We define and study Sobolev spaces associated with Jacobi expansions. We prove that these Sobolev spaces are isomorphic to Jacobi potential spaces. As a technical tool, we also show some approximation properties of Poisson-Jacobi integrals. n belong to all L p (0, π), 1 ≤ p ≤ ∞. However, if α < −1/2 or β < −1/2, then φ α,β n are in L p (0, π) if and only if p < p(α, β) := −1/ min(α+1/2, β+ 1/2). This leads to the so-called pencil phenomenon (cf. [18]) manifesting in the restriction Mathematics Subject Classification: primary 42C10; secondary 42C05, 42C20.

Following a symmetrization procedure proposed recently by Nowak and Stempak, we consider the setting of symmetrized Jacobi expansions. In this framework we investigate mapping properties of several fundamental harmonic analysis operators, including Riesz transforms, Poisson semigroup maximal operator, Littlewood-Paley-Stein square functions and multipliers of Laplace and Laplace-Stieltjes transform type. Our paper delivers also some new results in the original setting of classical Jacobi expansions.The central objects of our study are the following linear or sublinear operators associated with J α,β (strict definitions will be given in Section 2).(i) Symmetrized Riesz-Jacobi transforms of arbitrary order N(ii) The symmetrized Jacobi-Poisson semigroup maximal operator +2N−1 dt) , where M, N = 0, 1, 2, . . . and M + N > 0. (iv) Laplace and Laplace-Stieltjes transform type multiplierswhere m is as in Definition 1.1 below. Definition 1.1. Following E. M. Stein [10, p. 58, p. 121] we say that m is a multiplier of Laplace (transform) type associated with J α,β if m has the form m(z) = m φ (z) = ∞ 0

We investigate potential spaces associated with Jacobi expansions. We prove structural and Sobolev-type embedding theorems for these spaces. We also establish their characterizations in terms of suitably defined fractional square functions. Finally, we present sample applications of the Jacobi potential spaces connected with a PDE problem.