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\centerline{\bigheading Convolution operators on Lipschitz graphs}
\centerline{\bigheading with harmonic kernels}
\vglue 0.5 cm

\centerline{\bf Terence Tao}

\vglue 0.5 cm

\centerline{\heading Abstract}
\leftskip = 2 cm
\rightskip = 2 cm
We use a Clifford-algebra version of the $T(b)$ theorem to
prove the $L^2$-boundedness of convolution operators with harmonic
Calder\'on-Zygmund kernels on Lipschitz graphs.
This extends the results of [7].
\leftskip = 0 cm
\rightskip = 0 cm

{\small (AMS classification numbers: Primary: 42B20, Secondary: 31B05)}
{\small This research was supported by an APRA scholarship.}

{\heading 1.  Introduction}

Let $n$ be a positive integer and $0<\omega<\pi/2$ be a fixed angle.
Suppose that $\Sigma$ is a Lipschitz graph in $\R^{n+1}$, by 
which we mean 
a manifold of the form $\rho(\R^n)$, where $\rho(x) = x + A(x)e_0$ and 
$A$ is a real function on $\R^n$ such that $|A(x) - A(y)| \leq |x-y| \tan
\omega$ for all $x,y$.  We consider (Clifford-valued) right-linear operators 
of the form
$$ Tf(x) = \lim_{\varepsilon \to 0} \Bigl( \int_{\Sigma,
|x-y|>\varepsilon} \phi(x-y) n(y) f(y)\ dS_y +
\underline{\phi}(\varepsilon n(x)) n(x) f(x)\Bigr),\eqno
(1.1)$$ 
where $\phi$ is a Clifford-valued harmonic Calder\'on-Zygmund kernel on
a certain sector $\R^{n+1}$, and $\underline{\phi}$ is a bounded potential of
$\phi$ (this term can be omitted if $\phi$ is odd).  The Cauchy singular 
integral operator is the model example for operators of this type.

In the plane, $L^2$-boundedness results for operators of this type 
were first obtained using complex analytic methods in [4].  In [3] two new 
proofs of the result in [4] were given; one using square-function
methods and the extension of $\phi$ to upper and lower half-spaces, 
and the other using a martingale-based method.
In [7] the methods of the first proof were extended to Clifford
algebras, to give the boundedness of operators (1.1) in arbitrary dimension
under the assumption that $\phi$ was right-monogenic.
Meanwhile, Clifford-valued martingales were developed in [6] to prove 
a Clifford-algebraic version of the $T(b)$ theorem.  In an unpublished
note T. Qian has shown that this theorem can be used to show the
boundedness of the operator (1.1) under the relatively strong assumption
that $\phi$ is bi-monogenic.  Given the results of [7], it seems
natural to try to extend the result of T. Qian to right-monogenic and
harmonic kernels.  This is achieved in this paper.  In particular,
boundedness results for real-valued operators with harmonic kernels 
are obtained as a consequence of Clifford-analytic techniques.

Our method of proof is as follows.  We first prove the
result for bi-monogenic kernels, using a
similar method to that of T. Qian and an adaptation of
the $T(b)$ theorem (see also [1]).  Then we extend the result to
right-monogenic kernels by decomposing these kernels as a left-linear
combination of bi-monogenic kernels.  Finally, we show that every
scalar harmonic kernel is the scalar part of a right-monogenic kernel,
which proves $L^2$-boundedness for general harmonic kernels.

This paper is based on the thesis [12] by the same author.  The author 
thanks Garth Gaudry for his guidance, advice, and support.

{\heading 2.  Notation and preliminaries}

The notation we will use here is essentially the same as those used in [6]
and [7].  Detailed expositions of Clifford algebras, with complete definitions
and proofs of assertions, can be found in [2], [10] or [11].  

We use the symbol $C$ to denote various positive constants which depend only 
on geometric quantities such as $n$ and $\omega$.  If $F$ is a positive
quantity, we use $O(F)$ to denote an unspecified quantity whose magnitude
is at most $CF$ for some constant $C$.

The function $n(y)$ will always mean a consistent unit normal to the
surface of integration, and $dS$ will always mean surface measure.  If
$X$ is a set, we use $\partial X$ to denote the boundary of $X$, and
$|X|$ to denote the Lebesgue measure of $X$.

