#### For citation:

Simonov B. V., Simonova I. E., Ivanyuk V. A. Kolyada inequality for partial moduli of smoothness of functions with lacunary Fourier coefficients. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2022, vol. 22, iss. 4, pp. 447-457. DOI: 10.18500/1816-9791-2022-22-4-447-457, EDN: XVDVNQ

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

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.

- Ul’janov P. L. The imbedding of certain classes $H_p^\omega$.
*Izvestiya: Mathematics*, 1968, vol. 2, iss. 3, pp. 601–637. https://doi.org/10.1070/IM1968v002n03ABEH000650 - Ul’yanov P. L. Imbedding theorems and relations between best approximations (moduli of continuity) in different metrics.
*Mathematics of the USSR-Sbornik*, 1970, vol. 10, iss. 1, pp. 103–126. https://doi.org/10.1070/SM1970v010n01ABEH001589 - Kolyada V. I. On relations between the moduli of continuity in various metrics.
*Proceedings of the Steklov Institute of Mathematics*, 1989, vol. 181, pp. 127–148. - Netrusov Yu. V. Imbedding theorems for the spaces $H_p^{\omega, k}$ and $ H_p^{s,\omega, k}$.
*Zapiski Nauchnykh Seminarov LOMI*, 1987, vol. 159, pp. 83–102 (in Russian). - Gol’dman M. L. A criterion of imbedding for different metrics for isotropic Besov spaces with general moduli of continuity.
*Proceedings of the Steklov Institute of Mathematics*, 1994, vol. 201, iss. 2, pp. 155–181. - Trebels W. Inequalities for moduli of smoothness versus embeddings of function spaces.
*Archiv der Mathematik*, 2010, vol. 94, pp. 155–164. https://doi.org/10.1007/s00013-009-0078-4 - Potapov M. K., Simonov B. V., Tikhonov S. Yu.
*Drobnye moduli gladkosti*[Fractional Moduli of Smoothness]. Moscow, MAKS-Press, 2016. 337 p. (in Russian). - Besov O. V., Ilin V. P., Nikol’skii S. M.
*Integral Representations of Functions and Imbedding Theorems*. New York, Toronto, London, Halsted Press, 1978. 345 p. (Russ. ed.: Moscow, Nauka, 1975. 480 p.). - Hardy G. H., Littlewood J. E., Polya G.
*Inequalities*. London, Cambridge University Press, 1934. 336 p. (Russ. ed.: Moscow, IL, 1948. 456 p.). - Potapov M. K., Simonov B. V. Properties of mixed moduli of smoothness of functions with lacunary Fourier coefficients.
*Moscow University Mathematics Bulletin*, 2014, vol. 69, pp. 5–15. https://doi.org/https://doi.org/10.3103/S0027132214010021

- 607 reads