Bounded Mean Oscillation Function Spaces For Nondoubling Measures

Recently, meny mathematicians study the functions spaces and singular integeal operators theory for nondoubling measures. In this paper, we introduce three types of the definetion of the Bounded Mean Oscillation functions spaces and its basic properties for the nondoubling measures, those are BMO(μ), BMOρ(μ) and RBMO(μ). We discuss the relationship and difference of these spaces. Then we give some new characterizations for BMO(μ) and RBMO(μ).Firstly, we define generalized bounded mean oscillation functions spaces BMO φ(μ) and prove its John-Nirenberg inequality: Let f BMOφ(μ), then there exist constant bf such that for any λ > 0 and any cube Q, we havewhere constant B independent of f and Q. Using this result, we obtain some new equivalent norms for BMO(μ) and weaken the definition of BMO(μ).Lastly, Using similar method, we study the RBMO(μ) and gain similar results.