Аннотация
In recent years, a lot of attention has been paid to study of Morrey type spaces. Many applications in partial differential equation of Morrey spaces and Lizorkin-Triebel spaces have been given in the work of G. Di Fazioand, M. Ragua and the book of T. Mizuhara. The theory of generalized Triebel-Lizorkin-Morrey spaces is developed. Generalized Morrey spaces, proposed by Mizuhara and Nakai, are equipped with a parameter and a function. In the present paper we study vector-valued Fourier-multiplier theorem for smooth multipliers. Afterwards, we study vector valued version of the famous Riesz theorem. Here we concentrate on Lizorkin representations. Finally, we get an interesting characterization of the space εφ,p,qs(𝕋) by using differences of partial sums of the Fourier series.