We give $\R^{n+1} = {\rm span}\{e_0, e_1, \ldots, e_n\}$ and the
subspace $\R^n = {\rm span}\{e_1, \ldots, e_n\}$ the usual
Euclidean metric, and imbed them in the the Clifford algebra $\Ad$,
defined as the free associative algebra over $\R$ generated by $\R^{n+1}$, 
with $e_0$ as 
the multiplicative identity, modulo the relations $x^2 = -|x|^2 e_0$ for
all $x \in \R^n$.  Alternatively, one may use the equivalent set of
relations $e_ie_j + e_je_i = -2\delta_{ij} e_0$ for all $1 \leq i,j \leq
n$.
Unless otherwise specified, all functions and
elements of function spaces will take their values in this Clifford
algebra.

Real multiples of $e_0$ will be referred to as
scalars.  There is a canonical basis $\{e_I: I \in {\cal I}\}$
for $\Ad$ that extends the basis for $\R^{n+1}$; if $x$ is an element
of $\Ad$, we shall denote by $x_I$ or $[x]_I$ the $e_I$ component of
$x$.  We will use the standard definitions of norm, inner product, and 
conjugation in
the Clifford algebra (see [2]).  We note briefly that  $\langle x,y \rangle
= [x \overline{y}]_0$, $\overline {xy} = \overline{y}\ \overline{x}$, and
$|xy| \leq C |x|\ |y|$ for all $x, y \in \Ad$. 

Contrary to usual practice, we will use $fT$ or $(f)T$ to denote the action 
of a Clifford-left-linear operator $T$ on a function $f$; the more familiar 
$Tf$ or $T(f)$ will be reserved for operators that are Clifford-right-linear 
or real-linear.  In particular, we shall use this convention with the 
{\it Dirac operator} $D = \sum_{i=0}^n e_i {\partial \over \partial
x_i}$ and its adjoint
$\overline{D} = \sum_{i=0}^n \overline{e_i} {\partial \over \partial x_i}.$
which act both on the right and the left.  Thus, for example, we have
$fD(x) = \sum_{i=0}^n {\partial f \over \partial x_i}(x) e_i$.

If $f$ is a $C^1$ function such that $Df = 0$
on an open set $V$, then we say that $f$ is {\it
left-monogenic} on $V$.  Similarly, if $fD = 0$ on $V$,
then we say that $f$ is {\it right-monogenic} on $V$.  If
a function is both right-monogenic and left-monogenic on
$V$, then we say it is {\it bi-monogenic} on $V$.  From the formal
identity $\Delta = D\overline{D} = \overline{D}D$, where $\Delta =
\sum_{i=0}^n {\partial^2 \over \partial x_i^2}$ is the Laplacian, and a
Clifford analogue of the Cauchy integral formula, one can show that any
monogenic function will be smooth and harmonic.

The Dirac operator has a fundamental solution known as the
{\it Cauchy kernel} $E$, defined on $\R^{n+1} \backslash \{0\}$ by
$$ E(x) = {\Gamma((n+1)/2) \over 2\pi^{(n+1)/2}} {\overline{x} \over
|x|^{n+1}}.$$
This kernel is locally integrable and can therefore be thought
of as a distribution.  The conjugate kernel $\overline{E}$ is likewise a
fundamental solution of $\overline{D}$; in particular, $\overline{E}\
\overline{D} = \delta$, where $\delta$ is the Dirac distribution and the
identity is understood to be in a distributional sense.

Finally, we note the following Clifford-analytic version of Cauchy's
theorem, for left-monogenic functions.  

\proclaim Theorem {2.1}.  
If $f$ is any left-monogenic function on the neighbourhood of a compact set 
$S$ whose boundary is the union of Lipschitz graphs, then 
$\int_{\partial S} n(y)f(y)\ dS = 0.$

{\heading 3.  The $T(b)$ theorem for Clifford-valued
functions}

Here we state a version of the $T(b)$ theorem, which is a
modification of the theorem proved in [6].  More detail on the concepts
and results sketched here (but with somewhat different notation)
can be found in [5], [6].

In this section we shall fix a weight function $b$ defined on $\R^n$.
For simplicity, we shall make the relatively strong assumption that $b$
is {\it accretive}, by which we mean that
$C^{-1} \leq [b(x)]_0 \leq |b(x)| \leq C$ for all $x \in \R^n$; the
$T(b)$ theorem was proven in [6] using the more general condition of
para-accretivity.

