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We define and study Sobolev-type spaces W A s , p ℝ + associated with singular second-order differential operator on 0 , ∞ . 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.

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