Hardy Spaces, Singular Integrals and the

Hardy Spaces, Singular Integrals and the Geometry of Euclidean Domains of Locally Finite Perimeter ∗ Steve Hofmann, Emilio Marmolejo-Olea, Marius Mitrea, Salvador Pérez-Esteva and Michael Taylor

Abstract We study the interplay between the geometry of Hardy spaces and functional analytic properties of singular integral operators (SIO’s), such as the Riesz transforms as well as Cauchy-Clifford and harmonic double layer operator, on the one hand and, on the other hand, the regularity and geometric properties of domains of locally finite perimeter. Among other things, we give several characterizations of Euclidean balls, their complements, and half-spaces, in terms of the aforementioned SIO’s.

1

Introduction

Hardy spaces (of holomorphic functions) have originally been considered in domains in the plane, a context in which the two-dimensional Euclidean space is identified with the field of complex numbers. Defining Hardy spaces in the higher dimensional setting presupposes the existence of an additional structure, playing a role in relation to Rn somewhat analogous to the role played by C in relation to R2 . A natural choice for us is to consider the embedding Rn !→ C"n

(1.1)

where C"n is the Clifford algebra with n generators. That is, C"n is the minimal enlargement of Rn to an associative, unitary algebra with the property that X # X = −|X|2 for every X ∈ Rn (when n is odd, it is also understood that that C"n is not generated, as an algebra, by any proper subspace of Rn ). In this context, !n the role of the Cauchy-Riemann operator is played by the Dirac operator D, defined as Df = j=1 ej # (∂j f ) for C"n -valued functions f . Null-solutions of D, referred to as monogenic functions, play the role of holomorphic functions in C. Given a domain Ω in Rn , and p ∈ (1, ∞), let H±p (∂Ω) be the boundary Hardy (or Smirnov) spaces obtained by taking (pointwise) traces of inner and outer monogenics (having p-th power integrable nontangential maximal functions) on ∂Ω. When Ω is a two-sided NTA domain (cf. appendix) with an Ahlfors regular boundary, we show that H±p (∂Ω) are well-defined, closed subspaces of Lp (∂Ω, dσ) ⊗ C"n and the following direct sum decomposition holds: Lp (∂Ω, dσ) ⊗ C"n = H+p (∂Ω) ⊕ H−p (∂Ω). ∗

(1.2)

2000 Math Subject Classification. Primary: 49Q15, 42B20 Secondary 26B15, 30G35 Key words: Hardy spaces, double layer potential, Riesz transforms, Clifford algebras, Cauchy-Clifford operator, domains of locally finite perimeter, SKT domains, Clifford-Szeg¨ o projections, characterizations of balls, half-spaces

1

Here, the boundary measure σ is the restriction of the (n − 1)-dimensional Hausdorff measure Hn−1 to ∂Ω. Also, Lp (∂Ω, dσ) ⊗ C"n simply denotes the Lp -space of functions on ∂Ω with values in the Hilbert space C"n . This extends previous work in [2], [26], where the case of Lipschitz domains was treated, and in [3], [4], [35], [20] where the authors have dealt with the two-dimensional case. One significant feature of these considerations emphasized in this paper is that the geometry of Hardy spaces in Calderón’s decomposition (1.2) contains a remarkable amount of information about the regularity and shape of the domain Ω itself. A concrete result substantiating this claim is as follows. Let " # " )f+ , f− * # < ) H+2 (∂Ω) , H−2 (∂Ω) := arccos sup (1.3) f± ∈H 2 (∂Ω) +f+ ++f− + ±

be the angle between the closed subspaces H+2 (∂Ω) and H−2 (∂Ω) of L2 (∂Ω, dσ) ⊗ C"n (with )·, ·* and + · + denoting the natural inner product and norm in L2 (∂Ω, dσ) ⊗ C"n , respectively).

Theorem 1.1. In the class $of bounded, two-sided % NTA domains with Ahlfors regular boundaries, 2 2 the following holds. If θ :=< ) H+ (∂Ω) , H− (∂Ω) is sufficiently close to π/2 (relative to the NTA and Ahlfors constants of Ω) then Ω is a δ-SKT domain, for some δ = δ(θ) > 0 such that δ , 0 as θ - π/2 (cf. Appendix for definitions pertaining to terminology). In addition, " # π < ) H+2 (∂Ω) , H−2 (∂Ω) = ⇐⇒ Ω is a ball. (1.4) 2 This is remarkable, in that the specification θ = π/2 not only implies that Ω is smooth, but actually determines the shape of Ω. A condition that is equivalent to θ = π/2 and, at the same time, does not employ the Clifford algebra formalism nor does it involve the outward unit normal to Ω, is as follows. For 1 ≤ k ≤ n, recall that the Riesz transform Rk associated with Ω is the formal convolution operator on ∂Ω with xk 2 the kernel ωn−1 area of S n−1 . The most familiar setting |X|n , where ωn−1 stands for the surface ! is when Ω = Rn+ , in which case it is well-known that nk=1 Rk2 = −I and Rj Rk = Rk Rj for all j, k ∈ {1, ..., n} (called URTI, i.e., the usual Riesz transform identities). It is perhaps less known that the URTI are valid when Ω is a ball in Rn . What is, however, most remarkable about these considerations is that, under some mild measure theoretic background assumptions, the converse of the latter statement is also true. Specifically, we have the following. Theorem 1.2. If Ω ⊂ Rn is a two-sided NTA domain with an Ahlfors regular boundary, then ∂Ω is a sphere, or a (n − 1)-plane ⇐⇒ n & k=1

Rk2 = −I and Rj Rk = Rk Rj

∀ j, k ∈ {1, ..., n}.

(1.5)

Thus, in the setting of the above theorem, if the URTI hold then Ω is a ball if it is unbounded, the complement of a ball if it is unbounded and has compact boundary, and a half-space if it has an unbounded boundary. Another characterization of balls, complements of balls, and half-spaces which avoids involving the outward unit normal reads as follows: ' #Y −X+ ,Y −X− $ n−1 (Y ) = 0 and for every X± ∈ Ω± there hold |Y −X+ |n |Y −X− |n dH ∂Ω

'

∂Ω

(Y −X+ )j (Y −X− )k −(Y −X+ )k (Y −X− )j |Y −X+ |n |Y −X− |n

(1.6)

dHn−1 (Y ) = 0 2

∀ j, k ∈ {1, ..., n},

where we have set Ω+ := Ω and Ω− := Rn \ Ω. See Theorem 2.12. Hence, in the class of domains described in Theorem 1.1, ∂Ω is a sphere, or a (n − 1)-plane ⇐⇒ (1.6) holds.

