Convolution operators on Lipschitz graphs with harmonic kernels

Terence Tao

Abstract

We use a Clifford-algebra version of the T (b) theorem to prove the L^{2}-boundedness of convolution operators with harmonic Calderón-Zygmund kernels on Lipschitz graphs. This extends the results of [7].

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

1. Introduction

Let n be a positive integer and 0 < ω < π/2 be a fixed angle. Suppose that Σ is a Lipschitz graph in R^{n+1}, by which we mean a manifold of the form ρ(R^{n}), where ρ(x) = x + A(x)e_{0} and A is a real function on R^{n} such that |A(x) − A(y)| ≤ |x − y| tan ω for all x, y. We consider (Clifford-valued) right-linear operators of the form

T f(x) = lim_{ε→0}(∫_{Σ,|x−y|>ε} ϕ(x − y)n(y)f(y) dS_{y} + \underline{ϕ}(εn(x))n(x)f(x)),  (1.1)

where ϕ is a Clifford-valued harmonic Calderón-Zygmund kernel on a certain sector R^{n+1}, and \underline{ϕ} is a bounded potential of ϕ (this term can be omitted if ϕ 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 ϕ 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 ϕ 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 In an unpublished note T. Qian has shown that this theorem can be used to show the boundedness of the operator (1.1) under the relatively strong assumption that ϕ is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.

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 ω. 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 ∂X to denote the boundary of X, and |X| to denote the Lebesgue measure of X.

We give R^{n+1} = span{e_{0}, e_{1}, …, e_{n}} and the subspace R^{n} = span{e_{1}, …, e_{n}} the usual Euclidean metric, and imbed them in the the Clifford algebra A_{n}, 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 ∈ R^{n}. Alternatively, one may use the equivalent set of relations e_{i}e_{j}+e_{j}e_{i} = −2δ_{ij}e_{0} for all 1 ≤ i, j ≤ 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 ∈ I} for A_{n} that extends the basis for R^{n+1}; if x is an element of A_{n}, 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 ⟨x, y⟩ = [x\overline{y}]_{0}, \overline{xy} = \overline{y} \overline{x}, and |xy| ≤ C|x| |y| for all x, y ∈ A_{n}.

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 T f 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 Dirac operator D = ∑_{i=0}^{n} e_{i} \frac{∂}{∂x_{i}} and its adjoint \overline{D} = ∑_{i=0}^{n} \overline{e_{i}} \frac{∂}{∂x_{i}} . which act both on the right and the left. Thus, for example, we have fD(x) = ∑_{i=0}^{n} \frac{∂f}{∂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 left-monogenic on V. Similarly, if fD = 0 on V, then we say that f is right-monogenic on V. If a function is both right-monogenic and left-monogenic on V, then we say it is bi-monogenic on V. From the formal identity Δ = D\overline{D} = \overline{D}D, where Δ = ∑_{i=0}^{n} \frac{∂^{2}}{∂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 Cauchy kernel E, defined on R^{n+1}\{0} by

E(x) = \frac{Γ((n + 1)/2)}{2π^{(n+1)/2}} \frac{\overline{x}}{|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} = δ, where δ 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.

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 ∫_{∂S} n(y)f(y) dS = 0.

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 accretive, by which we mean that C^{−1} ≤ [b(x)]_{0} ≤ |b(x)| ≤ C for all x ∈ R^{n}; the T (b) theorem was proven in [6] using the more general condition of para-accretivity.

We define a simple function to be any finite Clifford-linear combination of characteristic functions of quasicubes; 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 ⟨f, g⟩ = ∫_{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_{loc}^{1}.

Definition. A function K(x, y) defined for all x, y ∈ R^{n}, x ≠ y is said to be a Calderón-Zygmund kernel if there exists 0 < δ ≤ 1 such that, for all x, y, t ∈ R^{n} with t < |y − x|/2, the following three estimates hold:

\begin{aligned}|K(x, y)| & ≤ C|x − y|^{−n}, \\ |K(x, y + t) − K(x, y)| & ≤ C|t|^{δ}|x − y|^{−n−δ}, and \\ |K(x + t, y) − K(x, y)| & ≤ C|t|^{δ}|x − y|^{−n−δ}. \end{aligned}

We now fix T to be a Clifford-right-linear operator from S to L_{loc}^{1} that is associated to a Calderón-Zygmund kernel K in the sense that T g(x) = ∫_{spt(f)} K(x, y)b(y)g(y) dy for all g ∈ S and almost every x not in the support of g. We also assume the existence of a Clifford-left-linear operator T^{∗} : S → L_{loc}^{1} which is the adjoint of T in the sense that ⟨f, T g⟩ = ⟨fT^{∗}, g⟩ for all f, g ∈ 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.

