Let 1 ⩽ p < ∞ and −n/p < α ⩽ 1. A distinguished subset \({\cal{C}}_{*}^{\alpha,p}(\omega)\) of the weighted Morrey-Campanato space \({\cal{C}}^{\alpha,p}(\omega)\) on ℝn is introduced and studied. This new class is a proper subset of \({\cal{C}}^{\alpha,p}(\omega)\) . We establish John-Nirenberg-type inequalities suitable for the Morrey-Campanato spaces \({\cal{C}}^{\alpha,p}(\omega)\) and \({\cal{C}}_{*}^{\alpha,p}(\omega)\) with ω ∈ A1. Based on this result, some new equivalent characterizations of the Morrey-Campanato spaces \({\cal{C}}^{\alpha,p}(\omega)\) and \({\cal{C}}_{*}^{\alpha,p}(\omega)\) are also given. Let T(f) denote the Littlewood-Paley square operators, including the Littlewood-Paley g-function \({\cal{G}}_{\psi}(f)\) , Lusin’s area integral \({\cal{S}}_{\psi}(f)\) and Stein’s function \({\cal{G}}_{\lambda,\psi}^{*}(f)\) with λ > 2. Here ψ is a Littlewood-Paley function on ℝn. We establish the boundedness of Littlewood-Paley square operators on weighted Morrey-Campanato spaces. It is proved that if T(f)(x0) is finite for a single point x0 ∈ ℝn, then T(f)(x) is finite almost everywhere in ℝn. Moreover, it is shown that T(f) is bounded from \({\cal{C}}^{\alpha,p}(\omega)\) into \({\cal{C}}_{*}^{\alpha,p}(\omega)\) for 1 ⩽ p < ∞ and 0 < α ⩽ 1, provided that ω ∈ A1.