(1.7)

The theory of the Hardy spaces H±2 (∂Ω) is also strongly intertwined with that of a certain principal-value singular integral operator C on ∂Ω, called the Cauchy-Clifford operator (cf. (2.19) for a formal definition). One basic aspect of this relationship is that the proximity of " # 2 2 < ) H+ (∂Ω) , H− (∂Ω) to π/2 turns out to be closely related to the degree of failure for C to be

be self-adjoint, as an operator on L2 (∂Ω, dσ) ⊗ C"n . Somewhat more precisely, " # π < ) H+2 (∂Ω) , H−2 (∂Ω) is close to ⇐⇒ +C − C ∗ +L(L2 (∂Ω,dσ)⊗C$n ) is small (1.8) 2 and " # π ⇐⇒ C = C ∗ . (1.9) < ) H+2 (∂Ω) , H−2 (∂Ω) = 2 As with the Hardy spaces themselves, the Cauchy-Clifford operator C also encodes a remarkable amount of information about the geometry of Ω. In the final section of [17], N. Kerzman is asking whether the spectral properties of C − C ∗ can be related in a significant way to the geometry of the underlying domain Ω. In a slightly more general context than that of Theorem 1.1, it has been established in [10] that C − C ∗ is compact on L2 (∂Ω, dσ) ⊗ C"n ⇐⇒ Ω is a regular SKT domain.

(1.10)

Here we augment this with the following:

Theorem 1.3. Let Ω ⊂ Rn be a UR (uniformly rectifiable) domain which satisfies ∂Ω = ∂Ω. Then

C − C ∗ = 0 ⇐⇒ Ω is either a ball, the complement of a (closed) ball, or a half-space. (1.11)

In fact, instead of C = C ∗ , i.e. C is self-adjoint on L2 (∂Ω, dσ)⊗C"n , we could assume the seemingly considerably weaker condition that C is normal, i.e. C C ∗ = C ∗ C . The class of UR domains is defined in the appendix. Here we only wish to mention that every two-sided NTA domain with Ahlfors regular boundary is a UR domain. In the two dimensional setting, when the Clifford algebra is replaced by the field of complex numbers and C is the classical Cauchy operator in complex analysis, we succeed in refining the above theorem by proving the following perturbation result: Theorem 1.4. Assume that Ω ⊂ R2 is a bounded, connected UR domain satisfying ∂Ω = ∂Ω. Then for every δ > 0 there exists ε > 0, which depends on δ > 0 and the geometrical characteristics of Ω, with the property that +C ∗ 1 − 12 +L2 (∂Ω,dσ)⊗C < ε =⇒

inf +χΩ − χD +L2 (R2 ) < δ.

D disk

As a corollary, for every δ > 0 there exists ε > 0 such that ( Ω is a δ-SKT domain and ∗ $ % < ε =⇒ +C − C + 2 L L (∂Ω,dσ)⊗C inf D disk +χΩ − χD +L2 (R2 ) < δ

(1.12)

(1.13)

in the class of all bounded, connected domains Ω ⊂ R2 which satisfy a two-sided local John condition and have Ahlfors regular boundaries, with a certain fixed common bound on the geometrical constants involved in describing these characteristics. 3

As is well-known, the scalar part (i.e., the coefficient of the multiplicative unit e0 of the Clifford algebra C"n ) of C is K, the harmonic double layer (cf. (2.25) for a formal definition) when both operators act on scalar-valued functions. More specifically, in the matrix representation of C in the standard basis in C"n in which all entries are operators mapping scalar-valued functions to scalar-valued functions, the entry (1, 1) (corresponding to the scalars in C"n ) is exactly the double layer potential. In particular, C = C ∗ =⇒ K = K ∗ , with K ∗ the adjoint of K. It is therefore surprising that the seemingly much weaker condition K = K ∗ contains, at least for bounded domains, essentially the same amount of information as C = C ∗ . Concretely, we have: Theorem 1.5. Assume that Ω ⊂ Rn , n ≥ 3, is a bounded UR domain with ∂Ω = ∂Ω and for which K = K ∗ . Then Ω is a ball. As a corollary, retaining the same background hypotheses as above, C = C ∗ ⇐⇒ K = K ∗ .

(1.14)

It is unclear to us whether perturbation results similar to those discussed in Theorem 1.4 hold in dimension n ≥ 3. See the discussion at the end of § 4 for more in this regard. As already alluded to above, C contains significant information regarding the geometry of Ω. In the class of two-sided NTA domains with Ahlfors regular boundaries, we prove the estimate % = 1 occurs if and % ≥ 1 and show that the extremal case +C + $ +C + $ 2 2 2 L L2 (∂Ω,dσ)⊗C$n L L (∂Ω,dσ)⊗C$n % ≥ 1 with equality only if ∂Ω is a sphere, or a (n − 1)-plane. Likewise, +C + C ∗ + $ L L2 (∂Ω,dσ)⊗C$n

precisely when ∂Ω is a sphere, or a (n − 1)-plane. These, along with several other closely related results, are described in Theorem 4.21. Let us consider for a moment Calderón’s decomposition (1.2) with p = 2 and denote by P± the Clifford-Szëgo projections of L2 (∂Ω, dσ) ⊗ C"n onto H+2 (∂Ω) and H−2 (∂Ω), respectively. In this paper we also show that for every δ > 0 there exist ε > 0 and R > 0 with the property that

+I − P+ − P− + $

L L2 (∂Ω,dσ)⊗C$n

% ≤ ε =⇒

sup X∈∂Ω, 0ε

where k ∈ {1, ..., n}, and f is a (possibly Clifford algebra-valued) function on ∂Ω. It is then clear from (2.19)-(2.23) that C =

1 2

n &

(2.24)

Rk Mek Mν .

k=1

Let also recall here the harmonic double layer and its formal adjoint ' 1 )ν(Y ), Y − X* Kf (X) := lim f (Y ) dσ(Y ), + |X − Y |n ε→0 ωn−1

X ∈ ∂Ω,

(2.25)

Y ∈∂Ω

|X−Y |>ε

∗

K f (X) := lim

ε→0+

1 ωn−1

'