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

(i) ∫_{I} |(T χ_{I})(x)| dx ≤ C|I|

(ii) ∫_{I} |(χ_{I}T^{∗})(x)| dx ≤ C|I|.

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 T 1 and 1T^{∗}, as well as a notion of BMO for these functionals.

The left-linear functional T 1 is defined on the subspace S_{0}^{b} = {f ∈ S : ⟨f, 1⟩ = 0} on S by requiring that

(f)T 1 = lim_{i→∞}⟨f, T χ_{J_{i}}⟩ = lim_{i→∞}⟨fT^{∗}, χ_{J_{i}}⟩

for all f ∈ S_{0}^{b} and all nested sequence of quasi-cubes J_{0} ⊂ J_{1} ⊂ …; it is straightforward to verify that T 1 is well defined. We shall say that T 1 is in BMO if one has |(h)T 1| ≤ C for every left-atom h, that is an element of S_{0}^{b} supported on a quasi-cube I such that ∥h∥_{∞} ≤ |I|^{−1}. In a symmetric fashion one can define 1T^{∗} as a right-linear functional on the space {g ∈ S : ⟨1, g⟩ = 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 T 1, 1T^{∗} are both in BMO, and that T is weakly bounded in the sense that |⟨χ_{I}, T χ_{I}⟩| ≤ C|I| for all quasi-cubes I. This last condition is obviously implied by both (i) and (ii). It remains to prove that T 1 is in BMO; the proof for 1T^{∗} is entirely analogous and will be omitted. We have to show that lim_{i→∞}⟨hT^{∗}, χ_{J_{i}}⟩ is bounded uniformly for all left-atoms h. We rewrite this expression as

|⟨hT^{∗}, χ_{2I}⟩ + lim_{i→∞} ∫_{J_{i}\2I} hT^{∗}(x)b(x) dx|,

where I is the support cube of h, and estimate the first term by

|⟨hT^{∗}, χ_{2I}⟩| = |⟨h, T χ_{2I}⟩| ≤ C|I|^{−1} ∫_{I} |T χ_{2I}| dx ≤ C,

by (i). The second term is majorized by ∫_{2I^{c}} |hT^{∗}| dx, but a simple integration by parts argument and the fact that ∫ h(x)b(x) dx = 0 shows that |hT^{∗}(x)| ≤ Cdiam(I)^{δ}d(x, I)^{−n−δ} when x ∉ 2I, and by inserting this bound we obtain a final estimate of C for the magnitude of the second term. Thus T 1 and similarly 1T^{∗} are in BMO, and so T is L^{2}-bounded, as required. □

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 ϕ is bi-monogenic. The method of proof is due to T. Qian.

Let Σ, ρ, A be as in the introduction. For each angle 0 < α < π/2 we define the sectors S_{α} = {x : |x_{0}| < |x| tan α} and the cones C_{α} = {x : x_{0} > |x| cot α}, and define the function ψ on R^{n} by ψ = e_{0} − ∇A; since A is a Lipschitz function, ψ will be defined almost everywhere, and is an accretive function taking values in C_{ω}. Furthermore, one has

∫_{Σ} g(y)n(y)f(y) dS = ∫_{R^{n}} g∘ρ(x)ψ(x)f ∘ρ(x) dx

whenever f and g are such that the above integrals are absolutely convergent.

Definition. Suppose that ϕ is a continuous function on S_{σ}, and \underline{ϕ} is a function on C_{σ}, where σ is some angle greater than ω. We say that \underline{ϕ} is a bounded potential function for \underline{ϕ} if \underline{ϕ} is uniformly bounded and satisfies

\underline{ϕ}(Rn) − \underline{ϕ}(rn) = ∫_{⟨x,n⟩=0,r<|x|<R} ϕ(x) dS

for all R > r > 0 and unit vectors n ∈ C_{σ}.

Definition. A function ϕ is said to be a harmonic kernel if, for some σ > ω, ϕ is harmonic in S_{σ}, one has |ϕ(x)| ≤ C|x|^{−n} for all x in S_{σ}, and ϕ has at least one bounded potential \underline{ϕ}. 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 ϕ is a harmonic kernel then for every α ≤ σ there is a constant C such that |∇ϕ(x)| ≤ C|x|^{−n−1} for all x in the cone C_{α}.

