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 Mat. Zametki, 1978, Volume 23, Issue 3, Pages 361–372 (Mi mz8151)

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.

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

Bibliographic databases:

UDC: 517.5

Citation: M. M. Lekishvili, “Conjugate functions of several variables in the class $\operatorname{Lip}_\alpha$”, Mat. Zametki, 23:3 (1978), 361–372; Math. Notes, 23:3 (1978), 196–203

Citation in format AMSBIB
\Bibitem{Lek78} \by M.~M.~Lekishvili \paper Conjugate functions of several variables in the class $\operatorname{Lip}_\alpha$ \jour Mat. Zametki \yr 1978 \vol 23 \issue 3 \pages 361--372 \mathnet{http://mi.mathnet.ru/mz8151} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=481923} \zmath{https://zbmath.org/?q=an:0467.42013|0418.42009} \transl \jour Math. Notes \yr 1978 \vol 23 \issue 3 \pages 196--203 \crossref{https://doi.org/10.1007/BF01651431} 

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This publication is cited in the following articles:
1. M. I. Dyachenko, “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47:5 (1992), 103–171
2. V. A. Okulov, “Multidimensional analogue of a theorem of Privalov”, Sb. Math., 186:2 (1995), 257–269
3. V. A. Okulov, “A multidimensional analog of a theorem due to Zygmund”, Math. Notes, 61:5 (1997), 600–608
4. M. M. Lekishvili, A. N. Danelia, “Multidimensional conjugation operators and deformations of the classes $Z(\omega^{(2)};C(T^m))$”, Math. Notes, 63:6 (1998), 752–759
5. M. M. Lekishvili, A. N. Danelia, “On deformation of certain functional classes in the spaces $C(T^m)$ and $L(T^m)$”, Sb. Math., 192:8 (2001), 1209–1224
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