Rate of strong differentiation of integrals (Q1358521)

The results of the present paper can be considered as the multiparameter analogues of those by \textit{K. I. Oskolkov} [Mat. Sb., Nov. Ser. 103(145), 563-589 (1977; Zbl 0371.41007)]. Four theorems are proved. As a specimen, we present one of them which relates to the isotropic case. Theorem 1. If \(n\geq 2\), \(\omega(\delta)\) is a modulus of continuity, and \(\psi(\delta)\) is a positive nondecreasing function such that \[ \int^1_0 {\omega(\delta)\over\delta} \Biggl(\log {1\over\delta}\Biggr)^{n- 2}d\delta< \infty,\quad \int^1_0 {\omega(\delta)\over\delta \psi(\delta)} \Biggl(\log{1\over\delta}\Biggr)^{n- 2} d\delta< \infty, \] then for every function \(f\) in \(H^\omega_1([0, 1]^n)\), we have \[ |I|^{-1} \int_{I\ni x}|f(y)- f(x)|dy= O_x\{\psi(\text{diam }I)\} \] almost everywhere, where \(I:= I_2\times\cdots\times I_n\) is any multidimensional interval of \([0,1]^n\).