Reletions between partial moduli of smoothness of functions with monotone Fourier coefficients

Abstract

The problem of estimating the moduli of smoothness of functions from Lq in terms
of moduli of smoothness from the broader Lebesgue class Lp has been known for a long time. At the
initial stage, in the works of Titchmarsh, Hardy, Littlewood, Nikolsky, the properties of functions from
Lipschitz classes were studied and the corresponding embeddings were obtained. For moduli of continuity
of functions of one variable P.L. Ulyanov proved an inequality later named after him - "Ulyanov’s
inequality". The classical Hardy-Littlewood embedding for Lipschitz spaces is a consequence of Ulyanov’s
inequality. As V.A. Andrienko showed, Ulyanov’s inequality is exact in the scale of classes H!
p : Further
development of this direction is connected with the works of V.A. Andrienko, E.A. Storozhenko, M.K.
Potapov, L. Leindler, V.I. Kolyada, P. Oswald, N. Temirgaliev, S.V. Lapin and other mathematicians.

Kolyada proved that Ulyanov’s inequality can be strengthened and proved the corresponding
"Kolyada’s inequality". Kolyada’s inequality is exact in the sense that there exists a function in Lp
with any given order of the modulus of continuity for which this estimate cannot be improved for any
value of : Yu.V. Netrusov, M.L. Goldman and W. Trebelz extended Kolyada’s inequality to the moduli
of smoothness of higher orders. Another direction of research was the study of fractional moduli of
smoothness in the works of M.K. Potapov, B.V. Simonov, S.Yu. Tikhonov. This made it possible to
strengthen the Ulyanov inequality and showed the specificity and special significance of using fractional
moduli of smoothness, without which, as it turned out, it was impossible to obtain final results.
In this article, we study partial moduli of smoothness of functions of two variables. Inequalities are
obtained that extend Kolyada’s inequality to partial moduli of smoothness for functions with monotone
Fourier coefficients. Estimates are also obtained for the partial moduli of smoothness of the derivative of
a function with monotone Fourier coefficients in terms of the partial moduli of smoothness of the original
function.