Let us now fix ϕ to be a bi-monogenic kernel. We consider the right-linear operator T = T_{ϕ} defined in (0.1), and also define the left-linear operator T^{∗} by

fT^{∗}(y) = lim_{ε→0}(∫_{Σ,|y−x|>ε} f(x)n(x)ϕ(x − y) dS_{x} + f(y)n(y)\underline{ϕ}(εn(y))).

A straightforward argument using Cauchy’s theorem and the almost-everywhere differentiability of A shows that T f and fT^{∗} are well-defined a.e. whenever f ∘ ρ is a simple function. When ϕ 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{ϕ}. However, this freedom has no impact on the question of L^{2}-boundedness, since the difference between two T_{ϕ} 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 Σ instead of R^{n}, one must conjugate them by ρ to place them in the form discussed in the previous section. More precisely, if we define the operators T^{ρ} and T^{∗ρ} by T^{ρ}(f ∘ρ) = (T f)∘ρ and (f ∘ρ)T^{∗ρ} = (fT^{∗})∘ρ, then T^{ρ} is associated with the Calderón-Zygmund kernel K(x, y) = ϕ(ρ(x) − ρ(y)) with δ = 1 and b = ψ, and T^{∗ρ} 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^{ρ}, and consequently T, is an L^{2}-bounded operator. To do this we will show that

∫_{I} |(T^{ρ}χ_{I})(x)| dx ≤ C|I|  (4.1)

for all quasi-cubes I; this estimate, together with a similarly proved one for T_{ρ}^{∗}, will yield L^{2}-boundedness by Theorem 3.1.

Fix a quasi-cube I, and consider the quantity (T^{ρ}χ_{I})(x) = (T χ_{ρ(I)})(ρ(x)) for x ∈ I. Denote by T_{ε} the term inside the limit in the definition of T χ_{ρ(I)}(ρ(x)). Then by elementary estimates, the term T_{ε} is bounded in modulus for ε ≤ d(x, ∂I) by O(log(diam(I)/ε)). By Cauchy’s theorem, the error between T_{ε_{1}} and T_{ε_{2}} is equal to the integral of ϕ over a portion of the sphere of radius ε_{1} and a similar portion on ε_{2}. By the bounds on ϕ, this error is bounded independently of ε_{1} and ε_{2}. Hence, by taking ε_{2} arbitrarily close to 0 and ε_{1} arbitrarily close to dist(x, ∂I), we obtain (T^{ρ}χ_{I})(x) = O(log(diam(I)/dist(x, ∂I))) for almost every x. Integrating this over I yields (4.1) as desired. Combining this with the analogous estimate for T^{∗ρ} and using Theorem 3.1, we have thus proved

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

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 bimonogenic 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 Poincaré’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 ϕ is a right-monogenic kernel defined on the sector S_{σ}. We will extend ϕ to all of R^{n+1} by declaring it to be zero outside of S_{σ}. Choose any angle α such that ω < α < σ and define the function F on S_{α} by

F (x) = ϕ ∗ \overline{E}(x) = ∫_{R^{n+1}} ϕ(y)\overline{E}(x − y) dy.

Since \overline{E} is locally integrable and ϕ 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 ∈ S_{α}. Since ∇F = (∇ϕ) ∗ \overline{E} and ∇ϕ decays like O(|x|^{−n−1}) on every sub-sector of S_{σ}, we have the estimate ∇F (x) = O(|x|^{−n}) for all x ∈ S_{α}.

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

We can decompose ϕ on S_{α} as ϕ = ∑_{I∈I} e_{I}g_{I}, where g_{I} = [F]_{I} \overline{D}. Since [F]_{I} is harmonic, g_{I} is rightmonogenic. 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_{α}. 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 ∈ I. Since T_{ϕ} will equal ∑_{I} e_{I}T_{g_{I}} after an unimportant adjustment in potentials, we would have thus proven that T_{ϕ} 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

∫_{⟨x,n⟩=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

∫_{x∈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

∫_{x∈R^{n},r<|x|<R} ϕ(x) dx = O(1)

for ϕ, which is true since ϕ has a bounded potential. Since ϕ = F \overline{D}, we thus have

∑_{i=0}^{n}(∫_{x∈R^{n},r<|x|<R} \frac{∂F (x)}{∂x_{i}} dx \overline{e}_{i}) = O(1).  (5.1)

However, by Stokes’ theorem and the decay of F, we have

∫_{x∈R^{n},r<|x|<R} \frac{∂F (x)}{∂x_{i}} dx = O(1)  (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