We define a {\it simple function} to be any finite Clifford-linear 
combination of characteristic functions of quasi-cubes; by a quasi-cube we 
mean a dyadic rectangular box such that no side is more than twice as long as 
any other side.  We use $S$ to denote the space of all simple functions,
and endow it with the Clifford-bilinear form
$ \langle f, g\rangle = \int_{\R^n} f(x) b(x) g(x)\ dx.$
This form extends in the obvious manner to the case when one of the
operands is a general element of $L^1_{loc}$.

\defin  A function $K(x,y)$ defined for all $x,y \in \R^n$, $x \not = y$
is said to be a {\it Calder\'on-Zygmund kernel} if there exists $0 <
\delta \leq 1$ such that, for all $x, y, t \in \R^n$ with $t < |y-x|/2$,
the following three estimates hold:
$$\eqalign{
|K(x,y)| &\leq C|x-y|^{-n},\cr
|K(x,y+t)-K(x,y)| &\leq C |t|^\delta |x-y|^{-n-\delta}\hbox{, and}\cr
|K(x+t,y)-K(x,y)| &\leq C |t|^\delta |x-y|^{-n-\delta}.\cr}$$

We now fix $T$ to be a Clifford-right-linear operator from $S$ to 
$L^1_{loc}$ that is
associated to a Calder\'on-Zygmund kernel $K$ in the sense that
$Tg(x) = \int_{spt(f)} K(x,y) b(y) g(y)\ dy$
for all $g \in S$ and almost every  $x$ not in the support of $g$.  We also assume
the existence of a Clifford-left-linear operator $T^*: S \to L^1_{loc}$ 
which is the adjoint of $T$ in the sense that
$\langle f, Tg \rangle = \langle fT^*, g\rangle$
for all $f, g \in S$.  Note that $T^*$ is associated to the transpose
$K^*(x,y) = K(y,x)$ of $K$.

We can now state our version of the $T(b)$ theorem.

\proclaim Theorem {3.1}.  If $b$, $T$, and $T^*$ are as above, then
$T$ is extensible to a bounded
linear operator from $L^2(\R^n)$ to $L^2(\R^n)$
if and only if, for some constant $C$ and all quasi-cubes $I$, one has
\parindent = 30 pt 
{
\item{(i)}  $\int_{I} |(T\ch_I)(x)|\ dx \leq C|I|$ 
\item{(ii)} $\int_{I} |(\ch_IT^*)(x)|\ dx \leq C|I|.$
}
\hfil\break
\parindent = 0 pt
\noindent

\proof  The ``only if'' implication is obvious from the Cauchy-Schwartz
inequality, so we shall assume that (i) and (ii) hold and show that $T$
is $L^2$-bounded on $S$.  We will invoke the $T(b)$ theorem proved in
[6] to do this, but first we must define the linear functional $T1$ and
$1T^*$, as well as a notion of BMO for these functionals.

The left-linear functional $T1$ is defined on the subspace 
$S^b_0 = \{f \in S: \langle f, 1\rangle = 0\}$
on $S$ by requiring that
$$(f)T1 = \lim_{i \to \infty} \langle f, T\ch_{J_i} \rangle = 
\lim_{i \to \infty} \langle fT^*, \ch_{J_i}\rangle$$
for all $f \in S^b_0$ and all nested sequence of quasi-cubes $J_0
\subset J_1 \subset \ldots$;
it is straightforward to verify that $T1$ is
well defined.  We shall say that $T1$ is in BMO if one has
$ |(h)T1| \leq C$
for every {\it left-atom} $h$, that is an element of $S^b_0$ supported
on a quasi-cube $I$ such that $\|h\|_\infty \leq |I|^{-1}$.  In a
symmetric fashion one can define $1T^*$ as a right-linear functional on
the space $\{g \in S: \langle 1, g\rangle = 0\}$ and formulate a similar
criterion for it to be in BMO.  This definition of BMO can be easily
shown to be equivalent to the one in [6] by the usual duality arguments.

