In this paper q-Sobolev type spaces are defined on R q by using the q-cosine Fourier transform and its inverse. In particular, embedding results for these spaces are established. Next we define the q-cosine potential and study some of its properties.

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A Wiener-Tauberian theorem is proven on the Laguerre hypergroup [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. As consequence of this theorem we establish a Pompeiu type-theorem and we study some of its applications.

We prove the analogue of Hörmander-Mikhlin multiplier theorem for the multidimensional Fourier-Bessel transform associated with the Poly-axially operator.

In this work, by using the Wiener-Tauberian Theorem, we establish a Pompeiu-type theorem (which consists to study the uniqueness of some convolution equations solutions) associated with a differentialdifference operator D α , called the Dunkl operator on the real line. Next, we give some applications.