Haar systems on real lines and adapted to an accretive function

By | October 17, 2018

This article is based on the paper of Coifman-Jones-Semmes (JAMS, 1989).

One of the fundamental problem is the solvability of the Dirichlet problem for the Poisson equation in a bounded domain(open and connected)
\Omega in \mathbb{R}^{n}:

    \[ \left(D\right)\,\begin{cases} \triangle u=0 & \text{in }\Omega,\\ u=g & \text{on }\partial\Omega. \end{cases} \]

Many authors have been studied the solvability of the problem \left(D\right) in a various settings. One possible way to solve the problem \left(D\right) is to use the method of layer potential. In this talk, we assume n\ge3 for simplicity. The fundamental solution for the Laplacian is defined by

    \[ \Gamma\left(x\right)=\frac{1}{\left(n-2\right)\omega_{n}\left|x\right|^{n-2}}, \]

where \omega_{n} is the volume of the unit ball in \mathbb{R}^{n}. Suppose first that \Omega has C^{2}-boundary. For f\in C\left(\partial\Omega\right), we define the double layer potential of f by

    \[ \mathscr{D}f\left(x\right)=\int_{\partial\Omega}\frac{\partial}{\partial\nu_{y}}\Gamma\left(x-y\right)f\left(y\right)d\sigma\left(y\right)\quad\text{for }x\in\Omega, \]

where \nu denotes the outward unit normal to the boundary \partial\Omega. Then it is easy to check that \mathscr{D}f is well-defined on \mathbb{R}^{n} and is harmonic in \mathbb{R}^{n}\setminus\partial\Omega. Also, \mathscr{D}f can be extended as a continuous function from \Omega to up to the boundary. Set G\left(x,y\right)=\frac{\partial}{\partial\nu_{y}}\Gamma\left(x-y\right) for x\neq y, x,y\in\partial\Omega. Since \partial\Omega is of C^{2}, one can easy to show that

(1)   \begin{equation*} \left|G\left(x,y\right)\right|\le\frac{C}{\left|x-y\right|^{n-2}}\quad\text{for }x,y\in\partial\Omega \end{equation*}

for some constant C>0.

For f\in C\left(\partial\Omega\right), we define

    \[ Kf\left(x\right)=\lim_{\varepsilon\rightarrow0}\int_{\left|x-y\right|>\varepsilon,y\in\partial\Omega}G\left(x,y\right)f\left(y\right)d\sigma\left(y\right)\quad x\in\partial\Omega. \]

Due to (1), the above one is well-defined. We have the jump relation formula for the double layer potential:
for fixed x_{0}\in\partial\Omega, we have

    \[ \lim_{x\rightarrow x_{0};x\in\Omega}\mathscr{D}f\left(x\right)=\left(\frac{1}{2}I+K\right)f\left(x_{0}\right). \]

Also, it can be shown that K:C\left(\partial\Omega\right)\rightarrow C\left(\partial\Omega\right) is compact, (and also K:\Leb p\left(\partial\Omega\right)\rightarrow\Leb p\left(\partial\Omega\right), 1<p<\infty is compact). Hence by the Riesz-Schauder theory, the operator \left(\frac{1}{2}I+K\right) is invertible if and only if it is injective. Using some argument, one can show that it is injective
and so the Dirichlet problem \left(D\right) has a solution. Uniqueness follows from the maximum principle. One can also obtain the solvability of the Dirichlet problem when the boundary data is in \Leb p\left(\partial\Omega\right). We interpret the boundary condition via nontangential limits. We also note that the above argument holds when \Omega is a C^{1,\alpha}-domain.

The difficulty in extending these results to the case of C^{1} and Lipschitz domains is that, in these cases, we have

    \[ \left|G\left(x,y\right)\right|\le\frac{C}{\left|x-y\right|^{n-1}}\quad x,y\in\partial\Omega,x\neq y. \]

So even the \Leb p-boundedness, much less the compactness of K is far from obvious. Another difficulty is that the operator we considered is of non-convolution type operators. So we cannot use the Fourier transform method to guarantee the \Leb 2-boundedness of the operator.

