Abstract. Analogous of Bessel and Flett potentials are defined and studied for the Dunkl transform associated with a family of weighted functions that are invariant under a reflection group. We show that the Dunkl-Bessel potentials, of positive order, can be represented by an integral involving the k-heat transform and we give some applications of this result.Also, we obtain an explicit inversion formula for the Dunkl-Flett potentials, which are interpreted as negative fractional powers of a certain operator expressed in terms of the Dunkl-Laplacian.

In this paper, we develop the left-definite spectral theory associated with the self-adjoint operator A µ in L 2 (R, |t| 2µ exp(−t 2 )), generated from the Dunkl second-order Hermite differential equationthat has the generalized Hermite polynomials {H µ m } ∞ m=0 as eigenfunctions and where T µ is a differential-difference operator called the Dunkl operator on R of index µ. More specifically, for each n ∈ N, we explicitly determine the unique left-definite Hilbert space W µ n and associated inner product (. , .) µ, n , which is generated from the nth integral power n µ [.] of µ [.]. Moreover, for each n ∈ N, we determine the corresponding unique left-definite self-adjoint operator A µ,n in W µ n and characterize its domain in terms of another left-definite space. As a consequence, we explicitly determine the domain of each integral power of A µ and in particular, we obtain a new characterization of the domain of the Dunkl right-definite operator A µ .

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.

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