Y ∈∂Ω

|X−Y |>ε

)ν(X), X − Y * f (Y ) dσ(Y ), |X − Y |n

X ∈ ∂Ω,

(2.26)

considered in the same geometric measure theoretic setting as before. Next, call Ω ⊆ Rn a regular Semmes-Kenig-Toro (SKT) domain if it is a two-sided NTA domain with an Ahlfors-regular boundary and for which the unit normal is in VMO (∂Ω, dσ), Sarason’s space of functions of vanishing mean oscillation. This piece of terminology, which originated in [10], replaces what was previously called chord-arc domains with vanishing constant. A somewhat different, albeit equivalent, point of view on this topic can be found in the Appendix. We now record a series of results recently established in [10] (the reader is again advised to consult the Appendix for the relevant definitions). To state the first theorem, recall that the commutator of two operators A, B is defined as [A, B] := AB − BA. Theorem 2.1. Assume that Ω ⊂ Rn is an open set satisfying a two-sided local John condition and such that ∂Ω is Ahlfors regular and compact. Also, denote by ν = (ν1 , ..., νn ) the outward unit normal to Ω. Then the following statements are equivalent: (i) the harmonic double layer K and the commutators [Mνj , Rk ], between the Riesz transforms and multiplication by the components of the unit normal, are compact operators on Lp (∂Ω, dσ) for some (and, hence, all) p ∈ (1, ∞); (ii) C − C ∗ is a compact operator on Lp (∂Ω, dσ) ⊗ C"n for some (hence all) p ∈ (1, ∞); (iii) Ω is a regular SKT domain. The result below, which has recently been obtained in [10], involves two new concepts, both defined in the Appendix, namely the δ-SKT domain and the John condition. Heuristically, the latter condition can be thought of as a curvilinear, scale-invariant version of starlikeness. As for the former, this means that the boundary of the domain in question is Ahlfors regular, wellapproximated by planes at scales ≈ δ, and the mean-oscillations of the unit normal are ≤ δ; see (5.5) in connection with the definition of VMO(∂Ω, dσ). 8

Theorem 2.2. Let Ω ⊆ Rn be a bounded open set that satisfies a two-sided John condition and whose boundary is Ahlfors regular. Then there exists a constant Co > 1, depending only on the John and Ahlfors constants on Ω, with the following significance. Assume that there exists δ > 0, sufficiently small relative to the John and Ahlfors constants of Ω, with the property that dist (ν , VMO (∂Ω, dσ)) < δ,

(2.27)

where the distance is taken in the John-Nirenberg space BMO (∂Ω, dσ), of functions of bounded mean oscillations. Then Ω is a δo -SKT domain, with δo = Co δ. In addition, there exists R > 0 with the property that for every X ∈ ∂Ω and r ∈ (0, R] ) ) ) σ(∆(X, r)) ) ) ) ≤ Co δ, (2.28) − 1 ) ωn−1 rn−1 ) where ∆(X, r) := B(X, r) ∩ ∂Ω.

In order to continue, we make one more definition. Specifically, given a Banach space (X , + · +), set L(X ) := the space of all bounded linear operators on X ,

Cp (X ) := the space of all linear compact operators on X .

(2.29)

As is well-known, L(X ) becomes a Banach space when equipped with the natural norm +T +L(X ) := sup {+T x+ : x ∈ X , +x+ ≤ 1}, and Cp (X ) is a closed subspace of L(X ). The theorem below appears in [10]. Theorem 2.3. Let Ω ⊂ Rn be an UR domain and assume that p ∈ (1, ∞). Then there exists C > 0, depending only on n, p, and the Ahlfors constants of ∂Ω, such that " # 0 " #11/n dist ν , VMO (∂Ω, dσ) ≤ C dist C − C ∗ , Cp (Lp (∂Ω, dσ) ⊗ C"n ) . (2.30)

The results in this subsection will play a prominent role in subsequent considerations. Although, strictly speaking, the Clifford algebra setting from §2.1 does not contain the standard field of complex numbers, very similar considerations apply to this latter context, leading to analogous results to the ones discussed above. In this scenario, the Clifford-Cauchy operator takes the familiar form ' f (ζ) 1 dζ, z ∈ ∂Ω, Ω ⊂ R2 ≡ C. (2.31) C f (z) := lim ζ −z ε→0+ 2πi ζ∈∂Ω

|z−ζ|>ε

This remark will be tacitly assumed in what follows. In fact, it is possible to slightly alter the Clifford algebra formalism discussed in §2.1 in order to ensure a more uniform theory, in which (2.19) becomes precisely (2.31) when the Clifford algebra in question has precisely one imaginary unit. See, e.g., [26] for the details of this construction.

2.3

Hardy spaces and Calder´ on’s decomposition

Assume that Ω ⊂ Rn be an open set and abbreviate Ω+ := Ω,

Ω− := Rn \ Ω. 9

(2.32)

In analogy with the classical setting of functions of one complex variable, for each p ∈ (1, ∞) define the Hardy spaces H p (Ω± ) by H p (Ω+ ) := {u : Ω+ → C"n : N (u) ∈ Lp (∂Ω, dσ), Du = 0 in Ω+ },

(2.33)

with the convention that if Ω+ is unbounded and ∂Ω is bounded, the decay condition u(x) = O(|x|1−n )

as

|x| → ∞

(2.34)

is also included. The space H p (Ω− ) is defined analogously. It is useful to observe that, thanks to (2.12), functions in H p (Ω± ) are harmonic. Next, the boundary Hardy spaces are defined as H±p (∂Ω) := {u|∂Ω : u ∈ H p (Ω± )}.

(2.35)

Above, for u : Ω± → C"n , we define the nontangential maximal function of u by N u(Z) := Nα u(Z) := sup {|u(X)| : X ∈ Ω± , |X − Z| < (1 + α) dist (X, ∂Ω)},

Z ∈ ∂Ω, (2.36)

where α > 0 is a fixed parameter, and we make the convention that N u(Z) = 0 whenever the supremum is taken over an empty set. In addition to the principal value Cauchy-Clifford operator C from (2.19) let us also recall here the Cauchy-Clifford operator ' 1 X −Y Cf (X) := # ν(Y ) # f (Y ) dσ(Y ), X ∈ Ω, (2.37) ωn−1 ∂Ω |X − Y |n

mapping a C"n -valued function f defined on ∂Ω into a C"n -valued function defined in Ω. Up to this point, the above considerations are purely formal, as the class of domains to which Ω belongs. From [10, Section 3], it follows that if Ω is a UR domain then for every f ∈ Lp (∂Ω, dσ)⊗C"n , p ∈ (1, ∞), C f is meaningfully defined a.e. on ∂Ω, +N (Cf )+Lp (∂Ω,dσ) ≤ C+f +Lp (∂Ω,dσ)⊗C$n , +C f +Lp (∂Ω,dσ) ≤ C+f +Lp (∂Ω,dσ)⊗C$n , (2.38) ) ) (2.39) D(Cf ) = 0 in Ω, and Cf ) = ( 12 I + C )f a.e. on ∂Ω, ∂Ω

where the boundary trace is taken in a nontangential pointwise sense. Let us also note here that if Ω ⊂ Rn is an NTA domain with an Ahlfors regular boundary and p ∈ (1, ∞), the following Fatou type theorem holds ) ) ∀ u ∈ H p (Ω+ ) =⇒ u) exists a.e. in the nontangential pointwise sense, (2.40) ∂Ω

and that the following Cauchy’s reproducing formula is valid

∀ u ∈ H p (Ω+ ) =⇒ u = C(u|∂Ω ) in Ω.

