Boundedness of fractional integrals on weighted Orlicz–Hardy spaces

Abstract

Let Iα be the fractional integral on the Euclidean space , where α ∈ (0, n). Assume that Φ and Ψ are two Orlicz functions satisfying i(Φ), i(Ψ) ∈ (0, 1], and for all t ∈ (0, ∞ ), where i(Φ) and i(Ψ) are, respectively, the critical lower type indices of Φ and Ψ, and Φ − 1 and Ψ − 1 their inverses. In this paper, the authors prove that Iα is bounded from the weighted Orlicz–Hardy space to the weighted Orlicz–Hardy space , whenever there exist 1 < p1 < q1 < ∞ , satisfying , such that belongs to the weight. To this end, the authors first establish the atomic and the molecular characterizations of these weighted Orlicz–Hardy spaces, which are defined via the nontangential grand maximal function. Thus, as a byproduct, the authors establish the nontangential grand maximal function and molecular characterizations of the weighted atomic Orlicz–Hardy space introduced by Harboure, Salinas, and Viviani. Copyright © 2013 John Wiley & Sons, Ltd.