Nikol’skii-Besov and Lizorkin-Triebel spaces constructed on the base of the multidimensional Fourier-Bessel transform

Abstract

In this paper we define the Nikol’skii-Besov and Lizorkin-Triebel spaces (B-Nikol’skii-Besov and B-Lizorkin-Triebel spaces) in the context of the Fourier-Bessel harmonic analysis. We establish some basic properties of the B-Nikol’skii-Besov and B-Lizorkin-Triebel spaces such as embedding theorems, the lifting property, and characterizing of the Bessel potentials in terms of the B-Lizorkin-Triebel spaces. We prove the inclusion and the density of the Schwartz space in the B-Nikol’skii-Besov and B-Lizorkin-Triebel spaces and prove an interpolation formula for these spaces by the real method. We also prove the Young inequality for the B-convolution operators in the B-Bessel potential spaces. Finally, we give some applications involving the Laplace-Bessel differential operator.