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## An introduction to BMO Space

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This note is taken in the PDE discussion group in 2012, on the topic of important spaces in fluid dynamics.

Lecture 1: An introduction to BMO Space

•  Definition (BMO =  Bounded Mean Oscillation)

$$f$$ is a BMO function if

$\|f\|_{BMO}:=\sup_B\frac{1}{|B|}\int_B|f-f_B|<+\infty,~~\text{where}~~f_B=\frac{1}{|B|}\int_B f,~~B\text{ is any ball.}$

• $$L^\infty(\mathbb{R}^n)\subsetneq BMO(\mathbb{R}^n)$$.

Clearly, $$\|f\|_{BMO(\mathbb{R}^n)}\leq 2\|f\|_{L^\infty(\mathbb{R}^n)}$$. Here, we prove $$\log|x|\in BMO(\mathbb{R})$$.

Proof: First, we observe

$\frac{1}{|B|}\int_B|f-f_B|\leq\frac{1}{|B|}\int_B|f-c|+ |c-f_B|\leq\frac{2}{|B|}\int_B|f-c|.$

For every ball $$B$$, we need to find a constant $$c_B$$ such that $$\frac{1}{|B|}\int_B|f-c_B|$$ is bounded independent of the choice of $$B$$.

For $$B=B(x_0,R)$$, do scaling $$z=x/R$$:

$\frac{1}{|B|}\int_{B(x_0,R)}(\log|x|-c)dx=\frac{1}{|B_1|}\int_{B(x_0/R,1)}\left(\log|z|-(c-\log R)\right)dz.$

Only need to consider the case $$R=1$$.

If $$|x_0|\leq 2$$, take $$c=0$$. $$\int_{|x-x_0|\leq1}\left|\log|x|\right|dx$$ is uniformly bounded. Note that no singularity at the origin.

If $$|x_0|\geq 2$$, take $$c=\log|x_0|$$.

$\int_{|x-x_0|\leq1}\left|\log|x|-\log|x_0|\right|dx=|x_0|\int_{|z-1|\leq1/|x_0|}\log|z|dz=|x_0|\cdot\frac{2}{|x_0|}\log2.$

Here, we substitute $$z=x/|x_0|$$ where $$1/2<z<3/2$$.

One remark, $$\log|x|\cdot\textbf{1}_{\{x>0\}}\not\in BMO(\mathbb{R})$$.

Proof: Consider $$B(0,\epsilon)$$. $$f_B=\frac{1}{2\epsilon}\int_0^\epsilon\log xdx=\frac{1}{2}(\log\epsilon-1)$$.

$\frac{1}{|B|}\int_B|f-f_B|=\frac{1}{2\epsilon}\left[\int_0^\epsilon\left|\log x-\frac{1}{2}(\log\epsilon-1)\right|+\int_{-\epsilon}^0\left|\frac{1}{2}(\log\epsilon-1)\right|\right],$

where the second part blows up as $$\epsilon$$ goes to 0.

• Same space if take supreme over all Euclidean cubes $$Q$$, with a different (but equivalent) norm

$\|f\|_{BMO_\square}:=\sup_Q\frac{1}{|Q|}\int_Q|f-f_Q|.$

• Scaling property on the means for BMO function

Proposition: Let $$f$$ be a BMO function. $$Q$$ is a cube. Then $$|f_Q-f_{2^mQ}|\leq 2^nm\|f\|_{BMO}$$.

Proof: It is sufficient to prove $$|f_Q-f_{2Q}|\leq 2^n\|b\|_{BMO}$$.

$|f_Q-f_{2Q}|\leq\frac{1}{|Q|}\int_Q|f-f_{2Q}|\leq\frac{|2Q|}{|Q|}\frac{1}{|2Q|}\int_{2Q}|f-f_{2Q}|\leq 2^n\|f\|_{BMO}.$

• John-Nirenberg inequality

Theorem (John-Nirenberg): Let $$f\in BMO(\mathbb{R}^n)$$. For any cube $$Q$$, any level $$\lambda$$,

$|\{x\in Q~:~|f(x)-f_Q|>\lambda\}|\lesssim_n e^{-C_n\lambda/\|f\|_{BMO_\square}}|Q|.$

• The size of the set with large oscillation decays exponentially. It indicates that locally the function looks like a $$\log$$ at most.

Proof (part 1): Without loss of generality, assume $$\|f\|_{BMO_\square}=1$$. Use Chebyshev inequality to get a rough estimate:

$|\{x\in Q~:~|f(x)-f_Q|>\lambda\}|\leq\frac{1}{\lambda}\int_Q|f-f_Q|\leq\frac{1}{\lambda}|Q|.$

Suppose the best constant is $$C(\lambda)$$. Clearly $$C(\lambda)\leq\min\{1,1/\lambda\}$$.

