% This paper is typeset in plain TeX, not AMS-TeX! \input amssym.def \input amssym.tex \parindent = 0 pt \parskip = 12 pt \baselineskip = 18 pt \font \heading = cmbx10 at 10 true pt \font \bigheading = cmbx7 at 14 true pt \font \medheading =cmbx7 at 10 true pt \font \small = cmr7 at 7 true pt \def \script{\cal} \def \R{{\bf R}} \def \r#1{{\R}^{#1}} \def \Ad{{{\bf A}_n}} \def \proof {{\it Proof. }} \def \diam {{\rm diam}} \def \dist {{\rm dist}} \def \ch {{\raise 0.5ex \hbox{$\chi$}}} \def \ldq {\lq\lq} \def \rdq {\rq\rq} \def \Circ {{\scriptstyle \circ}} \def \defin {{\bf Definition. }} \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 $$ \int_{x \in R^n, r<|x| \alpha > \omega$. We shall construct a right-monogenic kernel $g$ on $S_\alpha$ whose scalar part is equal to $\phi$. Before we do this, let us see how this would imply the $L^2$-boundedness of $T_\phi$. By the theorem just proved, $T_g$ is bounded on $L^2$. By linearity and the fact that the weight $\psi$ is invertible, we only need verify the boundedness of $T_\phi$ on functions $f$ for which $\psi f$ is scalar-valued. But for these functions $T_\phi f$ is equal to the scalar part of $T_g f$ (after an unimportant adjustment in potentials). Thus $T_\phi$ is bounded for these functions, and hence on all of $L^2$. To construct this $g$, we start by considering the auxillary function $f$ on $S_\alpha$ defined by $f(x) = (1-n) \phi(x) - x (\phi\overline{D}(x)).$ By the estimates of the decay of $\phi$, we see that $f(x) = O(|x|^{-n})$ for all $x$ in $S_\alpha$. Also, $f$ is right-monogenic; indeed, we have $$\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|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, 525--546, Stud. Appl. Math., CRC Press. 1994. [2] F. Brackx, R. Delanghe and F. Sommen, \ldq Clifford Analysis\rdq. Research Notes in Mathematics. Vol. 76. Pitman Advanced Publishing Company. 1982. [3] R. R. Coifman, P.W. Jones and S. Semmes, {\it Two elementary proofs of the $L_2$ boundedness of Cauchy integrals on Lipschitz curves}. J. Amer. Math. Soc. {\bf 2} (1989), 553--564. [4] R. R. Coifman, A. McIntosh and Y. Meyer, {\it L'int\'egrale de Cauchy d\'efinit un op\'erateur born\'e sur $L^2$ pour les courbes lipschitziennes}. Annals of Math. {\bf 116} (1982), 361--387. [5] G. David and J.-L. 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Springer-Verlag, Berlin, 1994. [11] M.A.M. Murray, {\it The Cauchy integral, Calder\'on commutators and conjugations of singular integrals in $\R^n$}. Trans. Amer. Math. Soc. {\bf 289} (1985), 497--518. [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