From the $T(b)$ theorem of [6], to prove the $L^2$-boundedness of $T$ it
is sufficient to show that $T1, 1T^*$ are both in
BMO, and that $T$ is {\it weakly bounded} in the sense that
$|\langle \ch_I, T\ch_I \rangle| \leq C |I|$
for all quasi-cubes $I$.  This last condition is obviously implied by
both (i) and (ii).  It remains to prove that $T1$ is in BMO; the proof
for $1T^*$ is entirely analogous and will be omitted.  We have to
show that 
$\lim_{i \to \infty} \langle hT^*, \ch_{J_i} \rangle$ is bounded
uniformly for all left-atoms $h$.
We rewrite this expression as
$$|\langle h T^*, \ch_{2I} \rangle + \lim_{i \to \infty} \int_{J_i\backslash
2I} hT^*(x) 
b(x)\ dx|,$$
where $I$ is the support cube of $h$, and estimate the first term by
$$|\langle hT^*, \ch_{2I} \rangle| = |\langle h, T\ch_{2I}\rangle| \leq
C|I|^{-1} \int_I |T\ch_{2I}|\ dx \leq C,$$
by (i).  The second term is majorized by $\int_{2I^c} |hT^*|\ dx$, but
a simple integration by parts argument and the fact that $\int h(x)
b(x)\ dx = 0$ shows that $|hT^*(x)| \leq C \diam(I)^\delta d(x,I)^{-n-
\delta}$ when $x \not \in 2I$,  and by inserting this bound we obtain a 
final estimate of $C$ for the magnitude of the second term.  Thus $T1$ and 
similarly
$1T^*$ are in BMO, and so $T$ is $L^2$-bounded, as required.
\hfill$\square$

{\heading 4.  $L^2$-boundedness of $T$ for
bi-monogenic kernels}

In this section, we define the convolution operators $T$
and $T^*$ of the type used in [7] and outlined in (1.1). 
We then use the modified $T(b)$ theorem to
prove $L^2$-boundedness of $T$ in the case in which the
kernel $\phi$ is bi-monogenic.  The method of proof is due to T. Qian.

Let $\Sigma, \rho, A$ be as in the introduction.  
For each angle $0<\alpha<\pi/2$ we define the sectors
$ S_\alpha = \{x: |x_0| < |{\bf x}| \tan \alpha\}$
and the cones
$ C_\alpha = \{x: x_0 > |{\bf x}| \cot \alpha\}$, and
define the function $\psi$ on $\R^n$ by
$\psi = e_0 - \nabla A$;
since $A$ is a Lipschitz function, $\psi$ will be defined almost everywhere, 
and is an accretive function taking values in $C_\omega$.
Furthermore, one has
$$\int_\Sigma g(y)n(y)f(y)\ dS =
\int_{\R^n} g\Circ \rho(x)\psi(x) f\Circ \rho(x)\
dx$$ 
whenever $f$ and $g$ are such that the above integrals are absolutely
convergent.

\defin  Suppose that $\phi$ is a continuous function
on $S_\sigma$, and $\underline{\phi}$ is a function on $C_\sigma$, where
$\sigma$ is some angle greater than $\omega$.  
We say that $\underline{\phi}$ is a {\it bounded potential
function} for $\underline{\phi}$ if $\underline{\phi}$ is
uniformly bounded and
satisfies 
$$ \underline{\phi}(Rn) - \underline{\phi}(rn) =
\int_{\langle x,n\rangle = 0, r < |x| < R} \phi(x)\ dS$$ 
for all $R>r>0$ and unit vectors $n \in
C_\sigma$.

\defin  A function $\phi$ is said to be a 
{\it harmonic kernel} if, for some $\sigma > \omega$, 
$\phi$ is harmonic in $S_\sigma$, one has
$|\phi(x)| \leq C |x|^{-n}$ for all $x$ in $S_\sigma$, and $\phi$ has at
least one bounded potential $\underline{\phi}$.  Similarly we define the
notions of left-monogenic, right-monogenic, and bi-monogenic kernel.

From the usual estimates on harmonic functions (e.g. the Poisson
integral formula) one can show that if $\phi$ is a harmonic
kernel then for every $\alpha \leq \sigma$ there is a
constant $C$ such that
$|\nabla \phi(x)| \leq C |x|^{-n-1}$ for all $x$ in the cone $C_\alpha$.