(2.41)

Indeed, as has been noted in [10], Theorem 6.4 on p. 112 of [14] gives that any function u which is harmonic in an NTA domain Ω and nontangentially bounded from below on E ⊂ ∂Ω has a nontangential limit ω Xo -a.e. on E (where ω Xo is the harmonic measure with pole at Xo ∈ Ω). Now, the fact that u|∂Ω exists for every u ∈ H p (Ω+ ), p ∈ (1, ∞), is a consequence of (2.12), the above local Fatou theorem applied to Ek := {X ∈ ∂Ω : N u(X) < 2k }, k = 1, 2..., and the mutual absolute continuity between the surface and harmonic measures proved in [5]. The key to establishing (2.41) is the following version of Green’s formula from [10]: 10

Theorem 2.4. Let Ω ⊂ Rn be an open set which is either bounded or has an unbounded boundary. Assume that ∂Ω is Ahlfors regular and satisfies (2.16). Denote by ν the outward unit normal to ∂Ω and set σ := Hn−1 7∂Ω. Then ' ' ) div .v dX = )ν, .v )∂Ω * dσ (2.42) Ω

∂Ω

holds for each vector field .v ∈ C 0 (Ω) that satisfies

N .v ∈ L1 (∂Ω, dσ) ∩ Lploc (∂Ω, dσ) for some p ∈ (1, ∞), ) and the pointwise nontangential trace .v )∂Ω exists σ-a.e. on ∂Ω.

|.v |, div .v ∈ L1 (Ω),

(2.43)

For Ω ⊂ Rn NTA domain with an Ahlfors regular boundary, we deduce from (2.38), (2.39) and (2.40) that the operator C : Lp (∂Ω, dσ) ⊗ C"n −→ H p (Ω+ )

(2.44)

is well-defined, bounded and onto for each p ∈ (1, ∞). From this and (2.38)-(2.40) we then obtain ( 12 I + C )2 = 12 I + C so that, ultimately, C 2 = 41 I

on

Lp (∂Ω, dσ) ⊗ C"n ,

1 < p < ∞.

(2.45)

As a consequence, if Ω is an NTA domain with an Ahlfors regular boundary, and p ∈ (1, ∞), then # " # " (2.46) Im 12 I + C : Lp (∂Ω, dσ) ⊗ C"n = H+p (∂Ω) = Ker − 12 I + C : Lp (∂Ω, dσ) ⊗ C"n . Likewise, if Ω− is an NTA domain with an Ahlfors regular boundary and p ∈ (1, ∞), then also " # " # Im − 21 I + C : Lp (∂Ω, dσ) ⊗ C"n = H−p (∂Ω) = Ker 12 I + C : Lp (∂Ω, dσ) ⊗ C"n . (2.47)

Proposition 2.5. Suppose Ω ⊂ Rn is a two-sided NTA domain with an Ahlfors regular boundary (which makes it a UR domain), and fix p ∈ (1, ∞). Then the spectrum of the operator C on Lp (∂Ω, dσ) ⊗ C"n is {− 21 , + 12 }, the numbers ± 12 are eigenvalues and H±p (∂Ω) are the corresponding eigenspaces. Proof. It suffices to observe that, thanks to (2.45), (λI − C )−1 =

1 (λI + C ), λ2 − 14

λ ∈ R \ {± 12 },

as operators in Lp (∂Ω, dσ) ⊗ C"n . Then everything follows from this and (2.46)-(2.47).

(2.48) !

We conclude this subsection with a result which is going to play a basic role for the goals we have in mind. As a preamble, we remind the reader a piece of terminology. Given a two closed subspaces X0 , X1 of a Banach space X, we say that X is the direct sum of X0 and X1 (and write X = X0 ⊕ X1 ) provided any x ∈ X can be uniquely written as x = x0 + x1 with xj ∈ Xj , j = 0, 1, Note that, as a consequence of the Open Mapping Theorem, the assignments X 5 x :→ xj ∈ Xj , j = 0, 1, are continuous (i.e., the direct sum decomposition X = X0 ⊕ X1 is ‘topological’). 11

Theorem 2.6. Assume that Ω ⊂ Rn is a two-sided NTA domain with an Ahlfors regular boundary (making it a UR domain). Then the following decomposition is valid for each p ∈ (1, ∞): Lp (∂Ω, dσ) ⊗ C"n = H+p (∂Ω) ⊕ H−p (∂Ω).

(2.49)

Proof. Any f ∈ Lp (∂Ω, dσ) ⊗ C"n can be written as f+ − f− where f± := (± 12 I + C )f ∈ H±p (∂Ω), by (2.46)-(2.47). Furthermore, thanks to the second estimate in (2.38) (recall that a two-sided NTA domain with an Ahlfors regular boundary is a UR domain), there exists C = C(Ω, p) > 0 such that +f± +Lp (∂Ω,dσ)⊗C$n ≤ C+f +Lp (∂Ω,dσ)⊗C$n . To see that the sum in (2.49) is direct, assume that f ∈ H+p (∂Ω) ∩ H−p (∂Ω). Then ( 12 I + C )f vanishes, by (2.47) and is equal to f by (2.46). Thus, H+p (∂Ω) ∩ H−p (∂Ω) = 0, finishing the proof of the theorem. !

2.4

Clifford-Szeg¨ o projections and the Kerzman-Stein operator

Fix a two-sided NTA domain Ω ⊂ Rn with an Ahlfors regular boundary, and define the CliffordSzeg¨ o projections P± : L2 (∂Ω, dσ) ⊗ C"n −→ H±2 (∂Ω) !→ L2 (∂Ω, dσ) ⊗ C"n

(2.50)

as the orthogonal projections of L2 (∂Ω, dσ) ⊗ C"n onto the closed subspace H±2 (∂Ω). It has been shown in [10] that there exists ε > 0 with the property that P± in (2.50) extend to P± : Lp (∂Ω, dσ) ⊗ C"n −→ H±p (∂Ω)

(2.51)

in a continuous and onto fashion for each p ∈ (2 − ε, 2 + ε). Furthermore, when Ω is a bounded regular SKT domain, then this is true for each p ∈ (1, ∞). We now discuss a version of Kerzman-Stein’s formula (cf. [18]) in the Clifford algebra setting from [10]. Specifically, we note that formulas (2.46)-(2.47) and the definition of P± readily imply that P± (± 12 I + C ) = ± 12 I + C ,

(∓ 12 I + C )P± = 0.

(2.52)

From the second formula above and duality we also obtain P± (∓ 12 I + C ∗ ) = 0.