In Calder\’on’s remarkable paper, he proved that the Cauchy integral operator along a Lipschitz curve is bounded from \Leb 2 to itself when the Lipschitz constant is small. This leads to the solvability of the Dirichlet problem for the Poisson equation in C^{1}-domain due to Fabes-Rievere-Joedit. Later, Coifman-McIntosh-Meyer resolved the restrictive condition of Caderon’s result.

The first proof of theorem of Coifman-McIntosh-Meyer uses some inductive argument from Calder\’on’s result. One famous proof is to use the variant of T1-theorem, the Tb-theorem due to David-Journ\’e-Semmes. But we do not pursuit in this direction. In 1989, Coifman-Jones-Semmes provide another proof using the theory of complex variables and some variant of the Littlewood-Paley theory. We also note that there is a geometric proof of theorem of Coifman-McIntosh-Meyer given by Melnikov and Verdera. Recently, Muscalu gives a new proof of Coifman-McIntosh-Meyer theorem whose methods can be extended to the case of multilinear operators.

In Section 1, we construct Haar system on the real line \mathbb{R}. In Section 2, we construct Haar system associated to an accretive function, which will be defined in later. This leads to the proof of Coifman-McIntosh-Meyer theorem.  For those who are interested in, see the paper of Coifman-Jones-Semmes.

1. Haar systems on the real lines

In this section, we construct the Haar system on the real lines. Recall that a dyadic interval in \mathbb{R} is an interval of the form

    \[ \left[\frac{m}{2^{k}},\frac{m+1}{2^{k}}\right), \]

where m and k are integers. For k\in\mathbb{Z} we denote by \mathcal{F}_{k} the set of all dyadic intervals in \mathbb{R} whose side length is 2^{-k}. We also denote by \mathcal{F} the set of all dyadic intervals in \mathbb{R}. Then we have

    \[ \mathcal{F}=\bigcup_{k\in\mathbb{Z}}\mathcal{F}_{k}. \]

Moreover, the \sigma-algebra \sigma\left(\mathcal{F}_{k}\right) of measurable subset of \mathbb{R} formed by countable unions and
complements of elements of \mathcal{F}_{k} is increasing as k increases, i.e., \sigma\left(\mathcal{F}_{k}\right) is a filtration. For simplicity, we write \Sigma_{k}=\sigma\left(\mathcal{F}_{k}\right). Observe that any two dyadic intervals of the same side length either coincide or their interior are disjoint. Moreover, either two given dyadic interval contains the other or their interiors are disjoint.

Given a locally integrable function f on \mathbb{R}, we let

    \[ f_{I}=\frac{1}{\left|I\right|}\int_{I}f\left(t\right)dt \]

denote the average of f over an interval I. The conditional expectation of a locally integrable function f on \mathbb{R} with respect to the increasing family of \sigma-algebras \sigma\left(\mathcal{F}_{k}\right) generated by \mathcal{F}_{k} is defined as

    \[ P_{k}\left(f\right)\left(x\right)=\sum_{I\in\mathcal{F}_{k}}f_{I}\chi_{I}\left(x\right). \]

We also define the dyadic martingale difference operator Q_{k} as follows:

    \[ Q_{k}\left(f\right)=P_{k+1}\left(f\right)-P_{k}\left(f\right), \]

also for k\in\mathbb{Z}.

Remark (Justification of the terminology). First, we justify the terminology “conditional expectation”. We first show that for any I\in\mathcal{F}_{k}, we have

    \[ \int_{I}fdx=\int_{I}\sum_{J_{k}\in\mathcal{F}_{k}}\chi_{J_{k}}\left(f_{J_{k}}\right)dx. \]

Since I\in\mathcal{F}_{k}, for any J\in\mathcal{Q}_{k}, either I=J or I\cap J=\varnothing. So

    \[ \int_{I}\sum_{J_{k}\in\mathcal{F}_{k}}\chi_{J_{k}}\left(f_{J_{k}}\right)dx=\int_{I}f_{I}dx=\int_{I}fdx \]

The identity holds for all I\in\mathcal{F}_{k}. Hence the identity holds for all B\in\Sigma_{k}. This shows that

    \[ P_{k}\left(f\right)=\mathbb{E}\left[f\mid\Sigma_{k}\right], \]

which justify the terminology “conditional expectation”.

