H¹-estimates of Littlewood-Paley and Lusin functions for Jacobi analysis

For α ≥ β ≥ -1/2 let denote the weight function on R₊ and L¹(δ) the space of integrable functions on R₊ with respect to δ(x)dx, equipped with a convolution structure. For a suitable φ ∈ L¹(δ), we put for t > 0 and define the radial maximal operator M_φ as usual manner. We introduce a real Hardy space H¹(δ) as the set of all locally integrable functions f on R₊ whose radial maximal function M_φ(f) belongs to L¹(δ). In this paper we obtain a relation between H¹(δ) and H¹(R). Indeed, we characterize H¹(δ) in terms of weighted H¹ Hardy spaces on R via the Abel transform of f. As applications of H¹(δ) and its characterization, we shall consider (H¹(δ),L¹(δ))-boundedness of some operators associated to the Poisson kernel for Jacobi analysis: the Poisson maximal operator M_P, the Littlewood-Paley g-function and the Lusin area function S. They are bounded on L^p(δ) for p > 1, but not true for p = 1. Instead, M_{P, g} and a modified S_a,γ are bounded from H¹(δ) to L¹(δ).