(2.53)

Subtracting this from the first formula in (2.52) then gives P± (±I + C − C ∗ ) = ± 12 I + C . Now, if A := C − C ∗ : Lp (∂Ω, dσ) ⊗ C"n −→ Lp (∂Ω, dσ) ⊗ C"n ,

1 < p < ∞,

(2.54)

it has been observed in [10] that there exists ε > 0 so that I ± A : Lp (∂Ω, dσ) ⊗ C"n −→ Lp (∂Ω, dσ) ⊗ C"n are invertible for 2 − ε < p < 2 + ε.

(2.55)

The above considerations then justify the following Kerzman-Stein type formula P± = ( 12 I ± C ) ◦ (I ± A )−1 ,

(2.56)

valid in Lp (∂Ω, dσ) ⊗ C"n , if 2 − ε < p < 2 + ε. This, (2.55) and (2.22) then show that P± extend as bounded operators on Lp (∂Ω, dσ) ⊗ C"n for 2 − ε < p < 2 + ε and, in fact, P± : Lp (∂Ω, dσ) ⊗ C"n −→ H±p (∂Ω) are onto for 2 − ε < p < 2 + ε. Several other properties of interest are collected in the proposition below. 12

(2.57)

Proposition 2.7. Let Ω ⊂ Rn be a two-sided NTA domain with an Ahlfors regular boundary. Then there exists ε > 0 such that if 2 − ε < p < 2 + ε then the following identities are valid: (P+ − P− )−1 = C + C ∗ ,

(2.58)

I − P+ − P− = (P+ P− − P− P+ )(C + C ∗ ),

(2.60)

(I + 4C ∗ C )−1 = (P+ − P− )C ∗ ,

(2.62)

C − C ∗ = P+ P− (C + C ∗ ) − (C + C ∗ )P− P+ ,

(2.59)

(I + 4C C ∗ )−1 = (P+ − P− )C ,

(2.61)

4(I + 4C ∗ C )−1 [C , C ∗ ](I + 4C C ∗ )−1 = −(P+ − P− )(C − C ∗ ),

(2.63)

as operators in Lp (∂Ω, dσ) ⊗ C"n . Proof. As is apparent from Theorem 2.6, (2.38)-(2.39) and (2.41), the operators P± := 12 I ± C are, respectively, the skew projection of L2 (∂Ω, dσ) ⊗ C"n onto H 2 (∂Ω± ) parallel to (with kernel) H 2 (∂Ω∓ ). Then, from an abstract point of view, (2.58)-(2.63) considered as operator identities on L2 (∂Ω, dσ) ⊗ C"n , are formulas relating the skew projections P± to the orthogonal projections P± . Given that there exists ε > 0 with the property that all operators involved are bounded Lp (∂Ω, dσ) ⊗ C"n if 2 − ε < p < 2 + ε, the same identities are then valid on Lp (∂Ω, dσ) ⊗ C"n , 2 − ε < p < 2 + ε, by density. For the convenience of the reader we, nonetheless, include here a direct argument. To get started, we note that (2.52)-(2.53) imply 1 1 0 0 (P+ − P− )(C + C ∗ ) = P+ ( 12 I + C ) + (− 12 I + C ∗ ) − P− (− 12 I + C ) + ( 12 I + C ∗ ) = P+ ( 21 I + C ) + P+ (− 12 I + C ∗ ) − P− (− 12 I + C ) − P+ ( 21 I + C ∗ ) = ( 21 I + C ) + 0 − (− 12 I + C ) − 0 = I.

(2.64)

Hence also (C + C ∗ )(P+ − P− ) = I, by duality. From this, (2.58) follows. To continue, note that observe that (2.52) implies # " P+ (− 12 I + C ) = P+ ( 12 I + C ) − I = 21 I + C − P+ , (2.65) and, further, that

P+ P− (− 12 I + C ) = 12 I + C − P+ .

(2.66)

From the second formula in (2.52) and duality we also obtain P− ( 12 I + C ∗ ) = 0, so that P+ P− ( 21 I + C ∗ ) = 0.

(2.67)

Adding (2.67) and (2.66) then yields 1 2I

+ C − P+ = P+ P− (C + C ∗ ),

(2.68)

so by subtracting (2.68) from its dual version we arrive at (2.59). Going further, analogously to (2.68) we have P− P+ (C + C ∗ ) = − 12 I + C + P− . 13

(2.69)

By subtracting this from (2.68) we arrive at (2.60). Turning our attention to (2.61), we write 0 1−1 = (C + C ∗ )−1 C −1 = 4(P+ − P− )C (2.70) ( 14 I + C C ∗ )−1 = C (C + C ∗ )

by (2.45) and (2.58). Finally, (2.62) is proved similarly and (2.63) follows by subtracting (2.62) from (2.61). This concludes the proof of the proposition. !

2.5

The geometry of Hardy spaces

Recall that given two closed subspaces H1 , H2 of a Hilbert space H with inner product )·, ·* and norm + · +, the angle between H1 and H2 , denoted by < ) (H1 , H2 ), is the unique number θ ∈ [0, π/2] for which . / cos θ = sup )f1 , f2 */+f1 ++f2 + : f1 ∈ H1 , f2 ∈ H2 . (2.71) In particular,

H1 ⊥H2 ⇐⇒ < ) (H1 , H2 ) =

π , 2

(2.72)

and it is straightforward to check that if Pj : H → H, j = 1, 2, are the orthogonal projections onto H1 , H2 , then +P1 P2 +L(H) = +P2 P1 +L(H) = cos (< ) (H1 , H2 )).

(2.73)

Theorem 2.8. Let Ω ⊂ Rn be a two-sided NTA domain with an Ahlfors regular boundary, and set " # θ :=< ) H+2 (∂Ω) , H−2 (∂Ω) . (2.74)

Then

+P+ P− + $


+C + $


+ ± 21 I + C + $

% = +P− P+ + $


% = +C ∗ + $


+(P+ − P− )−1 + $

L


for every λ ≥ 0, and

(2.75)

" # % = 1 cot θ , L2 (∂Ω,dσ)⊗C$n 2 2

%=


+λI + C ∗ C + $

% = cos θ,

(2.76)

1 , sin θ

% = +C + C ∗ + $


% = +λI + C ∗ C + $ L

cos θ ≤ +C − C ∗ + $


cos θ ≤ +I − P+ − P− + $

%=

1 , sin θ

" # % = λ + 1 cot2 θ , L2 (∂Ω,dσ)⊗C$n 4 2 % ≤ 2 cot θ,


% ≤ 2 cot θ,

cos θ % ≤ 2 cos θ . ≤ +[C , C ∗ ]+ $ 2 L L (∂Ω,dσ)⊗C$ n 2 sin2 θ 14

(2.77) (2.78) (2.79)

(2.80) (2.81) (2.82)

Proof. As was the case with Proposition 2.7 (cf. the comment at the beginning of its proof), the above formulas follow from general Hilbert space geometrical considerations. A direct argument goes as follows. Clearly, (2.75) is a direct consequence of (2.73). Also, standard functional analysis % = +C + $ % . Thus, as far as (2.76) is concerned, there remains gives +C ∗ + $ 2 2 L L (∂Ω,dσ)⊗C$n

L L (∂Ω,dσ)⊗C$n norm is 12 cot (θ/2).