For a dyadic interval I=\left[m2^{-k},\left(m+1\right)2^{-k}\right) and write I=I_{L}\cup I_{R}, where I_{L} and I_{R} are the left and right halves of I. The function

    \[ h_{I}\left(x\right)=\left|I\right|^{-\frac{1}{2}}\left(\chi_{I_{L}}-\chi_{I_{R}}\right) \]

is called the Haar function associated with the interval I. By construction, the Haar functions have \Leb 2 norm equal to 1. Moreover, if I\neq I', either I\cap I'=\varnothing or not. Clearly, if I\cap I'=\varnothing, then

(2)   \begin{equation*} \int_{\mathbb{R}}h_{I}h_{I'}\myd x=0. \end{equation*}

If I\cap I'\neq\varnothing, we may assume \left|I'\right|<\left|I\right|. Then either I' is contained in the left or in the right half of I, on either of which h_{I} is contant. Hence (2) follows. We introduce the notation

    \[ \action{f,g}=\int_{\mathbb{R}}fg\myd x \]

for f,g\in\Leb 2\left(\mathbb{R}\right). Then

    \[ \action{h_{I},h_{J}}=\delta_{IJ}=\begin{cases} 1 & \text{if }I=J,\\ 0 & \text{if }I\neq J. \end{cases} \]

The following proposition shows that the dyadic difference operator is a projection to the space which is spanned by Haar functions.

Proposition 1. Let f\in\Leb 2\left(\mathbb{R}\right). For all k\in\mathbb{Z}, we have

    \[ Q_{k}\left(f\right)=\sum_{I\in\mathcal{F}_{k}}\action{f,h_{I}}h_{I} \]

and also

    \[ \norm{Q_{k}\left(f\right)}{\Leb 2}^{2}=\sum_{I\in\mathcal{F}_{k}}\left|\action{f,h_{I}}\right|^{2}. \]

Proof. Observe that every interval J in \mathcal{F}_{k+1} is either an I_{L} or an I_{R} for some unique I\in\mathcal{F}_{k}. Thus, we write

    \begin{align*} P_{k+1}\left(f\right) & =\sum_{J\in\mathcal{F}_{k+1}}\frac{1}{\left|J\right|}\left(\int_{J}f\myd t\right)\chi_{J}\\ & =\sum_{I\in\mathcal{F}_{k}}\left[\left(\frac{1}{\left|I_{L}\right|}\int_{I_{L}}f\myd t\right)\chi_{I_{L}}+\left(\frac{1}{\left|I_{R}\right|}\int_{I_{R}}f\myd t\right)\chi_{I_{R}}\right]\\ & =\sum_{I\in\mathcal{F}_{k}}\left[\left(\frac{2}{\left|I\right|}\int_{I_{L}}f\myd t\right)\chi_{I_{L}}+\left(\frac{2}{\left|I\right|}\int_{I_{R}}f\myd t\right)\chi_{I_{R}}\right]. \end{align*}

Also,

    \begin{align*} P_{k}\left(f\right) & =\sum_{I\in\mathcal{F}_{k}}f_{I}\chi_{I}\\ & =\sum_{I\in\mathcal{F}_{k}}\left(\frac{1}{\left|I\right|}\int_{I_{L}}f\myd t+\frac{1}{\left|I\right|}\int_{I_{R}}f\myd t\right)\left(\chi_{I_{L}}+\chi_{I_{R}}\right). \end{align*}

So

    \begin{align*} Q_{k}\left(f\right) & =P_{k+1}\left(f\right)-P_{k}\left(f\right)\\ & =\sum_{I\in\mathcal{F}_{k}}\left[\left(\frac{1}{\left|I\right|}\int_{I_{L}}f\myd t\right)\chi_{I_{L}}-\left(\frac{1}{\left|I\right|}\int_{I_{R}}f\myd t\right)\chi_{I_{L}}\right.\\ & \relphantom{=}\left.+\left(\frac{1}{\left|I\right|}\int_{I_{R}}f\myd t\right)\chi_{I_{R}}-\left(\frac{1}{\left|I\right|}\int_{I_{L}}f\myd t\right)\chi_{I_{R}}\right], \end{align*}

which is easily checked to be equal to

    \[ \sum_{I\in\mathcal{F}_{k}}\left(\int_{I}fh_{I}\myd t\right)h_{I}=\sum_{I\in\mathcal{F}_{k}}\action{f,h_{I}}h_{I}. \]

