Characterizations of Weighted BMO Space and Its Application

Abstract

In this paper, we prove that the weighted BMO space

$${\rm{BM}}{{\rm{O}}^p}(\omega ) = \left\{ {f \in L_{{\rm{loc}}}^1:\mathop {\sup }\limits_Q \left\| {{\chi _Q}} \right\|_{{L^p}(\omega )}^{ - 1}{{\left\| {(f - {f_Q}){\omega ^{ - 1}}{\chi _Q}} \right\|}_{{L^p}(\omega )}} < \infty } \right\}$$

is independent of the scale p ∈ (0, ∞) in sense of norm when ω ∈ A₁. Moreover, we can replace L^p(ω) by L^p,∞(ω). As an application, we characterize this space by the boundedness of the bilinear commutators [b, T]_j(j = 1, 2), generated by the bilinear convolution type Calderón-Zygmund operators and the symbol b, from ${L^{{p_1}}}(\omega ) \times {L^{{p_2}}}(\omega )$ to L^p(ω^1−p) with 1 < p₁,p₂ < ∞ and 1/p =1/p₁ + 1/p₂. Thus we answer the open problem proposed by Chaffee affirmatively.