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 differential 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 R 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 vectorvalued 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 differences of partial sums of the Fourier series Finally, we get an interesting characterization of the space E^s_ϕ,p,q(T) by using differences of partial sums of the Fourier series and consequence for strong approximation of Fourier series on Morrey space.