Spaces of DLp type and a convolution product associated with the spherical mean operator

where Sn is the unit sphere {(η,ξ) ∈R×Rn : η2 +‖ξ‖2 = 1} in Rn+1 and σn is the surface measure on Sn normalized to have total measure one. This operator plays an important role and has many applications, for example, in image processing of so-called synthetic aperture radar (SAR) data (see [7, 8]), or in the linearized inverse scattering problem in acoustics [6]. In [10], the authors associate to the operator a Fourier transform and a convolution product and have established many results of harmonic analysis (inversion formula, Paley-Wiener and Plancherel theorems, etc.). In [11], the authors define and study Weyl transforms related to the mean operator and have proved that these operators are compact. The spaces DLp , 1 ≤ p ≤∞, have been studied by many authors [1, 2, 4, 5, 12, 13]. In this work, we introduce the function spaces p(R×Rn), 1 ≤ p ≤∞, similar to DLp , but replace the usual derivatives by the operator