Isometries of Privalov spaces of holomorphic functions of several variables

Abstract

Complete descriptions for the sets of linear isometries of the Privalov spaces of holomorphic functions on the sphere and the polydisk are obtained. These results are generalizations, to the many-dimensional case, of those previously obtained by Iida and Mochizuki. Moreover, the general form of surjective linear isometries is established for maximal analogues of the Privalov spaces in the case of positive integer exponents. It is shown that the sets of linear isometries of the Privalov spaces with the natural and the maximal metrics are different, while their groups of surjective linear isometries coincide (in the case of positive integer exponents).