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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 \(C_n^{-q}\Gamma(q+1)e^{2^{n/p}C_n}\).

Read 1780 times Last modified on Saturday, 28 March 2015 16:47

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