On conjugate operator in multiple Hölder classes on torus (Q1922031)

Let \(T^n= \{z\in \mathbb{C}^n: |z_j|=1\), \(j=1,\dots,n\}\) be the \(n\)-dimensional torus, and let \(\alpha= (\alpha_1,\dots, \alpha_n)\), \(0<\alpha_j\leq 1\), \(j=1,\dots, n\), be a multi-index. The multiple Hölder class \(\Lambda(\alpha)\) consists of all continuous functions \(f\) on \(T^n\) satisfying the following inequality for arbitrary \(k\), \(1\leq k\leq n\), \((i_1,\dots, i_k)\in \mathbb{Z}^k_+\), \((h_{i_1},\dots, h_{i_k})\in \mathbb{R}^k\): \[ |\Delta [h_{i_k},\dots, h_{i_1}] (f)|\leq C_f|h_{i_1} |^{\alpha_{i_1}} \dots|h_{i_k}|^{\alpha_{i_k}}, \] where \[ \Delta [h_{i_k},\dots, h_{i_1}](f)= \Delta[h_{i_k}] (\dots(\Delta [h_{i_1}](f) \dots), \] \[ \Delta[h_j](f) (e^{i\theta_1},\dots, e^{i\theta_n})= f(e^{i\theta_1}, \dots, e^{(\theta_j+h_j)}, \dots,e^{i\theta_n})- f(e^{i\theta_1}, \dots, e^{i\theta_n}). \] The conjugate operator on \(\Lambda(\alpha)\) is defined by \[ \begin{multlined} S[i_1,\dots, i_k](f) (e^{i\theta_1}, \dots,e^{i\theta_n})=\\ (-2\pi)^{-k} \int_{Q^k} f(e^{i(\theta_1+u_1)}, \dots,e^{i(\theta_n+ u_n)}) \prod^k_{m=1} \text{cot} (t_{i_m}/2) dt_{i_1}\dots dt_{i_k}, \end{multlined} \] where \(Q^k= [-\pi,\pi]^k\), \((\theta_1,\dots, \theta_n)\in \mathbb{R}^n\), \(u_m=t_m\) if \(m=i_m\), and \(u_m=0\) otherwise \(m\neq i_m\), \(1\leq m\leq k\). In the case \(n=1\) the class \(\Lambda(\alpha)\) and operator \(S[1]\) coincide with the usual Hölder class \(H^\alpha(T)\) and the ordinary conjugate operator, respectively. It is known that the conjugate operator operates on \(H^\alpha(T)\), and that this assertion breaks down in \(H^\alpha(T^n)\), \(n>1\). The author shows that the multiple Hölder class \(\Lambda(\alpha)\) is invariant with respect to operators \(S[i_1,\dots, i_k]\) for arbitrary \(k\), \(1\leq k\leq n\), and \((i_1,\dots, i_k)\in \mathbb{Z}^k_+\).