This coplemtes the first part. Now from the orthogonality of \left\{ h_{I}\right\}, we get

    \begin{align*} \norm{Q_{k}\left(f\right)}{\Leb 2}^{2} & =\int Q_{k}\left(f\right)Q_{k}\left(f\right)\myd t\\ & =\sum_{I\in\mathcal{F}_{k}}\left|\action{f,h_{I}}\right|^{2}\int_{\mathbb{R}}\left|h_{I}\right|^{2}\myd t\\ & =\sum_{I\in\mathcal{F}_{k}}\left|\action{f,h_{I}}\right|^{2}. \end{align*}

This completes the proof of Proposition 1.

Using the dyadic difference operator, we can decompose a function in \Leb 2\left(\mathbb{R}\right) as follows:

Theorem 2. Every function f\in\Leb 2\left(\mathbb{R}\right) can be written as

(3)   \begin{equation*} f=\sum_{k\in\mathbb{Z}}Q_{k}\left(f\right)=\sum_{I\in\mathcal{F}}\action{f,h_{I}}h_{I}, \end{equation*}

where the series converges almost everywhere and in \Leb 2. We also have

(4)   \begin{equation*} \norm f{\Leb 2}^{2}=\sum_{k\in\mathbb{Z}}\norm{Q_{k}\left(f\right)}{\Leb 2}^{2}=\sum_{I\in\mathcal{F}}\left|\action{f,h_{I}}\right|^{2}. \end{equation*}

Proof. It follows from the Lebesgue differentitation theorem that there exists a set N_{f} of measure zero on \mathbb{R} such that for all x\in\mathbb{R}\setminus N_{f}, we have

    \[ \lim_{j\rightarrow\infty}f_{I_{j}}=f\left(x\right), \]

whenever I_{j} is a sequence of decreasing intervals such that \bigcap_{j}\overline{I_{j}}=\left\{ x\right\}. Given x in \mathbb{R}\setminus N_{f}, there exists a unique sequence of dyadic intervals I_{j}\left(x\right)\in\mathcal{F}_{j} such that \bigcap_{j=0}^{\infty}I_{j}\left(x\right)=\left\{ x\right\}. Then for all x\in\mathbb{R}\setminus N_{f}, we have

    \[ \lim_{j\rightarrow\infty}P_{j}\left(f\right)\left(x\right)=\lim_{j\rightarrow\infty}\sum_{I\in\mathcal{F}_{j}}f_{I}\chi_{I}\left(x\right)=\lim_{j\rightarrow\infty}f_{I_{j}\left(x\right)}=f\left(x\right). \]

Hence P_{j}\left(f\right)\rightarrow f a.e. as j\rightarrow\infty. Since

    \[ \left|P_{j}\left(f\right)\right|\le M_{d}\left(f\right), \]

where M_{d} denotes the dyadic maximal function, we have that \left|P_{j}\left(f\right)-f\right|\le2M_{d}\left(f\right). Hence due to the \Leb 2-boundedness of the dyadic maximal operator, it follows from the dominated convergence theorem that P_{j}\left(f\right)\rightarrow f in \Leb 2. For a given x\in\mathbb{R} and I_{j}\left(x\right) as before, we have

    \begin{align*} \left|P_{j}\left(f\right)\left(x\right)\right| & =\left|\frac{1}{\left|I_{j}\left(x\right)\right|}\int_{I_{j}\left(x\right)}f\myd y\right|\\ & \le\left(\frac{1}{\left|I_{j}\left(x\right)\right|}\int_{I_{j}\left(x\right)}\left|f\right|^{2}\myd y\right)^{\frac{1}{2}}\le2^{\frac{j}{2}}\norm f{\Leb 2} \end{align*}

