SINGULAR INTEGRALS AND RECTIFIABILITY

We shall discuss singular integrals on lower dimensional subsets of Rn. A survey of this topic was given in [M4]. The first part of this paper gives a quick review of some results discussed in [M4] and a survey on some newer results and open problems. In the second part we prove some results on the Riesz kernels in Rn. As far I know, they have not been explicitly stated and proved, but they are very closely related to some earlier results and methods.


A survey
Not much has happened in this topic after [M4] except for the Cauchy kernel.For surveys on the development related to the Cauchy kernel, see [D2], [M5] and [V2], and for a recent fundamental contribution [T3].One of the main goals of this survey is to emphasize the huge gap in our knowledge between the Cauchy kernel and almost any other kernel, in particular, the higher dimensional Riesz kernels.We shall mainly concentrate on general kernels and give some comments on the Cauchy kernel.Our setting consists of an m-dimensional Radon measure µ in R n , where 0 < m < n is an integer, of an m-dimensional Calderón-Zygmund kernel K, and the related singular integral operator T = T µ,K .We shall assume that µ is Ahlfors-David (AD) regular.This means that there is 0 < C < ∞ such that r m /C ≤ µ B(x, r) ≤ Cr m for x ∈ S, 0 < r < diam(S), (1.1) where S = spt µ is the support of µ.Then µ is equivalent to the restriction H m | S of the m-dimensional Hausdorff measure H m to S. Thus we could as well start with a closed set S and assume that µ = H m | S satisfies (1.1).In various places (1.1) could be relaxed, for example, to requirements concerning the positivity and/or finiteness of the lower 200 P. Mattila and/or upper limits of r −m µ B(x, r) , but we stick to (1.1) to simplify the exposition.
Also we shall not try to cover the most general class of kernels, but we shall restrict to smooth homogeneous odd kernels.Thus we assume that K is of the form where Ω is a smooth function on the unit sphere S n−1 satisfying Ω(−v) = −Ω(v).
The singular integral operator T is formally given by but of course this does not usually exist for x ∈ S.There are several well-known ways to go around this to define T , but since all our questions reduce to the behaviour of the truncated operators T ε , ε > 0; we shall not try to define T itself.

Examples.
1) An AD-regular curve Γ ⊂ R n ; That is, C 1 is constructed so that at each step the mid-half of the construction intervals is deleted.
Basic properties studied.We shall be interested in relations between three properties: rectifiability, L 2 -boundedness and existence of principal values.
Rectifiability.µ is said to be m-rectifiable if there are m Here any reasonable definition of a surface will do, and C 1 could be replaced by Lipschitz.For the theory of rectifiability, see for example [M3].
There is a quantitative notion of rectifiability of David and Semmes, called uniform rectifiability, see [DS].We don't give the somewhat complicated general definition, but the 1-dimensional case is simple: µ, or S = spt µ, is uniformly 1-rectifiable if there is an AD-regular curve such that S ⊂ Γ.
This means that the operators T ε , ε > 0, are bounded in L 2 (µ) uniformly, and it agrees with any other reasonable definition.
Existence of principal values.The principal values exist µ almost everywhere for T = T µ,k if there exists Three questions.The main questions we are interested in are Q1) Does L 2 -boundedness imply uniform rectifiability?Q2) Does a.e.existence of pricipal values imply rectifiability?Q3) Does L 2 -boundedness imply a.e.existence of principal values?The converse questions have positive answers.Recall our basic assumptions (1.1) and (1.2).Uniform rectifiability implies L 2 -boundedness by the results of David and Semmes, [DS].That rectifiability implies a.e.existence of principal values was proved by Melnikov and myself in [MM] for 1-dimensional sets and the Cauchy kernel.A different proof of Verdera [V1] easily generalizes to our more general setting, cf.[M3,Theorem 20.28].

P. Mattila
Of course the converse of Q3 cannot quite be true, but the method of Nazarov, Treil and Volberg [NTV] gives that the a.e.existence of principal values yields the L 2 -boundedness of T on a subset of S with large µ measure.For this it is even enough to assume that In fact, this finiteness of the maximal function is probably more natural assumption than the existence of principal values, but sometimes we can use the latter and not the former.
The question Q3 is amusing because one is so used to that in classical situations the answer is yes.But that is so because we can easily check it for smooth functions, which are dense in L 2 (µ).For general AD-regular measures the existence of principal values is not automatic for any function.There are cases, discussed below, where the positive answer to Q2 is known but not to Q1.The hope is that Q3 could be used as a link.Some answers.Very little is known about these three questions for general kernels.For the Cauchy kernel the answer to all three questions is yes, even with much less stringent conditions on µ than (1.1).The reason is the identity discovered by Melnikov in [Me]: here the sum is over all six permutations of {1, 2, 3} and c(z 1 , z 2 , z 3 ) is the so-called Menger curvature of the triple {z 1 , z 2 , z 3 } ⊂ C, which, by definition, equals 1/R where R is the radius of the circle passing through z 1 , z 2 and z 3 (R = ∞ if and only if z 1 , z 2 , z 3 are collinear).The main thing about the sum in (1.4) is not its very geometric interpretation, but that it is non-negative.This seems to be very peculiar to the Cauchy kernel and to some very closely related kernels, such as Re(1/z).
The usefulness of (1.4) becomes apparent when one integrates it with respect to an AD-regular measure µ.Then by Fubini's theorem and easy estimates (assuming µ has compact support)

