Weighted BMO Type Spaces Associated to Admissible Functions and Applications

Abstract

Let ρ be an admissible function on an RD-space X. The main aim of this paper is to establish the theory of weighted bounded mean oscillation (BMO) type spaces $BMO^{\beta }(X,w), BMO^{\beta }_{\rho } (X,w)$ and their basic properties, including the John-Nirenberg inequality for these spaces. As applications, we also study the boundedness of some singular integrals such as the Hardy-Littlewood maximal operators, the radial maximal functions, the Poisson semigroup maximal functions, the Littlewood-Paley g-functions, and fractional integrals on these weighted BMO type spaces. Our findings extend well-known results in Bongioanni et al. (J. Math. Anal. Appl. 348, 12–27, 2008), Liu et al. (Math. Nachr. 357, 1–35, 2012), Yang and Zhou (Trans. Am. Math. Soc. 363, 1197–1239, 2011), Yang et al. (Commun. Pure Appl. Anal. 9, 779–812, 2010), and Yang et al. (Nagoya Math. J. 198, 77–119, 2010).