Analysis Related to All Admissible Type Parameters in the Jacobi Setting

Abstract

We derive an integral representation for the Jacobi-Poisson kernel valid for all admissible type parameters α, β in the context of Jacobi expansions. This enables us to develop a technique for proving standard estimates in the Jacobi setting that works for all possible α and β. As a consequence, we can prove that several fundamental operators in the harmonic analysis of Jacobi expansions are (vector-valued) Calderón-Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory. The new JacobiPoisson kernel representation also leads to sharp estimates of this kernel. The paper generalizes methods and results existing in the literature but valid or justified only for a restricted range of α and β.

On potential spaces related to Jacobi expansions

“…here B(θ, r) denotes the ball (interval) centered at θ and of radius r. As it was observed in [16,Section 4], even when K(θ, ϕ) is not scalar-valued, the difference conditions (14) and (15) can be replaced by the more convenient gradient condition…”

“…The following result not only implies Theorem 6.1, but certainly is also of independent interest. In particular, it enhances [14,Corollary 2.5] and [16,Corollary 5.2].…”

“…A useful integral representation of H α,β t (θ, ϕ) was established in [14, Proposition 4.1] for α, β ≥ −1/2 and in [16,Proposition 2.3] in the general case. This representation will implicitly play a crucial role in what follows, however we decided not to invoke it here due to its complexity.…”

“…We begin with a brief introduction of the Jacobi trigonometric polynomial setting. For all these and further facts we refer to [14][15][16].…”

“…Thus we are going to study weighted counterpart of Theorem 6.1 in the above-mentioned setting. Our main tool will be vector-valued Calderón-Zygmund operator theory and its implementation in the Jacobi context established in [14,16]. We begin with a brief introduction of the Jacobi trigonometric polynomial setting.…”