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Let f : R → R be a locally integrable function of bounded lower oscillation. The paper contains the proofs of sharp strong-type, weak-type and exponential estimates for the mean oscillation of f . In particular, this yields the precise value of the norm of the embedding BLO ⊂ BMO p , 1 ≤ p < ∞. Higher-dimensional analogues for anisotropic BLO spaces are also established. 2010 Mathematics subject classification: primary 42B37; secondary 49L20.doi:10.1017/s0004972712000226 fatcat:4ogq6reo35ekxjpamlinzolseu