A little while ago, while discussing scaling results in analysis, I had a conversation with a colleague who pointed out to me an elegantly brief proof of the embedding of the **critical Sobolev space** *W*^{1,n}(ℝ^{n}) into the space BMO(ℝ^{n}) of functions of **bounded mean oscillation** in spatial dimension *n*. By definition, BMO consists of those *L*^{1} functions *u* such that the semi-norm

is finite, where the supremum runs over all balls *B* in ℝ^{n} of finite radius, |*B*| denotes the volume of the ball *B*, and *u*_{B} denotes the average of *u* over *B*:

The usual norm on BMO is the sum of the *L*^{1} norm and the BMO semi-norm. For *p* < *n*, the Sobolev space *W*^{1,p}(ℝ^{n}) embeds into a suitable *L*^{q} space; when *p* > *n*, the Sobolev space *W*^{1,p}(ℝ^{n}) embeds into a space of Hölder-continuous functions. The space BMO provides a neat (although not complete — see e.g. this paper) description of what happens in the critical case *p* = *n*, namely that *W*^{1,n}(ℝ^{n}) embeds into BMO(ℝ^{n}); that is, there is a constant *c* depending only on *n* such that

**Proof.** Suppose that *u*: ℝ^{n} → ℝ is a function in both *L*^{1}(ℝ^{n}) and *W*^{1,n}(ℝ^{n}). Let *B* be any ball of radius *r*; denoting the volume of the unit ball by *ω*_{n}, *B* has volume *ω*_{n} *r*^{n}. For each 1 ≤ *p ≤ ∞, *

**Poincaré’s inequality**asserts that there is a constant

*C*, depending on

*n*and

*p*but independent of

*u*and

*r*, such that

Taking *p* = 1 in Poincaré’s inequality yields

by Hölder’s inequality

which, since the *r*^{n−1}’s cancel, is

Taking the supremum over all balls *B* yields that *u* lies in BMO and also the claimed inequality that

An easy modification of the above argument produces a continuous embedding of *W*_{0}^{1,n}(Ω) into BMO(Ω) for any Lipschitz domain Ω ⊆ ℝ^{n}.