Let us now fix $\phi$ to be a bi-monogenic kernel.
We consider the right-linear operator $T = T_{\phi}$ defined 
in (0.1), and also define the left-linear operator $T^*$ by
$$ fT^*(y) = \lim_{\varepsilon \to 0} \bigl( \int_{\Sigma, |y-x|>\varepsilon}
f(x) n(x) \phi(x-y)\ dS_x + f(y) n(y)
\underline{\phi}(\varepsilon n(y)) \bigr).$$
A straightforward argument using Cauchy's theorem and the
almost-everywhere differentiability of $A$ shows that $Tf$ and 
$fT^*$ are well-defined a.e. whenever $f \circ \rho$ is a simple 
function.
When $\phi$
is merely a harmonic kernel it is not obvious
that these operators are well-defined, but this will follow from the
decomposition results of the next section. 

We note that $T$ is not uniquely defined because there is some
arbitrariness in the choice of $\underline{\phi}$.  However, this
freedom has no impact on the question of $L^2$-boundedness, since the
difference between two $T_\phi$ given by different bounded potentials
when applied to the same function $f$ is majorized by a constant multiple 
of $|f|$.  We shall take advantage of this in the sequel by adjusting
the bounded potentials whenever necessary.

Since $T$ and $T^*$ act on
functions on $\Sigma$ instead of $\R^n$, one must conjugate them by
$\rho$ to place them in the form discussed in the previous section.
More precisely, if we define the operators $T^\rho$ and $T^{*\rho}$ by
$T^\rho(f\Circ\rho) = (Tf) \Circ \rho$ and
$(f\Circ\rho)T^{*\rho} = (fT^*) \Circ \rho$, then $T^\rho$ is
associated with the Calder\'on-Zygmund kernel 
$K(x, y) = \phi(\rho(x)-\rho(y))$
with $\delta = 1$ and $b = \psi$, and $T^{*\rho}$ is its adjoint in the 
sense of Section 3.  These observations are easily verified and in any
event are proven in [12].

We will now show that $T^\rho$, and consequently $T$, is an
$L^2$-bounded operator.  To do this we will show that 
$$\int_I |(T^\rho \ch_I)(x)|\ dx \leq C|I|\eqno(4.1)$$
for all quasi-cubes $I$; this estimate, together with a similarly proved
one for $T^*_\rho$, will yield $L^2$-boundedness by Theorem 3.1.

Fix a quasi-cube $I$, and consider the quantity $(T^\rho \ch_I)(x) =
(T\ch_{\rho(I)})(\rho(x))$ for $x \in I$.
Denote by $T_\varepsilon$ the term inside the limit in the definition
of $T\ch_{\rho(I)}(\rho(x))$.  Then by elementary estimates,
the term $T_\varepsilon$
is bounded in modulus 
for $\varepsilon \leq d(x, \partial I)$ by
$O(\log(\diam(I)/\varepsilon))$.
By Cauchy's theorem, the error between $T_{\varepsilon_1}$ 
and $T_{\varepsilon_2}$ is equal to the integral of $\phi$ over a 
portion of the sphere of radius $\varepsilon_1$ and a similar portion on
$\varepsilon_2$.  By the bounds on $\phi$,
this error is bounded independently of $\varepsilon_1$ and
$\varepsilon_2$.  Hence, by taking $\varepsilon_2$
arbitrarily close to 0 and $\varepsilon_1$ arbitrarily close to $\dist(x,
\partial I)$, we obtain
$(T^\rho \ch_I)(x) = O(\log(\diam(I)/\dist(x,\partial I)))$
for almost every $x$. 
Integrating this over $I$ yields (4.1) as desired.  Combining this with
the analogous estimate for $T^{*\rho}$ and using Theorem 3.1, we have
thus proved

\proclaim Theorem {4.1}.  If $\phi$ is a bi-monogenic kernel
then $T_\phi$ is extensible to an $L^2$-bounded 
operator.

{\heading 5.  Right-monogenic and harmonic decomposition}

In this section, we prove that any right-monogenic
kernel can be written as a
left-linear combination 
of bi-monogenic kernels, thus proving the $L^2$-boundedness of operators 
generated by
right-monogenic kernels as a consequence of the results of
Section 4.  The decomposition is based on the observation that, since
$\overline{D}$ has a fundamental solution $\overline{E}$, every smooth
function in $\R^{n+1}$ with a reasonable decay has a primitive with
respect to $\overline{D}$
(Alan McIntosh [9] has noted that one can also get this decomposition by
using the Clifford-valued Fourier transform developed in [8]).
We then express an arbitrary scalar harmonic kernel as the scalar part of
a right-monogenic kernel $g$, thus proving the
$L^2$-boundedness of operators generated by harmonic
kernels.  The construction of $g$ is based on a
form of Poincar\'e's second lemma which appears in [2]. 