A more general result of this type is proved to prove that the first operator in (2.98) below. Here we only wish to comment that, in the language of the skew projections P± , we have C = 12 (P+ − P− ), so matters can be easily reduced to computing the norm of a 2 × 2 matrix by considering all possible restrictions of C onto two dimensional subspaces span (h+ , h− ), h± ∈ H±2 (∂Ω), then taking the supremum of the norm. Moving on, formula (2.77) simply expresses the fact that the operator norm of the skew projections P± := ± 12 I + C is 1/ sin θ. Next, we write )(C + C ∗ )f, f * = )( 12 I + C )f, f * + )(− 12 I + C ∗ )f, f *

= )f+ , f+ − f− * + )f+ − f− , f− * = +f+ +2 − +f− +2 ,

(2.83)

for every f ∈ L2 (∂Ω, dσ) ⊗ C"n . Here, as usual, f± := (± 12 I + C )f ∈ H±2 (∂Ω). Since C + C ∗ is a normal operator (in fact, self-adjoint), we may then write . / % = sup |)(C + C ∗ )f, f *| / +f +2 : f ∈ L2 (∂Ω, dσ) ⊗ C"n +C + C ∗ + $ 2 L L (∂Ω,dσ)⊗C$n

. | +f +2 − +f +2 | / + − 2 : f ∈ H (∂Ω) ± ± +f+ − f− +2 2 3 | +f+ +2 − +f− +2 | 2 = sup : f± ∈ H± (∂Ω) . +f+ +2 − 2)f+ , f− * + +f− +2 = sup

(2.84)

Elementary calculus shows that for every α ∈ [0, π/2] max

λ1 ,λ2 >0

|λ21 − λ22 | 1 = . 2 2 sin α λ1 + λ2 − 2λ1 λ2 cos α

(2.85)

Thus, for any two vectors v1 , v2 in a Hilbert space max

λ1 ,λ2 >0

| +λ1 v1 +2 − +λ2 v2 +2 | 1 . = 2 2 +λ1 v1 + − 2)λ1 v1 , λ2 v2 * + +λ2 v2 + sin (< ) (v1 , v2 ))

(2.86)

Returning with this in (2.84) then yields +C + C ∗ + $


%=

1 " $ %# , sin < ) H+2 (∂Ω) , H−2 (∂Ω)

(2.87)

finishing the justification of (2.78). As far as (2.79) is concerned, this is a consequence of (2.76) and elementary functional analysis. Next, observe that when (2.75), (2.78) and (2.59) are used in concert, they entail +C − C ∗ + $


% ≤ 2+P+ P− + $


% +C + C ∗ + $

≤ 2 (cos θ)(sin θ)−1 = 2 cot θ. 15


%

(2.88)

This justifies the upper bound in (2.80). As for the lower bound in (2.80), note that for each f± ∈ H±2 (∂Ω) we have )(C − C ∗ )f+ , f− * = )C f+ , f− * − )f+ , C f− * =

1 2 )f+ , f− *

+ 12 )f+ , f− * = )f+ , f− *.

(2.89)

Consequently, +C − C ∗ + $ L

L2 (∂Ω,dσ)⊗C$

n

. %/ % = sup )(C − C ∗ )g, h*/+g++h+ : g, h ∈ L2 (∂Ω, dσ) ⊗ C"n . / ≥ sup )(C − C ∗ )f+ , f− */+f+ ++f− + : f± ∈ H±2 (∂Ω) . / = sup )f+ , f− */+f+ ++f− + : f± ∈ H±2 (∂Ω)

= cos θ,

(2.90)

as desired. Turning our attention to (2.81), we first note that, (2.60), (2.75) and (2.78) give +I − P+ − P− + $


% ≤ 2 (cos θ)(sin θ)−1 = 2 cot θ,

(2.91)

which is the upper bound in (2.81). Since for each f± ∈ H±2 (∂Ω) we have )(I − P+ − P− )f+ , f− * = −)f+ , f− *, the estimate cos θ ≤ +I − P+ − P− + $ (2.81). As for (2.82), we first note that


(2.92)

% follows much as in (2.89)-(2.90). This justifies

[C , C ∗ ] = (C − C ∗ )(C + C ∗ )

(2.93)

by (2.45) and its dual version. In turn, this and (2.58) also give [C , C ∗ ](P+ − P− ) = C − C ∗ .

(2.94)

Consequently, +[C , C ∗ ]+ $


% = +(C − C ∗ )(C + C ∗ )+ $


≤ +C + C ∗ + $


≤ 2

cos θ cot θ =2 , sin θ sin2 θ

%

% +C ∗ − C + $


%

(2.95)

by (2.93) and (2.78)-(2.80), and cos θ ≤ +C − C ∗ + $


≤ +[C , C ∗ ]+ $


≤ 2+[C , C ∗ ]+ $

%

% +P+ − P− + $


16

%,


%

(2.96)

by (2.94) and (2.80). In concert, (2.95)-(2.96) prove (2.82), thus concluding the proof of the theorem. ! A similar type of argument as in the proof of (2.78) can be used to obtain resolvent norm formulas for the Cauchy-Clifford operator. Proposition 2.9. Suppose that Ω ⊂ Rn is a two-sided NTA domain with an Ahlfors regular boundary, and let θ be as in (2.74). Then 4 2 1 −1 $ % ) 1+ √ (2.97) +(λI − C ) + = )) 2 ) L L (∂Ω,dσ)⊗C$n (sec θ) 4λ2 tan2 θ + 1 + 2λ tan2 θ − 1 )λ + 21 ) for every λ ∈ R \ {± 12 }.