which tends to zero as j\rightarrow-\infty, since the side length of each I_{j}\left(x\right) is 2^{-j}. Since \left|P_{j}\left(f\right)\right|\le M_{d}\left(f\right), by the dominated convergence theorem, we conclude that P_{j}\left(f\right)\rightarrow0 in \Leb 2 as j\rightarrow-\infty. Observe that

    \[ \sum_{k=M}^{N}Q_{k}\left(f\right)=E_{N+1}\left(f\right)-E_{M}\left(f\right)\rightarrow f \]

as N\rightarrow\infty and M\rightarrow-\infty a.e. and in \Leb 2. By Proposition ??,

    \[ f=\sum_{k}\sum_{I\in\mathcal{F}_{k}}\action{f,h_{I}}h_{I}=\sum_{I\in\mathcal{F}}\action{f,h_{I}}h_{I}. \]

This proves identity (3). To prove (4), we rewrite

    \begin{align*} Q_{k}\left(f\right) & =\sum_{I\in\mathcal{F}_{k+1}}f_{I}\chi_{I}-\sum_{J\in\mathcal{F}_{k}}f_{J}\chi_{J}\\ & =\sum_{J\in\mathcal{F}_{k}}\left[\sum_{I\in\mathcal{F}_{k+1},I\subset J}f_{I}\chi_{I}-f_{J}\chi_{J}\right]\\ & =\sum_{J\in\mathcal{F}_{k}}\left[\sum_{I\in\mathcal{F}_{k+1},I\subset J}f_{I}\chi_{I}-\frac{1}{2}\sum_{I\in\mathcal{F}_{k+1},I\subset J}f_{I}\chi_{J}\right]\\ & =\sum_{J\in\mathcal{F}_{k}}\sum_{I\in\mathcal{F}_{k+1},I\subset J}f_{I}\left(\chi_{I}-\frac{1}{2}\chi_{J}\right). \end{align*}

For k'>k, we have

    \begin{align*} & \int_{\mathbb{R}}Q_{k}\left(f\right)Q_{k'}\left(f\right)\myd x\\ & =\sum_{J\in\mathcal{F}_{k}}\sum_{\substack{I\in\mathcal{F}_{k+1}\\ J\subset I } }f_{I}\sum_{J'\in\mathcal{F}_{k'}}\sum_{\substack{I'\in\mathcal{F}_{k'+1}\\ I'\subset J' } }f_{I'}\int\left(\chi_{I}-\frac{1}{2}\chi_{J}\right)\left(\chi_{I'}-\frac{1}{2}\chi_{J'}\right)\myd x. \end{align*}

Since k'>k, the last integral is nonzero only when J'\subsetneq J. If this is the case, then J'\subset I_{J'} for some dyadic interval I_{J'}\in\mathcal{F}_{k+1}. Then the function \chi_{I'}-\frac{1}{2}\chi_{J'} is supported in the interval I_{J'} and the function \chi_{I}-\frac{1}{2}\chi_{J} is constant on any dyadic subinterval I of J. Then

    \[ \sum_{\substack{I'\in\mathcal{F}_{k'+1}\\ I'\subset J' } }f_{I'}\int_{I_{j'}}\left(\chi_{I'}-\frac{1}{2}\chi_{J'}\right)\myd x=\sum_{\substack{I'\in\mathcal{F}_{k'+1}\\ I'\subset J' } }f_{I'}\left(\left|I'\right|-\frac{1}{2}\left|J'\right|\right)=0, \]

since \left|J'\right|=2\left|I'\right|. Hence, \action{Q_{k}\left(f\right),Q_{k'}\left(f\right)}=0 whenever k\neq k'. This shows that

    \[ \norm f{\Leb 2}^{2}=\sum_{k\in\mathbb{Z}}\norm{Q_{k}\left(f\right)}{\Leb 2}^{2}. \]

Now the identity (4) follows from Proposition 1. This completes the proof of Theorem 2.

2. Modified haar systems with an accretive functions

The purpose of this section is to extend Proposition 1 and Theorem 2. This extension leads to a proof of the theorem of Coifman-McIntosh-Meyer. We introduce the following notion.

