Kolyada inequality for partial moduli of smoothness of functions with lacunary Fourier coefficients

B. V. Simonov ; I. E. Simonova ; V. A. Ivanyuk

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 22 (2022) no. 4, pp. 447-457

Résumé

The problem of estimating the moduli of smoothness of functions from $L_q$ in terms of moduli of smoothness from $L_p$ is well known. The initial stage in estimating the moduli of smoothness was the study of properties of functions from Lipschitz classes and obtaining the corresponding embeddings in the works of Titchmarsh, Hardy, Littlewood, and Nikolsky. P. L. Ulyanov for the moduli of continuity of functions of one variable proved an inequality later named after him  — ‘`Ulyanov’s inequality". From this inequality, as a corollary, we obtain the classical Hardy – Littlewood embedding for Lipschitz spaces. Ulyanov's inequality is exact in the class scale $H_p^\omega$. Kolyada showed that this inequality could be strengthened. Its strengthening is Kolyada inequality. It finds application in the study of certain maximal functions which measure local smoothness. Kolyada inequality is exact in the sense that there exists a function with any given order of the modulus of continuity in $L_p$ for which this estimate cannot be improved for any value of $\delta$. Kolyada inequality was extended to the moduli of smoothness of higher orders (natural) by Yu. V. Netrusov and M. L. Goldman. W. Trebels extended Kolyada inequality to moduli of smoothness of positive order. In this article, we study the partial moduli of the smoothness of functions of two variables. Inequalities are obtained that extend Kolyada inequality to partial moduli of smoothness in the mixed norm for functions with lacunar Fourier coefficients. Functions are constructed for which Kolyada inequality for partial moduli of smoothness of functions with lacunary Fourier coefficients has different orders as functions of $\delta$. Thus, it is shown that the obtained estimates are sharp in a certain sense. Some special properties of partial moduli of smoothness of functions with lacunary Fourier series in each variable are also proved.