We prove (in part 2) the following recursive inequality:

$C(\lambda)\leq\frac{C(\lambda-r_n\Lambda)}{\Lambda},$

where $$\Lambda>1$$ is a constant level, $$r_n$$ is a constant which is $$2^n$$ in our case.

Then, for $$\lambda\in(Nr_n\Lambda, (N+1)r_n\Lambda]$$, use the inequality $$N$$ times,

$C(\lambda)\leq\frac{C(\lambda-Nr_n\Lambda)}{\Lambda^N}\leq\frac{1}{r_n\Lambda^{N+1}}\leq r_n^{-1}\Lambda^{-\frac{\lambda}{r_n\Lambda}}=r_n^{-1}e^{-\frac{\log\Lambda}{r_n\Lambda}\lambda}.$

This ends the proof with $$C_n=\log\Lambda/(r_n\Lambda)$$.

• Calderon-Zygmund decomposition.

For any $$f\in L^1(\mathbb{R}^n)$$ and $$\alpha>0$$, there exists a (good) function $$g$$ and a (bad) function $$b$$ on $$\mathbb{R}^n$$ such that

1. $$f=g+b$$.
2. $$\|g\|_{L^1}\leq\|f\|_{L^1}$$ and $$\|g\|_{L^\infty}\leq2^n\alpha$$.
3. $$b=\sum_j b_j$$, where $$supp(b_j)\subset Q_j$$.
4. $$\int_{Q_j}b_j(x)dx=0$$.
5. $$\|b_j\|_{L^1}\leq 2^{n+1}\alpha |Q_j|$$.
6. $$\sum_j|Q_j|\leq \alpha^{-1}\|f\|_{L^1}$$.

It tells us for $$L^1$$ function, high oscillations happens in some cube area whose total volume is bounded. $$\alpha$$ is a control parameter. The higher oscillation allowed for good function, the smaller the support of bad function would have.

Proof (part 2): We use C-Z decomposition to prove the recursive inequality. Fix a cube $$Q_0$$. Set $$\alpha=\Lambda$$, $$r_n=2^{n+1}$$.  Apply C-Z decomposition on $$F(x):=(f(x)-f_{Q_0})\cdot\textbf{1}_{Q_0}$$. Let $$\mathcal{B}$$ denote the collection of bad cubes. Using properties of C-Z decomposition, we have the following.

•  If $$x\not\in \cup_{Q\in\mathcal{B}}Q$$, $$|F(x)|\leq (r_n/2)\Lambda$$. (By C-Z 2).
• For $$Q\in\mathcal{B}$$, $$\frac{1}{|Q|}\int_Q|F(x)|dx\leq r_n\Lambda$$. (By C-Z 5). Then
$|f_Q-f_{Q_0}|=\left|\frac{1}{|Q|}\int_Q(f-f_{Q_0})\right|\leq\frac{1}{Q}\int_Q|F|\leq r_n\Lambda.$
• $$\sum_{Q\in\mathcal{B}}|Q|\leq\Lambda^{-1}\|F\|_{L^1}\leq\Lambda^{-1}|Q_0|$$. (By C-Z 6 and definition of BMO norm.)

Consider $$\lambda>r_n\Lambda$$.

\begin{align*}\left|\{x\in Q_0~:~|f-f_{Q_0}|>\lambda\}\right|\stackrel{1.}{=}&\left|\{x\in \bigcup_{Q\in\mathcal{B}}Q~:~|f-f_{Q_0}|>\lambda\}\right|\\ \leq&\sum_{Q\in\mathcal{B}}|\left\{x\in Q~:~|f-f_Q|+|f_Q-f_{Q_0}|>\lambda\}\right|\\ \stackrel{2.}{\leq}&\sum_{Q\in\mathcal{B}}|\left\{x\in Q~:~|f-f_Q|>\lambda-r_n\Lambda\}\right|\\ \stackrel{def}{\leq}&\sum_{Q\in\mathcal{B}}C(\lambda-r_n\Lambda)|Q|\\ \stackrel{3.}{\leq}&\frac{C(\lambda-r_n\Lambda)}{\Lambda}|Q_0|. \end{align*}

As $$C(\lambda)$$ is the best constant, the recursive inequality is proved.

• Corollaries of John-Nirenberg inequality

Corollary 1: Every BMO function is exponentially integrable over any cube. Namely, there exists a $$\gamma>0$$ such that

$\frac{1}{|Q|}\int_Q e^{\gamma|f(x)-f_Q|/\|f\|_{BMO}}dx\leq C_{n,\gamma}.$

Proof:

\begin{align*}\text{LHS}~=~&\frac{1}{|Q|}\int_0^\infty e^\alpha \left|\left\{x\in Q~:~\gamma|f(x)-f_Q|/\|f\|_{BMO}>\alpha\right\}\right|d\alpha\\ \stackrel{J-N}{\lesssim_n}&\int_0^\infty e^{\alpha-C\alpha/\gamma}d\alpha<+\infty\quad(\text{Pick}~ \gamma<C.)\end{align*}

