Generalized Weinstein Sobolev-Gevrey spaces and pseudo-differential operators

The first subject of this paper is to analyze and introduce a class of symbols and their associated pseudo-differential operators. In this case, we consider the generalized Weinstein operator Delta(d,alpha,n)(W) (for n=0, we regain the classical Weinstein operator Delta(alpha,d)(W)). The Weinstein operator, mostly referred to as the Laplace-Bessel differential operator is now known as an important operator in analysis, because of its applications in pure and applied mathematics, especially in fluid mechanics. We introduce and study the Sobolev-Gevrey spaces associated with the generalized Weinstein operator and investigate their properties. Next, we introduce certain classes of symbols and their associated pseudo-differential operators. We show that these pseudo-differential operators naturally act on the generalized Weinstein Sobolev-Gevrey spaces.