Range truncations preserve the BMO seminorm up to a constant

For a locally integrable $g$, a cube $Q$ and a constant $d$, one has $\mathrm{\mid }{g}_Q-d\mathrm{\mid }\le \mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }g-d\mathrm{\mid }$, so $\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }g-g_Q\mathrm{\mid }\le \mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }g-d\mathrm{\mid }+\mathrm{\mid }{g}_Q-d\mathrm{\mid }\le 2\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }g-d\mathrm{\mid }$; taking the supremum over $Q$ gives $\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le 2{\sup }_Q\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }g-d_Q\mathrm{\mid }$ for any choice of constants $d_Q$.

Let $b$ be real-valued, fix $c\in \mathbb{R}$ and put $g=\mathrm{\mid }b-c\mathrm{\mid }$. For every cube $Q$, [F2] with $d=\mathrm{\mid }{b}_Q-c\mathrm{\mid }$ gives $\mid \mathrm{\mid }b-c\mathrm{\mid }-\mathrm{\mid }{b}_Q-c\mathrm{\mid }\mid \le \mathrm{\mid }b-b_Q\mathrm{\mid }$ pointwise on $Q$, so choosing $d_Q=\mathrm{\mid }{b}_Q-c\mathrm{\mid }$ in step 1.1 yields $\mathrm{\parallel }\mathrm{\mid }b-c\mathrm{\mid }{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le 2\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$.

By [F2], $\max (b,c)=c+\frac{1}{2}(b-c)+\frac{1}{2}\mathrm{\mid }b-c\mathrm{\mid }$ and $\min (b,c)=c+\frac{1}{2}(b-c)-\frac{1}{2}\mathrm{\mid }b-c\mathrm{\mid }$; translation by constants leaves the seminorm unchanged by [F1] and the triangle inequality for the supremum gives $\mathrm{\parallel }\max (b,c){\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le \frac{1}{2}\mathrm{\parallel }b-c{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}+\frac{1}{2}\mathrm{\parallel }\mathrm{\mid }b-c\mathrm{\mid }{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le \frac{1}{2}\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}+\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}=\frac{3}{2}\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$, and the same computation applies to $\min (b,c)$.

If $-\mathrm{\infty }<L\le K<\mathrm{\infty }$, then $b_{[L,K]}=\min (K,\max (L,b))$ and applying step 3.1 twice gives $\mathrm{\parallel }{b}_{[L,K]}{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le \frac{3}{2}\mathrm{\parallel }\max (L,b){\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le \frac{9}{4}\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$.

Let $b=u+iv$ be complex-valued and put $u_M=\min (M,\max (-M,u))$ and $v_M=\min (M,\max (-M,v))$, so $b_M=u_M+i{v}_M$. Then $\mathrm{\mid }{u}_M\mathrm{\mid }\le \mathrm{\mid }u\mathrm{\mid }$ and $\mathrm{\mid }{v}_M\mathrm{\mid }\le \mathrm{\mid }v\mathrm{\mid }$, hence $\mathrm{\mid }{b}_M\mathrm{\mid }=(u_M^2+v_M^2{)}^{1\mathrm{/}2}\le (u^2+v^2{)}^{1\mathrm{/}2}=\mathrm{\mid }b\mathrm{\mid }$, and $u_M\to u$, $v_M\to v$ pointwise as $M\to \mathrm{\infty }$. Moreover $\mathrm{\parallel }\mathrm{Re}b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le \mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$ and $\mathrm{\parallel }\mathrm{Im}b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le \mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$, because $\mathrm{\mid }\mathrm{Re}(b-b_Q)\mathrm{\mid }\le \mathrm{\mid }b-b_Q\mathrm{\mid }$ and similarly for the imaginary part; step 4.1 applied to $u$ and $v$ gives $\mathrm{\parallel }{b}_M{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le \mathrm{\parallel }{u}_M{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}+\mathrm{\parallel }{v}_M{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}\le \frac{9}{4}(\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}+\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}})\le \frac{9}{2}\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}$.

Steps 3.1, 4.1 and 5.1 are the three assertions of the statement. No choice principle is used: only linearity of the integral, the definition of the seminorm and elementary real and complex inequalities.