Characterizations of Weighted BMO Space and Its Application

In this paper, we prove that the weighted BMO space $${\rm{BM}}{{\rm{O}}^p}(\omega ) = \left\{ {f \in L_{{\rm{loc}}}^1:\mathop {\sup }\limits_Q \left\| {{\chi _Q}} \right\|_{{L^p}(\omega )}^{ - 1}{{\left\| {(f {f_Q}){\omega ^{ 1}}{\chi \right\|}_{{L^p}(\omega )}} < \infty } \right\}$$ is independent of scale p ∈ (0, ∞) in sense norm when ω A1. Moreover, can replace Lp(ω) by Lp,∞(ω). As an application, characterize boundedness bilinear commutators [b, T]j(j 1, 2), generated convolution type Calderón-Zygmund operators and symbol b, from $${L^{{p_1}}}(\omega \times {L^{{p_2}}}(\omega )$$ to Lp(ω1−p) with 1 p1,p2 ∞ 1/p =1/p1 + 1/p2. Thus answer open problem proposed Chaffee affirmatively.