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 Mφ 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 Mφ(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 MP, the Littlewood-Paley g-function and the Lusin area function S. They are bounded on Lp(δ) for p > 1, but not true for p = 1. Instead, MP, g and a modified Sa,γ are bounded from H1(δ) to L1(δ).
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