Strong approximation of Fourier series on generalized periodic Morrey spaces

Abstract

In recent years, a lot of attention has been paid to study of Morrey type spaces. Many applications in partial di˙erential equation of Morrey spaces and Lizorkin-Triebel spaces have been given in work G.Di Fazioand, M. Ragusa and the book of T. Mizuhara. The theory of generalized Triebel-Lizorkin-Morrey spaces is developed. Generalized Morrey spaces, with T. Mizuhara and E. Nakai proposed, are equipped with a parameter and a function. First we give definition of Morrey and generalized Morrey spaces. Then we recall the boundedness of periodic Hilbert transform. This will be our main tool for all wtht follows. In a more or less elementary way, we carry over the known boundedness assertions for the Hilbert transform on Morrey spaces defined on 􀀀 to periodic Morrey spaces. Boundedness of the Hilbert transform implies uniform estimates of the operator norms of the partial sumd of the Fourier series. Then we study vector-valued Fourier-multiplier theorem for smooth multipliers. Afterwards, we study vector valued version of famous Riesz theorem. Here we concentrate on Lizorkin representations. Finally, we get an interesting characterization of the space 􀀀􀀄􀀀􀀁􀀂􀀁􀀃 􀀁􀀀 􀀀 by using di˙erences of partial sums of the Fourier series Finally, we get an interesting characterization of the space 􀀀􀀄􀀀􀀁􀀂􀀁􀀃 􀀁􀀀 􀀀 by using di˙erences of partial sums of the Fourier series and consequence for strong approximation of Fourier series on Morrey space.