Sobolev-Type Spaces on the Dual of the Chébli-Trimèche Hypergroup and Applications

Abstract

We define and study Sobolev-type spaces $W_A^{s,p}\left({ℝ}_{+}\right)$ associated with singular second-order differential operator on $\left(0,\infty \right)$. Some properties are given; in particular we establish a compactness-type imbedding result which allows a Reillich-type theorem. Next, we introduce a generalized Weierstrass transform and, using the theory of reproducing kernels, some applications are given.