Proof. Thanks to (2.48), it suffices to show that 4 ) ) ) ) % = )λ + 1 ) 1 + +λI + C + $ L L2 (∂Ω,dσ)⊗C$n

2

2 √ 2 2 (sec θ) 4λ tan θ + 1 + 2λ tan2 θ − 1

(2.98)

for every λ ∈ R, if λ @= − 12 (incidentally, the case λ = − 12 is covered by (2.77)). With this goal in mind, we observe that for each number λ ∈ R and each function f ∈ L2 (∂Ω, dσ) ⊗ C"n , we may write (λI + C )f = (λ − 21 )f + ( 12 I + C )f = (λ − 12 )(f+ − f+ ) + f+ = (λ + 21 )f+ − (λ − 12 )f− where, as before, f± := (± 12 I + C )f . Thus, . / % = sup +(λI + C )f +2 /+f +2 : f ∈ L2 (∂Ω, dσ) ⊗ C"n +λI + C +2 $ 2 L L (∂Ω,dσ)⊗C$n

. / = sup +(λ + 21 )f+ − (λ − 21 )f− +2 /+f+ − f− +2 : f± ∈ H±2 (∂Ω) (2.99) ( 5 (λ + 12 )2 +f+ +2 + 2(λ2 − 14 ))f+ , f− * + (λ − 12 )2 +f− +2 = sup : f± ∈ H±2 (∂Ω) . +f+ +2 − 2)f+ , f− * + +f− +2

For f± ∈ H±2 (∂Ω), nonzero, let α be such that cos α = )f+ , f− */+f+ ++f− + and set t := +f+ +/+f− +. An elementary (yet tedious) analysis shows that λ @= − 12 and α ∈ (0, π/2) max t>0

(λ + 12 )2 t2 + 2(λ2 − 14 )(cos α)t + (λ − 21 )2 t2 − 2 (cos α)t + 1 " #2 . / 2 √ = λ + 12 1+ . (sec α) 4λ2 tan2 α + 1 + 2λ tan2 α − 1

(2.100) !

Then formula (2.98) follows from (2.99) and (2.100). Next, we record an useful consequence of Theorem 2.8.

Corollary 2.10. Assume that Ω ⊂ Rn is a two-sided NTA domain with an Ahlfors regular boundary. The following are equivalent: (i) There holds L2 (∂Ω, dσ) ⊗ C"n = H+2 (∂Ω) ⊕ H−2 (∂Ω), 17

orthogonal sum.

(2.101)

(ii) The Cauchy-Clifford operator satisfies either of the following six conditions C − C ∗ = 0,

C C ∗ − C ∗ C = 0,

+ 12 I + C + $

% = 1, + − 1 I + C + $ 2

L L2 (∂Ω,dσ)⊗C$n % = 1, +C + $ 2 2 L L (∂Ω,dσ)⊗C$n

+C + C ∗ + $



% = 1,

(2.102)

% = 1.

(iii) The Clifford-Szeg¨ o projections satisfy either of the following four conditions P+ P− = 0,

P− P+ = 0,

P+ + P− = I, +(P+ − P− )−1 + $ L

L2 (∂Ω,dσ)⊗C$

n

% = 1.

(2.103)

Proof. This is a consequence of Theorem 2.8, which shows that each of the conditions (i) − (iii) above is equivalent to having θ = π/2. ! Remark. As is apparent from Theorem 2.8, in general we have + ± 12 I + C + $ +C + C ∗ + $ L


L2 (∂Ω,dσ)⊗C$n

% ≥ 1, +C + $

% ≥ 1,


% ≥ 1, 2

+(P+ − P− )−1 + $ L

L2 (∂Ω,dσ)⊗C$

n

% ≥ 1.

(2.104)

What the above corollary shows is that $ % extremal case in each of these four inequalities occurs precisely when < ) H+2 (∂Ω) , H−2 (∂Ω) = π2 . Later on, we shall show that, in turn, this latter condition actually determines the shape of Ω. Next, consider

  N &  X − Xj R± (∂Ω) := # a : N ∈ N, X ∈ Ω , a ∈ C" . j j ∓ j n   |X − Xj |n

(2.105)

j=1

That is, R± (∂Ω) are the right Clifford module spanned by rational functions of the form X ∈ ∂Ω, when Z ∈ Ω∓ .

X−Z |X−Z|n ,

Theorem 2.11. Suppose that Ω ⊂ Rn is a two-sided NTA domain whose boundary is Ahlfors ¯ σ := Hn−1 7 ∂Ω. Then the spaces R± (∂Ω) are regular and, as usual, set Ω+ := Ω, Ω− := Rn \ Ω, 2 dense in H± (∂Ω), respectively. Hence, as a consequence of this and (2.71), one has " $ %# cos < ) H+2 (∂Ω) , H−2 (∂Ω) (2.106) '       )P (X), P (X)* dσ(X)   + − ∂Ω : P ∈ R (∂Ω) = sup "9 . ± ± # " # 1/2 9 1/2     2 dσ(X)   ∂Ω |P+ (X)|2 dσ(X) |P (X)| − ∂Ω

Proof. We proceed in a series of steps, starting with

Step I. For every Λ ∈ (H+2 (∂Ω))∗ there exists g ∈ L2 (∂Ω, dσ) ⊗ C"n satisfying ( 12 I + C ∗ )g = g and for which ' )f, g* dσ, ∀ f ∈ H+2 (∂Ω). (2.107) Λ(f ) = ∂Ω

18

Indeed, as a consequence of Riesz’s representation theorem, there exists h ∈ H+2 (∂Ω) with the property that ' Λ(f ) = )f, h* dσ, ∀ f ∈ H+2 (∂Ω). (2.108) ∂Ω

Since every f ∈ H+2 (∂Ω) can be expressed as f = ( 12 I + C )f , it follows that (2.107) holds with g := ( 21 I + C ∗ )h. Step II. A functional Λ ∈ (H+2 (∂Ω))∗ vanishes identically if and only if # " X −· # a = 0, ∀ a ∈ C"n , ∀ X ∈ Ω− . Λ |X − ·|n

(2.109)

X−· 2 To prove this claim, we first observe that |X−·| n # a ∈ H+ (∂Ω) for every a ∈ C"n and every X ∈ Ω− . We may therefore use the representation formula (2.107), which holds for some function g ∈ L2 (∂Ω, dσ) ⊗ C"n with ( 12 I + C ∗ )g = g, in order to conclude that ' : ; :' ; X −Y X −Y 0= # a , g(Y ) dσ(Y ) = # g(Y ) dσ(Y ) , a , (2.110) n n ∂Ω |X − Y | ∂Ω |X − Y |

for every X ∈ Ω− (here, (2.9) is also used). In turn, this implies that C(ν # g) = 0 in Ω− and, further, that (− 12 I + C )(ν # g) = 0 on ∂Ω by going nontangentially to the boundary. By (2.21), it follows that ( 21 I + C ∗ )g = 0 on ∂Ω, hence ultimately g = 0 given that ( 12 I + C ∗ )g = g. This and (2.107) then prove that Λ ≡ 0.

