Elementary characterizations of generalized weighted Morrey-Campanato spaces

Abstract

Let α ∈ (0,∞), p, q ∈ [1,∞), s be a nonnegative integer, and ω ∈ A ₁(R ⁿ) (the class of Muckenhoupt’s weights). In this paper, we introduce the generalized weighted Morrey-Campanato space L(α, p, q, s, ω; R ⁿ) and obtain its equivalence on different p ∈ [1, β) and integers s ≥ └nα┘ (the integer part of nα), where β = ($\frac{1} {q}$ − α)⁻¹ when α < $\frac{1} {q}$ or β = ∞ when α ≥ $\frac{1} {q}$. We then introduce the generalized weighted Lipschitz space Λ(α, q, ω; R ⁿ) and prove that L(α, p, q, s, ω; R ⁿ) ⊂ Λ(α, q, ω; R ⁿ) when α ∈ (0,∞), s ≥ └nα┘, and p ∈ [1, β).