On Bary–Stechkin theorem

Abstract

In the beginning of the past century, N.N. Luzin proved almost everywhere convergence of an improper integral representing the function $\bar f$ conjugated to a $2\pi$-periodic summable with a square function $f(x)$. A few years later I.I. Privalov proved a similar fact for a summable function. V.I. Smirnov showed that if $\bar f$ is summable, then its Fourier series is conjugate to the Fourier series for $f(x)$. It is easy to see that if $f(x)\in\mathrm{Lip}\,\alpha$, $0<\alpha<1$, then $\bar f(x)\in\mathrm{Lip}\,\alpha$. The Hilbert transformation for $f(x)$ differs from $\bar f(x)$ by a bounded function and has a simpler kernel. It is easy to show that the Hilbert transformation of $f(x)\in\mathrm{Lip}\,\alpha$, $0<\alpha<1$, also belongs to $\mathrm{Lip}\,\alpha$. In 1956 N.K. Bari and S.B. Stechkin found the necessary and sufficient condition on the modulus of continuity $f(x)$ for the function $\bar f(x)$ to have the same modulus of continuity. In 2016, the author introduced the concept of conjugate function as Hilbert transformation for functions defined on a dyadic group. In the present paper we show an analogue of the Bari–Stechkin (and Privalov) theorem fails that for a conjugated in this sense function.