Definition. A bounded complex-valued function b on \mathbb{R} is said to be accretive if there is a constant c>0 such that \re b\left(x\right)\ge c for almost all x\in\mathbb{R}.

Let b be an accretive function. For a measurable set E in \mathbb{R} with \left|E\right|<\infty, we define

    \[ b\left(E\right)=\frac{1}{\left|E\right|}\int_{E}b\myd x. \]

Observe that

    \[ \left|b\left(E\right)\right|^{2}=\frac{1}{\left|E\right|^{2}}\left|\int_{E}b\myd x\right|^{2}\ge\frac{1}{\left|E\right|^{2}}\left|\int_{E}\re b\myd x\right|^{2}\ge c \]

since b is accretive. For each I\in\mathcal{F}, we define

    \[ \beta_{I}=\frac{1}{\left|I\right|^{1/2}}\left(\frac{b\left(I_{L}\right)b\left(I_{R}\right)}{b\left(I\right)}\right)^{1/2}\left\{ b\left(I_{L}\right)^{-1}\chi_{I_{L}}-b\left(I_{R}\right)^{-1}\chi_{I_{R}}\right\} \]

with a fixed choice of the square roots. If b\equiv1, then \beta_{I}=h_{I}. For f,g\in\Leb 2\left(\mathbb{R}\right), we introduce a pseudo-inner product

    \[ \action{f,g}_{b}=\int_{\mathbb{R}}fgb\myd x. \]

on \Leb 2\left(\mathbb{R}\right). By definition, each \beta_{I} is supported on I and is constant on I_{L} and I_{R}. Moreover,

    \[ \action{\beta_{I},\beta_{J}}_{b}=\delta_{IJ}\quad\text{and}\quad\action{\beta_{I},1}_{b}=0. \]

The following theorem is a generalization of Theorem 2.

Theorem 3. Let f\in\Leb 2\left(\mathbb{R}\right). Then

    \[ f=\sum_{I\in\mathcal{F}}\action{f,\beta_{I}}_{b}\beta_{I}, \]

where the sum converges in \Leb 2. Moreover,

    \[ c^{-1}\norm f{\Leb 2}^{2}\le\sum_{I\in\mathcal{F}}\left|\action{f,\beta_{I}}_{b}\right|^{2}\le c\norm f{\Leb 2}^{2} \]

for some constant c>0.

Proof. We define

    \[ E_{k}\left(f\right)\left(x\right)=\frac{1}{b\left(I\right)\left|I\right|}\int_{I}fb\myd t,\quad x\in I\in\mathcal{F}_{k} \]

and

    \[ \Delta_{k}\left(f\right)=E_{k+1}\left(f\right)-E_{k}\left(f\right). \]

Then we first show that

    \[ f=\sum_{k=-\infty}^{\infty}\Delta_{k}\left(f\right). \]

Since b is accretive, \int_{I}b\myd t\neq0 for any I\in\mathcal{F}. Since fb\in\Leb 2_{\loc}, it follows from the Lebesgue differentiation theorem that

    \[ \lim_{k\rightarrow\infty}E_{k}\left(f\right)\left(x\right)=\lim_{k\rightarrow\infty}\frac{\frac{1}{\left|I_{k}\right|}\int_{I_{k}}fb\myd t}{\frac{1}{\left|I_{k}\right|}\int_{I_{k}}b\myd t}=\frac{f\left(x\right)b\left(x\right)}{b\left(x\right)}=f\left(x\right)\quad\text{a.e.}, \]

where I_{k}\in\mathcal{F}_{k} is the unique dyadic interval such that x\in I_{k} for all k and \bigcap_{k=0}^{\infty}I_{k}=\left\{ x\right\}.

