From the introduction: Let \(U^n=\{z=(z_1, \dots, z_n):|z_j |<1\), \(j=1,\dots, n\}\) be the unit polydisk in the complex \(n\)-dimensional space \(\mathbb{C}^n\), and let \(\mathbb{T}^n\) be the frame of \(U^n\). Let \(\vec p=(p_1, \dots,p_n)\), \(\vec\omega(t) =(\omega_1(t), \dots,\omega_n (t))\), where \(0<p_j <+\infty\), and \(\omega_j(t)\) are positive slowly varying functions on \((0,1]\). We denote by \(L^{\vec p}(\vec\omega)\) the space of functions \(f\) measurable in \(U^n\) and satisfying \[ \begin{split} \|f\|_{L^{\vec p}(\vec \omega)}= \left(\int_U \omega_n\bigl(1- |\zeta_n |\bigr) \left(\int_U \omega_{n-1} \bigl(1-|\zeta_{n-1} |\bigr) \dots\left(\int_U \bigl|f(\zeta_1, \dots, \zeta_n)\bigr |^{p_1}\times \right.\right.\right.\\ \times \left. \left. \left.\omega_1\bigl(1-|\zeta_1 |\bigr) dm_2(\zeta_1) \right)^{p_2 \over p_1}\dots dm_2(\zeta_{n-1}) \right)^{p_n \over p_{n-1}}dm_2 (\zeta_n) \right)^{1\over p_n}< +\infty, \end{split} \] where \(dm_2\) is the planar Lebesgue measure on \(U\). The subspace of \(L^{\vec p}(\vec\omega)\) consisting of functions holomorphic in \(U^n\) is denoted by \(H^{\vec p}(\vec \omega)\), and the subspace consisting of \(n\)-harmonic functions is denoted by \(h^{\vec p}(\vec \omega)\).
We construct a bounded linear operator mapping \(L^{\vec p}(\vec\omega)\) to \(H^{\vec p}(\vec\omega)\) for all \(\vec p=(p_1, \dots, p_n)\), \(1\leq p_j< +\infty\), and also a mapping \(h^{\vec p}(\vec\omega)\) of \(L\vec p(\vec\omega)\) onto \(H^{\vec p}(\vec\omega)\) for \(0<p_j<1\), \(j=1, \dots,n\). Also, we obtain a complete description of continuous linear functionals on \(H^{\vec p}(\vec\omega)\) and characterize the traces of functions belonging to \(H^{\vec p}(\vec \omega)\) on the diagonal of \(U^n\) for all \(\vec p\) and \(\vec\omega\). Moreover, we construct a linear extension from the diagonal of the polydisk to the entire polydisk.