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

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.