This article is cited in 5 papers

Conjugate functions of several variables in the class $\operatorname{Lip}_\alpha$

M. M. Lekishvili

Tbilisi State University

Abstract: It is known that if a function $f$ of a single variable belongs to the class $\operatorname{Lip}(\alpha,C(\mathbf T))$ $(0<\alpha<1)$, then its conjugate function also belongs to the same class; in other words, the class $\operatorname{Lip}(\alpha,C(\mathbf T))$ $(0<\alpha<1)$ is invariant with respect to the operator of conjugation acting in it. In the two-dimensional case the class $\operatorname{Lip}(\alpha,C(\mathbf T^2))$ $(0<\alpha<1)$ 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 $\operatorname{Lip}(\alpha,C(\mathbf T^N))$ $(0<\alpha<1)$ with respect to the multidimensional conjugation operator acting in it is established.

UDC: 517.5

Received: 24.06.1976

 English version: Mathematical Notes, 1978, 23:3, 196–203

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