Some characterizations of BLO space

Abstract

For 0<p<∞, a real‐valued function f∈L loc 1 belongs to BLOp space if ∥f∥BLOp:=trueprefixsupQ()1false|Qfalse|∫Q0trueffalse(xfalse)− ess inf y∈Qffalse(yfalse)pdx1/p<∞.In this paper, we establish a version of John–Nirenberg inequality suitable for the BLOp space with 0<p≤1. As a corollary, it is proved that BLOp spaces are independent of the scale p∈(0,∞) in sense of norm. Also, we characterize the space BLO through weighted Lebesgue spaces and variable Lebesgue spaces, respectively.