Abstract
The modulus of continuity of a function (, ) 1-periodic in each variable is defined by The following estimate is established for the nonincreasing rearrangement of a function (, ; ): (1)Also, analytic functions of Hardy class in the unit disk are considered. It is proved that the inequality (1) () holds for the rearrangements of their boundary values also when (this is false for real functions of class ).Inequality (1) is used to find necessary and sufficient conditions for the space () of functions with a given majorant of the -modulus of continuity to be imbedded in the Orlicz classes , where satisfies the -condition and on . For the solution of this problem follows from estimates obtained earlier by the author (Zbl. 314#46031). An analogous result is established for classes of functions in the Hardy space ().The imbeddings with limiting exponent (Sobolev and Hardy-Littlewood theorems) are limiting cases of the results in this article.Bibliography: 27 titles.
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