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We investigate a scale of $BM{O}_{\psi }$-spaces defined with the help of certain Lorentz norms. The results are applied to extrapolation techniques concerning operators defined on adapted sequences. Our extrapolation works simultaneously with two operators, starts with $BM{O}_{\psi }$-$L_{\infty }$-estimates, and arrives at $L_p$-$L_p$-estimates, or more generally, at estimates between K-functionals from interpolation theory.

For an injective map τ acting on the dyadic subintervals of the unit interval [0,1) we define the rearrangement operator $T_s$, 0 < s < 2, to be the linear extension of the map $\left(h_I\right)/{\left(\right|I|}^{1/s})\mapsto \left(h_{\tau \left(I\right)}\right)\left(\right|\tau \left(I\right){|}^{1/s})$, where $h_I$ denotes the $L^{\infty }$-normalized Haar function supported on the dyadic interval I. We prove the following extrapolation result: If there exists at least one 0 < s₀ < 2 such that $T_{s₀}$ is bounded on $H^{s₀}$, then for all 0 < s < 2 the operator $T_s$ is bounded on $H^s$.