Construction of the extremal function for a functional on the class \(H_\omega^{(n)}\) (Q1274020)

If \(n\in\mathbb{N}\) and \(a<b\) are of \(\mathbb{R}^n\) then \([a,b]\) is an \(n\)-dimensional rectangle and \[ H_{n,\omega}[a,b]= \Biggl\{f\in C([a, b]):| f(x)- f(y)|\leq \sum^n_{i= 1}\omega_i(| x_i- y_i|)\Biggr\}, \] where \(\{x,y\}\subset [a,b]\) and \(\omega_i\), \(i\in 1,\dots, n\) are one-dimensional moduli of continuity. Also \(\psi(x)= \prod^n_{i=1} \psi(x_i)\), where \(\psi_i\), \(i\in 1,\dots, n\), are integrable functions on \([a_i,b_i]\) with the property that \(\psi_i(x_i)< 0\) almost everywhere on \([a_i, c_i]\) and \(\psi_i(x_i)> 0\) almost everywhere on \([c_i,b_i]\) with \(c= (c_1,\dots, c_n)\in [a,b]\) and \(\int^{b_i}_{a_i} \psi_i(x_i)dx_i= 0\). For \({\mathcal E}(\psi,\omega)= \sup\left\{\left| \int_{[a,b]} \psi(x)f(x)dx\right|: f\in H_{n,\omega}[a,b]\right\}\), A. I. Stepanets obtained the following upper bound \[ {\mathcal E}(\psi,\omega)\leq 2^{n-1} \int_{[a,c]} |\psi(x)| \min_i \omega_i(\rho_i(x_i)- x_i) dx,\tag{\(*\)} \] where \(\rho_i(x_i)\) are the functions defined on \([a_i,c_i]\) by the relations \[ \int^{x_i}_{a_i} \psi_i(t_i)dt_i= \int^{\rho_i(x_i)}_{a_i} \psi_i(t_i)dt_i,\;a_i\leq x_i\leq c_i\leq \rho_i(x_i)\leq b_i\text{ for every }i\in 1,\dots, n. \] In this paper, the author constructs an extremal function with estimate \((*)\) for an arbitrary kernel \(\psi\) and arbitrary convex moduli of continuity \(\omega_i\), \(i\in 1,\dots, n\).