Since b is accretive, we have

    \begin{align*} \left|E_{k}\left(f\right)\left(x\right)\right| & =\frac{\left|\int_{I}fb\myd t\right|}{\left|\int_{I}b\myd t\right|}\\ & \le\frac{\norm b{\Leb{\infty}}}{\left|I\right|}\int_{I}\left|f\right|\myd t\frac{\left|I\right|}{\left|\int_{I}b\myd t\right|}\\ & \le\frac{1}{c}\norm b{\Leb{\infty}}M_{d}\left(f\right)\left(x\right), \end{align*}

where I\in\mathcal{F}_{k} with x\in I. Hence by the dominated convergence theorem, we get E_{k}\left(f\right)\rightarrow f in \Leb 2 as k\rightarrow\infty. Following the exactly same argument as in the proof of Theorem 2, we can also show that E_{k}\left(f\right)\rightarrow0 a.e. and in \Leb 2 as k\rightarrow-\infty. Also,

    \[ \sum_{k\in\mathbb{Z}}\Delta_{k}\left(f\right)=f \]

in \Leb 2 and almost everywhere. Following the exact same argument in the proof of Proposition ??, we can show that

    \[ \Delta_{k}\left(f\right)=\sum_{I\in\mathcal{F}_{k}}\action{f,\beta_{I}}_{b}\beta_{I}. \]

So the first conclusion of the lemma is verified.

Expanding out \Delta_{k}, we see that

    \begin{align*} \left|\Delta_{k}\left(f\right)\right| & =\left|E_{k+1}f-E_{k}f\right|\\ & =\left|P_{k+1}\left(b\right)^{-1}Q_{k}\left(bf\right)-\frac{Q_{k}\left(b\right)}{P_{k}\left(b\right)P_{k+1}\left(b\right)}P_{k}\left(bf\right)\right|\\ & \le C\left|Q_{k}\left(bf\right)\right|+C\left|Q_{k}\left(b\right)\right|\left|P_{k}\left(bf\right)\right| \end{align*}

since b is accretive. So

    \[ \sum_{k=-\infty}^{\infty}\norm{\Delta_{k}\left(f\right)}{\Leb 2}^{2}\le C\sum_{k=-\infty}^{\infty}\norm{Q_{k}\left(bf\right)}{\Leb 2}^{2}+C\sum_{k=-\infty}^{\infty}\norm{Q_{k}\left(b\right)P_{k}\left(bf\right)}{\Leb 2}^{2}. \]

Since b\in\Leb{\infty}\left(\mathbb{R}\right), it follows from Theorem 1 that

    \[ \sum_{k=-\infty}^{\infty}\norm{Q_{k}\left(bf\right)}{\Leb 2}^{2}=\norm{bf}{\Leb 2}^{2}\le\norm b{\Leb{\infty}}^{2}\norm f{\Leb 2}^{2}. \]

It remains to show that

    \[ \sum_{k=-\infty}^{\infty}\norm{Q_{k}\left(b\right)P_{k}\left(bf\right)}{\Leb 2}^{2}\le C\norm f{\Leb 2}^{2}. \]

Recall that \norm{M_{d}f}{\Leb 2}\le C\norm f{\Leb 2} for some constant C>0.

For J\in\mathcal{F}, it follows from Theorem 1 that

    \[ \sum_{I\subset J}\left|\action{b,h_{I}}\right|^{2}=\sum_{I\in\mathcal{F}}\left|\action{b\chi_{J},h_{I}}\right|^{2}=\norm{b\chi_{J}}{\Leb 2}^{2}\le\norm b{\Leb{\infty}}^{2}\left|J\right|. \]

Hence for any set \mathcal{O}\subset\mathbb{R}, we have

    \[ \sum_{I\subset\mathcal{O}}\left|\action{b,h_{I}}\right|^{2}\le\norm b{\Leb{\infty}}^{2}\left|\mathcal{O}\right|. \]

Recall that for 0<p<\infty, we have

(5)   \begin{equation*} \sum_{k\in\mathbb{Z}}2^{kp}\left|\left\{ \left|f\right|>2^{k}\right\} \right|\approx\norm f{\Leb p}^{p}. \end{equation*}

Let \mathcal{O}_{n}=\left\{ x:M_{d}\left(bf\right)>2^{n}\right\}. Then

(6)   \begin{equation*} \sum_{k\in\mathbb{Z}}2^{2k}\left|\mathcal{O}_{n}\right|\apprle\norm{M_{d}\left(bf\right)}{\Leb 2}^{2}\le C\norm b{\Leb{\infty}}^{2}\norm f{\Leb 2}^{2}. \end{equation*}

