Geodesic
Conjugate functions of several variables in the class $\operatorname{Lip}_\alpha$
Matematičeskie zametki, Tome 23 (1978) no. 3, pp. 361-372.

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It is known that if a function $f$ of a single variable belongs to the class $\operatorname{Lip}(\alpha,C(\mathbf T))$ $(0\alpha1)$, then its conjugate function also belongs to the same class; in other words, the class $\operatorname{Lip}(\alpha,C(\mathbf T))$ $(0\alpha1)$ is invariant with respect to the operator of conjugation acting in it. In the two-dimensional case the class $\operatorname{Lip}(\alpha,C(\mathbf T^2))$ $(0\alpha1)$ is no longer invariant with respect to conjugate functions of two variables. Here a final result elucidating the full character of violation of invariance of the class $\operatorname{Lip}(\alpha,C(\mathbf T^N))$ $(0\alpha1)$ with respect to the multidimensional conjugation operator acting in it is established. 
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     author = {M. M. Lekishvili},
     title = {Conjugate functions of several variables in the class $\operatorname{Lip}_\alpha$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {361--372},
     publisher = {mathdoc},
     volume = {23},
     number = {3},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_3_a3/}
}
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M. M. Lekishvili. Conjugate functions of several variables in the class $\operatorname{Lip}_\alpha$. Matematičeskie zametki, Tome 23 (1978) no. 3, pp. 361-372. http://geodesic.mathdoc.fr/item/MZM_1978_23_3_a3/