We begin with the treatment of right-monogenic kernels.  If 
the Clifford algebra is commutative, then every right-monogenic kernel is 
bi-monogenic, and we may 
use the results of the previous section.  Thus we can restrict our
attention to the case $n>1$.  Suppose that
$\phi$ is a right-monogenic kernel defined on the sector $S_\sigma$.  We will
extend $\phi$ to all of $\R^{n+1}$ by declaring it to be zero outside of
$S_\sigma$.  Choose any angle $\alpha$ such that $\omega < \alpha <
\sigma$ and define the function $F$ on $S_\alpha$ by 
$$F(x) = \phi*\overline{E}(x) = \int_{\R^{n+1}} \phi(y)
\overline{E}(x-y)\ dy.$$
Since $\overline{E}$ is locally integrable and $\phi$ and $\overline{E}$
both decay as $O(|x|^{-n})$, one can show that $F$ is well-defined and
smooth on this cone; in fact, one can use crude estimates to show that $F(x) =
O(|x|^{-n+1})$ for all $x \in S_\alpha$.  Since $\nabla F = (\nabla \phi)
* \overline{E}$ and $\nabla \phi$ decays like $O(|x|^{-n-1})$ on every
sub-sector of $S_\sigma$, we have the estimate $\nabla F(x) =
O(|x|^{-n})$ for all $x \in S_\alpha$.

Furthermore, since $\overline{E}$ is the fundamental solution of
$\overline{D}$, we have that $F\overline{D} = \phi$.  Since $\phi$ was
assumed to be right-monogenic, this means that $\Delta F =
F\overline{D}D = \phi D = 0$, so that $F$ is harmonic.  In
particular, all the components $[F]_I$ of $F$ are harmonic.

We can decompose $\phi$ on $S_\alpha$ as $\phi = \sum_{I \in \cal I} e_I g_I$, 
where $g_I = [F]_I \overline{D}$.  Since
$[F]_I$ is harmonic, $g_I$ is right-monogenic.  Since $[F]_I$ is scalar
and thus commutes with all elements of the Clifford algebra, we have
that $g_I = \overline{D} [F]_I$, so that $g_I$ is also left-monogenic.
Thus each $g_I$ is bi-monogenic.  From the estimates on the derivative
of $F$, we also see that $g_I(x) = O(|x|^{-n})$ for all $x$ in
$S_\alpha$.  Thus, if we can demonstrate that each $g_I$ has at least
one bounded potential, then the $g_I$ will all be bi-monogenic kernels
and so $T_{g_I}$ will be $L^2$-bounded for each $I \in {\cal I}$.  Since
$T_\phi$ will equal $\sum_I e_I T_{g_I}$ after an unimportant adjustment
in potentials, we would have thus proven that $T_\phi$ is bounded on
$L^2$.

It remains only to show that each $g_I$ has a bounded potential.  From
the definition of a potential, it is clearly sufficient to show that
$$ \int_{\langle x,n\rangle = 0, r < |x| < R} g_I(x)\ dS$$
is bounded uniformly in $n, r, R$.  From Cauchy's theorem and the fact
that $g_I$ is right-monogenic and is $O(|x|^{-n})$, it suffices to
prove the above estimate in the special case $n = e_0$.  In other words,
we need only show that
$$ \int_{x \in \R^n, r < |x| < R} g_I(x) dS = O(1),$$
where $O(1)$ denotes a quantity bounded in magnitude by a constant $C$
that is independent of $r,R$.
To prove this, we start with the corresponding estimate
$$ \int_{x \in \R^n, r < |x| < R} \phi(x)\ dx = O(1)$$
for $\phi$, which is true since $\phi$ has a bounded
potential.  Since $\phi = F \overline{D}$, we thus have
$$ \sum_{i = 0}^n \bigl(\int_{x \in \R^n, r < |x| < R}
{\partial F(x) \over \partial x_i} \ dx\ \overline{e}_i\bigr)
= O(1).\eqno(5.1)$$
However, by Stokes' theorem and the decay of $F$, we have
$$\int_{x \in \R^n, r < |x| < R}
{\partial F(x) \over \partial x_i} \ dx
= O(1)\eqno(5.2)$$
whenever $i$ is non-zero.  By comparing this with (5.1), we see that (5.2)
must also be true when $i$ is zero.  If we now take the $e_I$ components
of (5.2) for all $i$, multiply them by $\overline{e_i}$, and add, we obtain
$$ \int_{x \in R^n, r<|x|<R} g_I(x)\ dx =
\int_{x \in \R^n, r < |x| < R} \sum_{i = 0}^n
{\partial F_I(x) \over \partial x_i}
\overline{e_i}\ dx = O(1),$$
as desired.  Thus each $g_I$ has a bounded potential, and we have
proved