Since

    \[ \left|Q_{k}\left(b\right)\right|^{2}=\sum_{I\in\mathcal{F}_{k}}\left|\action{b,h_{I}}\right|^{2}\left|h_{I}\right|^{2}=\sum_{I\in\mathcal{F}_{k}}\frac{1}{\left|I\right|}\left|\action{b,h_{I}}\right|^{2}\chi_{I}, \]

by (5),we have

    \begin{align*} \sum_{k=-\infty}^{\infty}\int_{\mathbb{R}}\left|Q_{k}\left(b\right)P_{k}\left(bf\right)\right|^{2}\myd x & \le\sum_{k=-\infty}^{\infty}\sum_{I\in\mathcal{F}_{k}}\frac{1}{\left|I\right|}\left|\action{b,h_{I}}\right|^{2}\int_{\mathbb{R}}\chi_{I}\left|P_{k}\left(bf\right)\right|^{2}\myd t\\ & \le4\sum_{k=-\infty}^{\infty}\sum_{I\in\mathcal{F}_{k}}\frac{1}{\left|I\right|}\left|\action{b,h_{I}}\right|^{2}\int_{\mathbb{R}}\chi_{I}\left|M_{d}\left(bf\right)\right|^{2}\myd t\\ & \le C\sum_{k=-\infty}^{\infty}\sum_{I\in\mathcal{F}_{k}}\left|\action{b,h_{I}}\right|^{2}\sum_{n=-\infty}^{\infty}2^{2n}\frac{\left|I\cap\mathcal{O}_{n}\right|}{\left|I\right|}. \end{align*}

Since

    \[ \mathcal{O}_{n}=\bigcup_{I\in\mathcal{F},I\subset\mathcal{O}_{n}}I, \]

we get

    \[ \sum_{k=-\infty}^{\infty}\int_{\mathbb{R}}\left|Q_{k}\left(b\right)P_{k}\left(bf\right)\right|^{2}\myd x\le C\sum_{k=-\infty}^{\infty}\sum_{I\in\mathcal{F}_{k}}\sum_{n=-\infty}^{\infty}2^{2n}\sum_{I\subset\mathcal{O}_{n}}\left|\action{b,h_{I}}\right|^{2}. \]

Thus by (5) and (6), we get

    \begin{align*} & \sum_{k=-\infty}^{\infty}\sum_{I\in\mathcal{F}_{k}}\sum_{n=-\infty}^{\infty}2^{2n}\sum_{I\subset\mathcal{O}_{n}}\left|\action{b,h_{I}}\right|^{2}\\ & =\sum_{n=-\infty}^{\infty}2^{2n}\sum_{I\in\mathcal{F},I\subset\mathcal{O}_{n}}\left|\action{b,h_{I}}\right|^{2}\\ & \le\sum_{n=-\infty}^{\infty}2^{2n}\left|\mathcal{O}_{n}\right|\\ & \le C\norm b{\Leb{\infty}}^{2}\norm f{\Leb 2}^{2}. \end{align*}

This completes the first part of the proof. For the converse, let \norm f{\Leb 2}=1 and set g=\overline{b}\overline{f} , a_{I}^{\prime}=\action{g,\beta_{I}}_{b}.
Then

    \[ \sum_{I}a_{I}a_{I}'=\action{f,g}_{b}=\norm{fb}{\Leb 2}^{2}. \]

Since

    \begin{align*} \sum_{I}a_{I}a_{I}' & =\sum_{I}\action{f,\beta_{I}}_{b}\action{g,\beta_{I}}_{b}\\ & \le\left(\sum_{I}\left|\action{f,\beta_{I}}_{b}\right|^{2}\right)^{\frac{1}{2}}\left(\sum_{I}\left|\action{g,\beta_{I}}_{b}\right|^{2}\right)^{\frac{1}{2}}\\ & \le C\left(\sum_{I}\left|\action{f,\beta_{I}}_{b}\right|^{2}\right)^{\frac{1}{2}}. \end{align*}

Since b is accretive, we get the desired result. This completes the proof.

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