On Hardy and Bellman transforms of series with respect to multiplicative systems in symmetric spacesО преобразованиях Харди и Беллмана рядов по мультипликативным системам в симметричных пространствах

Hardy (C, 1) transform of Fourier-Vilenkin coefficients and its conjugate Bellman transform are studied in symmetric spaces E. Sufficient conditions are proved for the action of these transforms in spaces HEω of functions with given majorant of modulus of continuity in E. In the case of functions with generalized monotone Fourier-Vilenkin coefficients, criterions for f ∈ E are given in terms of their Fourier coefficients or in terms of their Hardy and Bellman transforms.