BMO seminorm and the quotient by constants

Complex scalars, Lebesgue measure on ${\mathbb{R}}^n$, and the complex $L^p$ and test-function conventions of Complex Lp classes and Euclidean test-function conventions. A cube is a nondegenerate axis-parallel cube (Axis-parallel rectangles in ${\mathbb{R}}^m$ and their volume). For $b\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n)$ and a cube $Q$ put $b_Q:=\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_{Q}b$, so that $b_Q$ is the mean of $b$ over $Q$, and define the $\mathrm{B}\mathrm{M}\mathrm{O}$ seminorm $\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}:={\sup }_Q\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }b-b_Q\mathrm{\mid }\in [0,\mathrm{\infty }]$, the supremum over all cubes, and $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n):=\{b\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1:\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}<\mathrm{\infty }\}$. The mean is optimal up to a factor of $2$ in the average: $\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }b-b_Q\mathrm{\mid }\le 2{\inf }_c\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }b-c\mathrm{\mid }\le 2\mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }b-b_Q\mathrm{\mid }$ where the infimum is over $c\in \mathbb{C}$. Moreover $\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{B}\mathrm{M}\mathrm{O}}=0$ exactly when $b$ is constant almost everywhere, so the page works with the quotient $\mathrm{B}\mathrm{M}\mathrm{O}({\mathbb{R}}^n)\mathrm{/}\mathbb{C}$ of equivalence classes modulo constants; a class is written $b\in \mathrm{B}\mathrm{M}\mathrm{O}\mathrm{/}\mathbb{C}$ and the zero class is the class of the constants. Replacing cubes by balls in the supremum defines an equivalent seminorm with comparison constants depending only on $n$.

These assertions follow directly from the averages. For every $c\in \mathbb{C}$, $\mathrm{\mid }{b}_Q-c\mathrm{\mid }\le \mathrm{\mid }Q{\mathrm{\mid }}^{-1}{\int }_Q\mathrm{\mid }b-c\mathrm{\mid }$, so the triangle inequality gives the factor-$2$ upper bound; taking $c=b_Q$ gives the other inequality. For locally integrable functions $b,d$, homogeneity and $(b+d{)}_Q=b_Q+d_Q$ give the seminorm laws. If the seminorm is zero, $b=b_{Q_m}$ almost everywhere on each $Q_m=[-m,m{]}^n$, $m\ge 1$. The constants agree on their positive-measure overlaps, and the exceptional set is the union of the explicit measurable null sets $Q_m\cap \{b\mathrm{\ne }{b}_{Q_1}\}$, hence is null by countable additivity of Lebesgue measure. Conversely an almost-everywhere constant has zero oscillation. Thus the quotient also identifies almost-everywhere equal representatives. Finally, if $E\subseteq F$ are measurable sets of finite positive measure and $b\in L^1(F)$, then $\mathrm{\mid }E{\mathrm{\mid }}^{-1}{\int }_E\mathrm{\mid }b-b_E\mathrm{\mid }\le 2(\mathrm{\mid }F\mathrm{\mid }\mathrm{/}\mathrm{\mid }E\mathrm{\mid })\mathrm{\mid }F{\mathrm{\mid }}^{-1}{\int }_F\mathrm{\mid }b-b_F\mathrm{\mid }$. Every ball is contained in a concentric cube of comparable volume, and every cube is contained in a concentric ball of comparable volume. Applying this inequality in both directions proves the cube/ball comparison.