∫_{x∈R^{n},r<|x|<R} g_{I}(x) dx = ∫_{x∈R^{n},r<|x|<R} ∑_{i=0}^{n} \frac{∂F_{I}(x)}{∂x_{i}} \overline{e_{i}} dx = O(1),

as desired. Thus each g_{I} has a bounded potential, and we have proved

Theorem 5.1. If ϕ is a right-monogenic kernel, then T_{ϕ} 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 ϕ. Fix any α such that σ > α > ω. We shall construct a right-monogenic kernel g on S_{α} whose scalar part is equal to ϕ. Before we do this, let us see how this would imply the L^{2}-boundedness of T_{ϕ}. By the theorem just proved, T_{g} is bounded on L^{2}. By linearity and the fact that the weight ψ is invertible, we only need verify the boundedness of T_{ϕ} on functions f for which ψf is scalar-valued. But for these functions T_{ϕ}f is equal to the scalar part of T_{g}f (after an unimportant adjustment in potentials). Thus T_{ϕ} 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_{α} defined by f(x) = (1 − n)ϕ(x) − x(ϕ\overline{D}(x)). By the estimates of the decay of ϕ, we see that f(x) = O(|x|^{−n}) for all x in S_{α}. Also, f is right-monogenic; indeed, we have

fD(x) = (1 − n)ϕD(x) − (xϕ\overline{D}(x))D

= (1 − n)ϕD(x) − ∑_{i=0}^{n} e_{i}ϕ\overline{D}(x)e_{i} − x(ϕ\overline{D}D)

= (1 − n)ϕD(x) − ∑_{i=0}^{n} e_{i}ϕ\overline{D}(x)e_{i}

= (1 − n)ϕD(x) − ∑_{j=0}^{n} \frac{∂ϕ(x)}{∂x_{j}} ∑_{i=0}^{n} e_{i}\overline{e_{j}}e_{i}

= 0, where we have used the identity ∑_{i=0}^{n} e_{i}\overline{e}_{j}e_{i} = (1 − n)e_{j} for all j = 0, 1, … n, which is easily verifiable from the Clifford multiplication laws. Note that ∂ϕ(x)/∂x_{j} is scalar-valued and hence commutes with all elements of the Clifford algebra.

We now define g on S_{α} by the formula

g(x) = ∫_{1}^{∞} 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 −\frac{d}{dt}(t^{n−1}ϕ(tx)), so that ϕ 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_{ϕ}, once we show that g has a bounded potential.

To do this, it will suffice as before to show that ∫_{x∈R^{n},r<|x|<R} g(x) dx is bounded. Fix r and R, and denote the preceding integral by P. Since [g]_{0} = ϕ 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 = ∫_{⟨x,n⟩=0,r<|x|<R} g(x)n dS + O(1)

whenever n ∈ C_{α} is such that |n| = 1, and use the fact that ϕ 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 ⟨P, m⟩ = 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) = f(x) + ∑ ∑_{1≤i<j≤n} f_{ij}(x)e_{i}e_{j},

where f ∈ R^{n+1}, and f_{ij}(x) = x_{j} \frac{∂ϕ}{∂x_{i}} (x) − x_{i} \frac{∂ϕ}{∂x_{j}} (x). Thus, we have

P = P + ∑ ∑_{1≤i<j≤n} P_{ij}e_{i}e_{j},

where P ∈ R^{n+1} and

P_{ij} = ∫_{x∈R^{n},r<|x|<R} g_{ij}(x) dx

= ∫_{r}^{R} ∫_{S^{d−1}} s^{d−1}g_{ij}(sω) dωds

= ∫_{r}^{R} ∫_{S^{d−1}} s^{d−1} ∫_{1}^{∞} t^{d−2}f_{ij}(tsω) dtdωds = ∫_{r}^{R} ∫_{S^{d−1}} ∫_{s}^{∞} t^{d−2}f_{ij}(tω) dtdωds = ∫_{r}^{R} ∫_{x∈R^{n},|x|>s} |x|^{−1}f_{ij}(x) dxds

= ∫_{r}^{R} ∫_{x∈R^{n},|x|>s} \frac{x_{j}}{|x|} \frac{∂ϕ}{∂x_{i}} (x) − \frac{x_{i}}{|x|} \frac{∂ϕ}{∂x_{j}} (x) dxds

= ∫_{r}^{R} ∫_{|x|>s} div(\frac{(x_{j}e_{i} − x_{i}e_{j})ϕ(x)}{|x|}) dxds

= ∫_{r}^{R} 0 ds

as desired. By the earlier discussion, we have thus proved Theorem 5.2. If ϕ is a harmonic kernel, then T_{ϕ} is extensible to an L^{2}-bounded operator.

6. References

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