A multidimensional analog of a theorem due to Zygmund
In [Prace Mat.-Fiz. 33, 125-132 (1924; JFM 52.0274.03)] \textit{A. Zygmund} proved that if \[ \int^\pi_0 {\omega(t)\over t} dt= \infty, \] then there exists a continuous function \(f\) such that \(\omega(f, \delta)= O(\omega(\delta))\) and the adjoint function \(\overline f\) is even unbounded. He also proved that if \(f\in C(T)\), then \[ \omega(\overline f, \delta)\leq A\Biggl\{ \int^\delta_0 {\omega(f,t)\over t} dt+ \delta\int^\pi_\delta {\omega(f,t)\over t^2} dt\Biggr\}.\tag{1} \] \textit{N. K. Bari} and \textit{S. B. Stechkin} [Trudy Moskovsk. Mat. Obsc. 5, 483-522 (1956; Zbl 0072.05702)] verified that the estimate given in (1) is asymptotically sharp. The author proves the multidimensional analogs of the above-stated results.
scientific article; zbMATH DE number 1238048