shown that the modulus of continuity of such functions is $\mathcal{O}(\delta(\log\frac{1}{\delta})^{1-\gamma})$ if $\gamma \in (0,1)$ and $\mathcal{O}(\delta(\log\log\frac{1}{\delta}))$ if $\gamma=1.$ Moreover, these functions are differentiable if $\gamma>1.$ These results extend the results in literatures [4], [5].

History

Received: August 30, 2016
Revised: November 21, 2016
Published: February 9, 2017

AMS Classification, Key Words

AMS Subject Classification: 26A15, 42A55, 46E35
Key Words and Phrases: Zygmund functions, modulus of continuity, differentiability

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Bibliography

1
H. Akhadkulov, M.S. Noorani, S. Akhatkulov, Renormalization of circle diffeomorphisms with a break-type singularity. Preprint, (2016), 26-pages, Avaliable on: https://arxiv.org/pdf/1510.03202v2.pdf

2
F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14, (1961), 415-426.

3
J. Hu, D. Sullivan, Topological conjugacy of circle diffeomorphisms, Ergodic Theory Dynam. Systems, 17, No. 1 (1997), 173-186.

4
M. Weiss, A. Zygmund, A note on smooth functions, Indag. Math., 62 (1959), 52-58.

5
A. Zygmund, Trigonometric Series, Third Edition, Volumes I and II, Cambridge University Press, London, 2002.

How to Cite?

DOI: 10.12732/ijpam.v112i3.3 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 112
Issue: 3
Pages: 481 - 488


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