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)
Many authors have been studied the solvability of the problem in a various settings. One possible way to solve the problem is to use the method of layer potential. In this talk, we assume for simplicity. The fundamental solution for the Laplacian is defined by
where is the volume of the unit ball in . Suppose first that has -boundary. For , we define the double layer potential of by
where denotes the outward unit normal to the boundary . Then it is easy to check that is well-defined on and is harmonic in . Also, can be extended as a continuous function from to up to the boundary. Set for , . Since is of , one can easy to show that
for some constant .
For , we define
Due to (1), the above one is well-defined. We have the jump relation formula for the double layer potential:
for fixed , we have
Also, it can be shown that is compact, (and also , is compact). Hence by the Riesz-Schauder theory, the operator is invertible if and only if it is injective. Using some argument, one can show that it is injective
and so the Dirichlet problem 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 . We interpret the boundary condition via nontangential limits. We also note that the above argument holds when is a -domain.
The difficulty in extending these results to the case of and Lipschitz domains is that, in these cases, we have
So even the -boundedness, much less the compactness of 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 -boundedness of the operator.
In Calder\’on’s remarkable paper, he proved that the Cauchy integral operator along a Lipschitz curve is bounded from to itself when the Lipschitz constant is small. This leads to the solvability of the Dirichlet problem for the Poisson equation in -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 -theorem, the -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 . 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 is an interval of the form
where and are integers. For we denote by the set of all dyadic intervals in whose side length is . We also denote by the set of all dyadic intervals in Then we have
Moreover, the -algebra of measurable subset of formed by countable unions and
complements of elements of is increasing as increases, i.e., is a filtration. For simplicity, we write . 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 on , we let
denote the average of over an interval . The conditional expectation of a locally integrable function on with respect to the increasing family of -algebras generated by is defined as
We also define the dyadic martingale difference operator as follows:
also for .
Remark (Justification of the terminology). First, we justify the terminology “conditional expectation”. We first show that for any , we have
Since , for any , either or . So
The identity holds for all . Hence the identity holds for all . This shows that
which justify the terminology “conditional expectation”.
For a dyadic interval and write , where and are the left and right halves of . The function
If , we may assume . Then either is contained in the left or in the right half of , on either of which is contant. Hence (2) follows. We introduce the notation
for . Then
The following proposition shows that the dyadic difference operator is a projection to the space which is spanned by Haar functions.
Proposition 1. Let . For all , we have
Proof. Observe that every interval in is either an or an for some unique . Thus, we write
which is easily checked to be equal to
This coplemtes the first part. Now from the orthogonality of , we get
This completes the proof of Proposition 1.
Using the dyadic difference operator, we can decompose a function in as follows:
Proof. It follows from the Lebesgue differentitation theorem that there exists a set of measure zero on such that for all , we have
whenever is a sequence of decreasing intervals such that . Given in , there exists a unique sequence of dyadic intervals such that . Then for all , we have
Hence a.e. as . Since
where denotes the dyadic maximal function, we have that . Hence due to the -boundedness of the dyadic maximal operator, it follows from the dominated convergence theorem that in . For a given and as before, we have
which tends to zero as , since the side length of each is . Since , by the dominated convergence theorem, we conclude that in as . Observe that
as and a.e. and in . By Proposition ??,
For , we have
Since the last integral is nonzero only when . If this is the case, then for some dyadic interval . Then the function is supported in the interval and the function is constant on any dyadic subinterval of . Then
since . Hence, whenever . This shows that
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 on is said to be accretive if there is a constant such that for almost all .
Let be an accretive function. For a measurable set in with , we define
since is accretive. For each , we define
with a fixed choice of the square roots. If , then . For , we introduce a pseudo-inner product
on . By definition, each is supported on and is constant on and . Moreover,
The following theorem is a generalization of Theorem 2.
Theorem 3. Let . Then
where the sum converges in . Moreover,
for some constant .
Proof. We define
Then we first show that
Since is accretive, for any . Since , it follows from the Lebesgue differentiation theorem that
where is the unique dyadic interval such that for all and .
Since is accretive, we have
where with . Hence by the dominated convergence theorem, we get in as . Following the exactly same argument as in the proof of Theorem 2, we can also show that a.e. and in as . Also,
in and almost everywhere. Following the exact same argument in the proof of Proposition ??, we can show that
So the first conclusion of the lemma is verified.
Expanding out , we see that
since is accretive. So
Since , it follows from Theorem 1 that
It remains to show that
Recall that for some constant .
For , it follows from Theorem 1 that
Hence for any set , we have
by (5),we have
This completes the first part of the proof. For the converse, let and set , .
Since is accretive, we get the desired result. This completes the proof.