Corollary 2: For all $$0<p<\infty$$, there exists a constant $$B_p$$ such that

$\sup_Q\left(\frac{1}{|Q|}\int_Q|f-f_Q|^pdx\right)^{1/p}\leq B_p\|f\|_{BMO_\square}.$

Proof: Without loss of generality, assume $$\|f\|_{BMO_\square}=1$$.

\begin{align*}\text{LHS}~=~&\sup_Q\left(\frac{1}{|Q|}\int_0^\infty p\alpha^{p-1} \left|\left\{x\in Q~:~|f(x)-f_Q|>\alpha\right\}\right|d\alpha\right)^{1/p}\\ \stackrel{J-N}{\lesssim_n}&\left(\int_0^\infty p\alpha^{p-1} e^{-C_n\alpha}d\alpha\right)^{1/p}=C_n^{-1}\Gamma(p+1)^{1/p}.\end{align*}

Corollary 3(Interpolation): Let $$1\leq p<q<\infty$$ and $$f\in L^p(\mathbb{R}^n)\cap BMO(\mathbb{R}^n)$$. Then, $$f\in L^q(\mathbb{R}^n)$$ with

$\|f\|_{L^q(\mathbb{R}^n)}\lesssim\|f\|_{L^p(\mathbb{R}^n)}^{p/q}\|f\|_{BMO(\mathbb{R}^n)}^{1-p/q}.$

Proof: Without loss of generality, take $$\|f\|_{BMO(\mathbb{R}^n)}=1$$. The correct scaling determine the power $$1-p/q$$. We are left to prove that $$\int_{\mathbb{R}^n}|f|^q\lesssim\int_{\mathbb{R}^n}|f|^p$$. Note that if $$|f|\leq\lambda$$, then $$|f|^q\leq\max(1,\lambda^{q-p})|f|^p$$. We only need to concern the part where $$|f|$$ is big.

Form a C-Z covering lemma for $$|f|^p$$ with $$\alpha=1$$. There exists a collection of disjoint cubes $$\mathcal{B}$$ such that

• $$|f(x)|^p\leq 1$$ for all $$x\not\in \cup_{Q\in\mathcal{B}}Q$$.
• $$1<\frac{1}{|Q|}\int_Q|f(x)|^pdx\leq 2^n$$, for all $$Q\in\mathcal{B}$$.

A direct consequence is that $$\sum_{Q\in\mathcal{B}}|Q|\leq\|f\|_{L^p}^p$$.

Also, we have the estimate (second inequality is Holder)

$|f_Q|\leq\frac{1}{|Q|}\int_Q|f(x)|dx\leq\left(\frac{1}{|Q|}\int_Q|f(x)|^pdx\right)^{1/p}\leq2^{n/p}.$

Now, consider $$\lambda> 2^{n/p}$$.

\begin{align*}|\{|f|>\lambda\}|~\stackrel{1.}{=}~&\left|\bigcup_{Q\in\mathcal{B}}\{x\in Q:|f|>\lambda\}\right|\leq\sum_{Q\in\mathcal{B}}\big|x\in Q:|f-f_Q|>\lambda-|f_Q|\big|\\ \stackrel{J-N}{\lesssim_n}&\sum_{Q\in\mathcal{B}}e^{-C_n(\lambda-2^{n/p})}|Q|\leq e^{-C_n(\lambda-2^{n/p})}\|f\|_{L^p}^p.\end{align*}

Finally, compute

$\int_{\mathbb{R}^n}|f|^q=\int_0^\infty q\lambda^{q-1}|\{|f|>\lambda\}|d\lambda=\int_0^{2^{n/p}}q\lambda^{q-1}|\{|f|>\lambda\}|d\lambda+\int_{2^{n/p}}^\infty q\lambda^{q-1}|\{|f|>\lambda\}|d\lambda.$

and estimate the two terms seperately

\begin{align*}\int_0^{2^{n/p}}q\lambda^{q-1}|\{|f|>\lambda\}|d\lambda=&\frac{q}{p}(2^{n/p})^{q-p}\int_0^{2^{n/p}}p\lambda^{p-1}|\{|f|>\lambda\}|d\lambda\leq2^{\frac{n(q-p)}{p}}\frac{q}{p}\int_{\mathbb{R}yy^n}|f|^p.\\ \int_{2^{n/p}}^\infty q\lambda^{q-1}|\{|f|>\lambda\}|d\lambda=&\int_{2^{n/p}}^\infty q\lambda^{q-1}|\{|f|>\lambda\}|d\lambda\leq\|f\|_{L^p}^p\int_{2^{n/p}}^\infty q\lambda^{q-1}e^{-C_n(\lambda-2^{n/p})}d\lambda,\end{align*}

where the last integral is bounded by $$C_n^{-q}\Gamma(q+1)e^{2^{n/p}C_n}$$.