A very quick Sobolev embedding

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 W1,n(ℝn) into the space BMO(ℝn) of functions of bounded mean oscillation in spatial dimension n. By definition, BMO consists of those L1 functions u such that the semi-norm

\displaystyle \bigl\| u \bigr\|_{\mathrm{BMO}(\mathbb{R}^{n})} := \sup_B \frac1{|B|} \int_B \bigl| u(x) - u_B \bigr| \, \mathrm{d} x

is finite, where the supremum runs over all balls B in ℝn of finite radius, |B| denotes the volume of the ball B, and uB denotes the average of u over B:

\displaystyle u_B := \frac1{|B|} \int_B u(x) \, \mathrm{d} x .

The usual norm on BMO is the sum of the L1 norm and the BMO semi-norm. For p < n, the Sobolev space W1,p(ℝn) embeds into a suitable Lq space; when p > n, the Sobolev space W1,p(ℝn) 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 W1,n(ℝn) embeds into BMO(ℝn); that is, there is a constant c depending only on n such that

\displaystyle \bigl\| u \bigr\|_{\mathrm{BMO}(\mathbb{R}^{n})} \leq c \bigl\| u \bigr\|_{W^{1,n} (\mathbb{R}^n)}.

Proof. Suppose that u: ℝn → ℝ is a function in both L1(ℝn) and W1,n(ℝn). Let B be any ball of radius r; denoting the volume of the unit ball by ωn, B has volume ωn rn. 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

\displaystyle \bigl\| u - u_B \bigr\|_{L^p (B)} \leq C r \bigl\| \mathrm{D} u \bigr\|_{L^p (B)}.

Taking p = 1 in Poincaré’s inequality yields

\displaystyle \frac1{|B|} \int_B \bigl| u(x) - u_B \bigr| \, \mathrm{d} x
\displaystyle \leq \frac{C r}{|B|} \int_B \bigl| \mathrm{D} u(x) \bigr| \, \mathrm{d} x
\displaystyle \leq \frac{C}{\omega_n r^{n-1}} \left[ \int_B 1 \, \mathrm{d} x \right]^{\frac{n-1}{n}} \left[ \int_B \bigl| \mathrm{D} u(x) \bigr|^{n} \, \mathrm{d} x \right]^\frac{1}{n} by Hölder’s inequality
\displaystyle = \frac{C}{\omega_n r^{n-1}} \omega_n^{\frac{n-1}{n}} r^{n-1} \bigl\| \mathrm{D}u \bigr\|_{L^n (B)},

which, since the rn−1’s cancel, is

\displaystyle = C \omega_n^{1/n} \bigl\| \mathrm{D}u \bigr\|_{L^n (B)}
\displaystyle \leq C \omega_n^{1/n} \bigl\| u \bigr\|_{W^{1,n} (\mathbb{R}^n)} .

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

\displaystyle \bigl\| u \bigr\|_{\mathrm{BMO}(\mathbb{R}^{n})} \leq C \omega_n^{1/n} \bigl\| u \bigr\|_{W^{1,n} (\mathbb{R}^n)}.

An easy modification of the above argument produces a continuous embedding of W01,n(Ω) into BMO(Ω) for any Lipschitz domain Ω ⊆ ℝn.

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