Bounded mean oscillation

In Harmonic analysis, a branch of mathematics, the space of functions of bounded mean oscillation (BMO) play an important role.

A function u in $L^1_{\ell oc}(R^n)$ is said to be a BMO function if

$\sup_{x,r} \inf_a \frac{1}{|B_{x,r}|}\int_{B_{x,r}}|u(y)-a|\,dy=\|u\|_{BMO}<\infty.$

The infimum over a can be replaced by the choice of

$a=(u)_{B_{x,r}}=\frac{1}{|B_{x,r}|}\int_{B_{x,r}} u(y)\,dy.$

where $B x, r$ is the ball centered at $x$ with radius $r$ .

The importance of BMO spaces lies in the fact that it is dual to $H 1$ , the Hardy space with $p = 1$ .

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