\proclaim Theorem {5.1}.  If $\phi$ is a right-monogenic kernel,
then $T_\phi$ is extensible to an $L^2$-bounded operator.

We now consider the analogue of Theorem 5.1 for harmonic kernels.  By
linearity, we can restrict our attention to scalar-valued kernels $\phi$.
Fix any $\alpha$ such that $\sigma > \alpha >
\omega$.  We shall construct a right-monogenic kernel $g$ on
$S_\alpha$ whose scalar part is equal 
to $\phi$.  Before we do this, let us see how this would
imply the $L^2$-boundedness of $T_\phi$.  By the theorem just proved,
$T_g$ is bounded on $L^2$.  By linearity and the fact that
the weight $\psi$ is invertible, we only need
verify the boundedness of $T_\phi$ on functions $f$ for which $\psi f$
is scalar-valued.  But for these functions $T_\phi f$ is equal to the
scalar part of $T_g f$ (after an unimportant adjustment in potentials).
Thus $T_\phi$ is bounded for these functions, and hence on all of
$L^2$.

To construct this $g$, we start by considering the auxillary function
$f$ on $S_\alpha$ defined by
$f(x) = (1-n) \phi(x) - x (\phi\overline{D}(x)).$ 
By the estimates of the decay of $\phi$, we see that $f(x) =
O(|x|^{-n})$ for all $x$ in $S_\alpha$.
Also, $f$ is right-monogenic; indeed, we have
$$\eqalign{
fD(x) &= (1-n) \phi D(x) - (x \phi\overline{D}(x))D\cr
&=  (1-n) \phi D(x) - \sum_{i = 0}^n e_i \phi \overline{D}(x)e_i - x
(\phi\overline{D}D)\cr
&=  (1-n) \phi D(x) - \sum_{i = 0}^n e_i
\phi \overline{D}(x)e_i\cr
&= (1-n) \phi D(x) - \sum_{j = 0}^n {\partial \phi(x) \over \partial x_j}
\sum_{i = 0}^n e_i\overline{e_j}e_i\cr
&= 0,\cr}$$
where we have used the identity $\sum_{i = 0}^n e_i \overline{e}_j
e_i = (1-n) e_j$ for all $j = 0, 1, \ldots n$,
which is easily verifiable from the Clifford
multiplication laws.  Note that $\partial \phi(x) /
\partial x_j$ is scalar-valued and hence commutes with
all elements of the Clifford algebra.

We now define $g$ on $S_\alpha$ by the formula
$$g(x) = \int_1^\infty t^{n-2} f(tx)\ dt.$$
It is a routine matter to check that the integrand in the definition of
$g(x)$ has a scalar part equal to $-{d \over dt}(t^{n-1}\phi(tx))$, so
that $\phi$ is the scalar part of
$g$.  Furthermore, from the decay and right-monogenicity of $f$ we see that 
$g$ is also right-monogenic with the same decay.  Thus $g$ will have all
the properties needed to prove the $L^2$-boundedness of $T_\phi$, once
we show that $g$ has a bounded potential.