Singular Integrals and Rectifiability 203
This was used by Melnikov and Verdera in [MV] to give another proof of the boundedness of the Cauchy integral operator on Lipschitz graphs and by Melnikov, Verdera and myself in [MMV] to prove that the answer to Q1 is yes for the Cauchy kernel.That the answer to Q2 is yes was proved in [M2] (without the use of the identity (1.4)), and the yes-answer to Q3 follows from that of Q1 and the converse of Q2.So all is settled for the Cauchy kernel and AD-regular measures, but this goes much beyond them.In Q2 (1.1) can be relaxed to which is essentially the same thing as to study measures µ = H 1 | E where E is a Borel set with H 1 (E) < ∞.This is due to the results of Léger [L] and Tolsa [T2].By Tolsa's results in [T1] Q3 has positive answer for the Cauchy kernel and for any Radon measure µ.But this is not an abstract argument, it goes through rectifiability: The L 2 -boundedness implies, by a result of David, that µ B(z, r) ≤ Cr for all z ∈ C, r > 0. The set where lim sup r→0 r −1 µ B(x, r) > 0 (1.5) has σ-finite H 1 measure, and thus it is rectifiable by Léger's result (this uses the identity (1.4)).In the rest where (1.5) fails one can use that µ B(x, r) is small for small r.
For other kernels much less is known.We assume again (1.1) on µ.We shall discuss some partial results related to Q1 for the Riesz kernels K m at the end of this paper.Q2 has answer yes for K m by Preiss and myself, [MPr].It has also positive answer for 1-dimensional measures µ in C and the kernels z 2k−1 /|z| 2k , k = 2, 3 . . ., by Huovinen [H1].Both Q1 and Q2 have negative answers for example for the kernel Re(z/|z| 2 − z 3 /|z| 4 ) by Huovinen's result in [H2].By a modification of this kernel Huovinen (unpublished) has found a homogeneous kernel yielding a negative answer to Q3.Earlier David has used the Cantor set in Example 2 to construct non-homogeneous kernels giving similar counter-examples as Huovinen.
Returning to Huovinen's kernel Re(z/|z| 2 −z 3 /|z| 4 ), we observe that it vanishes both on real and imaginary axis.This is helpful in constructing the measure µ which gives the counter-example.But the situation is somewhat delicate: Re(1/z) vanishes on the imaginary axis, but the same results hold for it as for the Cauchy kernel 1/z, since it satisfies exactly the same identity (1.4).

P. Mattila
Farag [F2] has shown for the kernel where 1 ≤ p < ∞, that the sum of the permutations is comparable with that of the Cauchy kernel and hence leads to the same results.On the other hand he has also shown in [F1] that when m ≥ 2 practically any reasonable modification of such sums of permutations for the Riesz kernel K m , and any of the kernels x |x| −m−1 p , p ≥ 1, takes both positive and negative values and hence seems to be useless.This is the main reason for the huge gap between the results for the Cauchy kernel and for the Riesz kernels K m for m ≥ 2.

Partial results on the Riesz kernels
The investigations discussed above for the Cauchy kernel are closely related to the removable sets of bounded analytic functions, see, e.g., [M3,Section 19], and [D1] for the best result so far.Similarly, the Riesz kernel K n−1 in R n is related to the removable singularities of Lipschitz harmonic functions, see [MP].Some results obtained in the study of such removable singularities in [M1] (see also [M3,Theorem 19.17]) and in [MP], and also the results and methods of [Vi] indicate that the Theorems 2.1 and 2.2 below should be true.First we need the definition of tangent measures.
Let µ be a Radon measure in R n .For a ∈ R n and r > 0, let µ a,r be the blow-up of µ from B(a, r) to B(0, 1): A Radon measure ν is called a tangent measure of µ at a, denoted ν ∈ Tan(µ, a), if ν(R n ) > 0 and there are sequences c i > 0 and r i ↓ 0 such that c i µ a,ri → ν.
If µ satisfies (1.1), one can restrict to the sequences (c r −m i ).Knowing the measures in Tan(µ, a), we know how µ looks locally.For more on these, see [P], where they were introduced, or [M3, Section 14].
Theorem 2.1.Let µ be an AD-regular m-dimensional Radon measure in R n .If T µ,Km is bounded in L 2 (µ), then for µ almost all a ∈ R n , Tan(µ, a) contains some measure ν of the form ν = H m | V , where V is an affine m-plane.
In other words, this means that around µ almost all points a at some (but not necessarily all) arbitrarily small scales S = spt µ is very well approximable by m-planes.In particular, S cannot be a kind of m-dimensional analogue of the Cantor set C in Example 2.
A duality method and some standard estimates, cf.e.g.[M3,Section 20], show that there is g ∈ L ∞ (µ), g = 0, g ≥ 0, such that the maximal function (recall (1.3)) T * µ,Km g is bounded.Thus Theorem 3.1 follows from Theorem 2.2.Let µ be a Radon measure in R n such that for µ almost all x ∈ R n , where c > 0 and V is an m-plane.
Proof: It is not hard to see, cf.Lemmas 14.7 and 19.15 in [M3], that for µ almost all a ∈ R n , every ν ∈ Tan(µ, a) is an m-dimensional AD-regular measure such that Taking one more tangent measure at a point z where h(z) = 0 and h is approximately continuous, we find the required tangent measure of µ at a.