To do this, it will suffice as before to show that
$\int_{x \in \R^n, r < |x| < R} g(x)\ dx$
is bounded.  Fix $r$ and $R$, and denote the preceding integral by $P$.  Since
$[g]_0 = \phi$ has a bounded potential, we know that the scalar part of
$P$ is bounded.  More generally, we can use Cauchy's theorem and the
decay of $g$ to rewrite $P$ as
$$ P = \int_{\langle x,n\rangle = 0, r<|x|<R} g(x) n\ dS + O(1)$$
whenever $n \in C_\alpha$ is such that $|n|=1$, and use the
fact that $\phi$ has a bounded potential to conclude that the
scalar part of $P\overline n$ is bounded for all such $n$.  By linearity 
this means that
$\langle P, m\rangle = O(|m|)$ for all $m$ in $\R^{n+1}$.  Now, if $P$ were 
in $\R^{n+1}$, then the reverse Cauchy-Schwartz inequality would give us
that $P$ was bounded, and we would be done.  It turns out that $P$ is
indeed in $\R^{n+1}$; we shall prove this by explicitly computing the
components of $P$.  We start by writing $f$ as
$$ f(x) = {\bf f}(x) + \sum\sum_{1 \leq i < j \leq n} f_{ij}(x) e_i e_j,$$
where ${\bf f} \in \R^{n+1}$, and 
$ f_{ij}(x) = x_j {\partial \phi \over \partial x_i}(x) -
x_i {\partial \phi \over \partial x_j}(x).$
Thus, we have
$$ P = {\bf P} + \sum\sum_{1 \leq i < j \leq n} P_{ij} e_i e_j,$$
where ${\bf P} \in \R^{n+1}$ and
$$\eqalign{
P_{ij} &= \int_{x \in \R^n, r<|x|<R} g_{ij}(x)\
dx\cr
&= \int_r^R \int_{S^{d-1}} s^{d-1} g_{ij}(s\omega)\
d\omega ds\cr
&= \int_r^R \int_{S^{d-1}} s^{d-1} \int_1^\infty t^{d-2}
f_{ij}(ts\omega)\ dt d\omega ds\cr
&=\int_r^R \int_{S^{d-1}} \int_s^\infty t^{d-2}
f_{ij}(t\omega)\ dtd\omega ds\cr
&= \int_r^R \int_{x \in \R^n, |x|>s} |x|^{-1} f_{ij}(x)\
dx ds\cr
&= \int_r^R
\int_{x \in \R^n, |x|>s} {x_j \over |x|} {\partial
\phi \over \partial x_i}(x) - {x_i \over |x|} {\partial \phi
\over \partial x_j}(x)\ dx ds\cr
&= \int_r^R \int_{|x|>s} {\rm div}\bigl({(x_j e_i - x_i e_j)\phi(x) \over
|x|}\bigr)\ dxds\cr
&= \int_r^R 0\ ds\cr}$$
as desired.  By the earlier discussion, we have thus proved

\proclaim Theorem {5.2}.  If $\phi$ is a harmonic kernel, then $T_\phi$ is
extensible to an $L^2$-bounded operator.

{\heading 6.  References}

[1] L. Andersson, B. Jawerth, M. Mitrea, {\it The Cauchy Singular Integral
Operator and Clifford Wavelets}.  Wavelets: mathematics and applications,
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[2] F. Brackx, R. Delanghe and F. Sommen, \ldq Clifford
Analysis\rdq.  Research Notes in Mathematics.  Vol.
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[3] R. R. Coifman, P.W. Jones and S. Semmes, {\it Two
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[4] R. R. Coifman, A. McIntosh and Y. Meyer, {\it
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[5] G. David and J.-L. Journ\'e, {\it A boundedness
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[6] G.I. Gaudry, R. Long and T. Qian, {\it A martingale
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[7] C. Li, A. McIntosh and S. Semmes, {\it Convolution
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[8] C. Li, A. McIntosh and T. Qian, {\it Clifford Algebra, Fourier
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[9] A. McIntosh, {\it Clifford algebras, Fourier theory, singular
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Clifford algebras in analysis and related topics (Fayetteville, AR, 1993), 
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[10] M. Mitrea, {\it Clifford wavelets, singular integrals, and Hardy
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[11] M.A.M. Murray, {\it The Cauchy integral, Calder\'on
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[12] T. Tao, {\it Convolution operators generated by right-monogenic and harmonic kernels}, M. Sc. Thesis, Flinders University, 1992.

\vglue 2 cm

\leftline{Terence C. Tao}
\leftline{Department of Mathematics}
\leftline{UCLA}
\leftline{Los Angeles, CA 90095-1555}
\leftline{U.S.A.}
\leftline{tao@math.ucla.edu}

\end