Step III. The spaces R± (∂Ω) are dense in H±2 (∂Ω), respectively. For the choice plus of the sign, this is a direct consequence of Hahn-Banach’s extension theorem and the result proved in Step II. The choice minus of the sign is handled similarly. ! To state our next result, for any two vectors X = (xj )j , Y = (yj )j in Rn we define " # X × Y := xj yk − xk yj . 1≤j,k≤n

(2.111)

Theorem 2.12. Assume that Ω ⊂ Rn is a two-sided NTA domain whose boundary is Ahlfors ¯ σ := Hn−1 7 ∂Ω. Then regular and, as before, set Ω+ := Ω, Ω− := Rn \ Ω, ' $ 2 % π (Y − X+ ) # (Y − X− ) 2 < ) H+ (∂Ω) , H− (∂Ω) = ⇐⇒ dσ(Y ) = 0 ∀ X± ∈ Ω± . (2.112) n n 2 ∂Ω |Y − X+ | |Y − X− |

Coordinate-wise,

 9    ∂Ω $ 2 % π 2 < ) H+ (∂Ω) , H− (∂Ω) = ⇐⇒ 9  2   ∂Ω

#Y −X+ ,Y −X− $ |Y −X+ |n |Y −X− |n

dσ(Y ) = 0 and

(Y −X+ )×(Y −X− ) |Y −X+ |n |Y −X− |n

dσ(Y ) = 0

∀ X± ∈ Ω± .

Proof. Note that (2.10) implies that, for every X± ∈ Ω± and every a± ∈ C"n , ' : ; Y − X+ Y − X− # a , # a + − dσ(Y ) n |Y − X− |n ∂Ω |Y − X+ | : ' (Y − X ) # (Y − X ) ; + − =− dσ(Y ) , a # a . − + n n ∂Ω |Y − X+ | |Y − X− | 19

(2.113)

(2.114)

This identity shows that if the integral in the right hand-side of (2.114) vanishes X± %∈ Ω± $ 2 for every 9 2 then ∂Ω )P+ , P− * dσ = 0 for every P± ∈ R± (∂Ω). Hence, in this case, < ) H+ (∂Ω) , H− (∂Ω) = π2 by (2.106). $ % Conversely, if < ) H+2 (∂Ω) , H−2 (∂Ω) = π2 then H+2 (∂Ω)⊥H−2 (∂Ω) by (2.72). In particular, for −X+ 2 every X± ∈ Ω± and a± ∈ C"n , the Clifford algebra-valued functions |YY−X n # a+ ∈ H+ (∂Ω) and +| Y −X− |Y −X− |n

# a− ∈ H−2 (∂Ω) are orthogonal. Thus, by (2.114), the integral in the right hand-side of (2.112) necessarily vanishes for every X± ∈ Ω± . Finally, (2.113) is a consequence of (2.112) and the fact that the components of X # Y are recovered in )X, Y * and X × Y . !

3

Hardy spaces and SKT domains

We start with some motivation for the material in this section. Assume that Ω is a simply connected, open subset of R2 (naturally identified with the complex field C) and whose boundary ∂Ω is a rectifiable (orientable) curve. Substituting the Cauchy-Riemann operator ∂¯ in place of the Dirac operator D in the considerations in § 2.3, it is then possible to define Hardy spaces H±2 (∂Ω) in a similar fashion as before. In this setting, the Clifford-Szegö projections P± map from L2 (∂Ω, dσ)⊗C onto H±2 (∂Ω). G. David has established the following result (which appears as Theoreme 2 on p. 236 in [4]): Theorem 3.1. Given δ > 0 there exists ε > 0 with the property that +I − P+ − P− + $

L L2 (∂Ω,dσ)⊗C

% ≤ ε =⇒ ∂Ω is chord-arc with constant ≤ 1 + δ,

(3.1)

in the class of chord-arc domains Ω ⊆ R2 .

Recall that a curve Σ is called chord-arc, with constant ≤ C, if length of the arc in between z1 and z2 is ≤ C|z1 − z2 |,

∀ z1 , z2 ∈ Σ.

(3.2)

An equivalent way of describing (3.2) is to demand that |s − t| ≤ C|z(s) − z(t)| for s, t ∈ R, where z(s) is a parametrization of ∂Ω with |z ' (s)| = 1. One of our goals in this section is to prove a higher dimensional extension of Theorem 3.1. David’s proof of this result uses the arc-length parametrization of ∂Ω in order to reduce matters to an estimate from below of the norm of the commutator between the Hilbert transform (on R) and a unitary operator (on L2 (R)) associated with a certain change of variables. This strategy makes essential use of the two-dimensional setting, so a different approach is required in higher dimensions. To handle this problem, we shall make use of the recent results from [10]. First we establish the following result, which has independent interest. Theorem 3.2. Assume that Ω ⊂ Rn is a bounded, two-sided NTA domain whose boundary is Ahlfors regular. Then there exists C > 0, depending only on the NTA and Ahlfors constants of Ω, such that " # √ n dist ν , VMO (∂Ω, dσ) ≤ C cot θ (3.3)

$ % where θ :=< ) H+2 (∂Ω) , H−2 (∂Ω) . Furthermore, if θ is sufficiently close to π/2 (relative to the NTA and Ahlfors constants of Ω) then Ω is a δ-SKT domain, for some δ = δ(θ) > 0 such that δ , 0 as θ - π/2. 20

Proof. By (2.80), " # dist C − C ∗ , Cp (L2 (∂Ω, dσ) ⊗ C"n ) ≤ +C − C ∗ + $


% ≤ 2 cot θ,

(3.4)

so (3.3) follows from this and (2.30). Finally, the last part in the statement of the theorem is a consequence of (3.3) and Theorem 2.2. ! $ 2 % The above theorem shows that θ :=< ) H+ (∂Ω) , H−2 (∂Ω) encodes significant information about the regularity of Ω. In fact, the closer θ is to π/2, the more regular Ω becomes. As we shall see in Section 4, the limit case θ = π/2 is special in that not only does this ensure that Ω is smooth, but such a specification actually determines the shape of Ω. Our next result can be viewed as the higher dimensional analogue of Theorem 3.1. To state it, recall that ∆(X, r) := B(X, r) ∩ ∂Ω and that ωn−1 stands for the surface area of the unit sphere in Rn . Theorem 3.3. For every δ > 0 there exist ε > 0 and R > 0 with the property that ) n−1 ) )H ) (∆(X, r)) $ % ) +I − P+ − P− + ≤ ε =⇒ sup − 1)) ≤ δ, ) ωn−1 rn−1 L L2 (∂Ω,dσ)⊗C$n X∈∂Ω, 0 0 and R > 0 such that ) ) " # ) σ(∆(X, r)) ) ) ) ≤ δ. dist ν , VMO (∂Ω, dσ) ≤ ε =⇒ sup − 1 (3.6) ) ) n−1 X∈∂Ω, 0 0.

Next, with B as above, define the potential energy of µ ∈ P(B) as ' ' E(µ) := − E(X − Y ) dµ(X)dµ(Y ), B

(4.15)

B

where E is as in (2.13). According to Theorem 6.3 in [33], a minimizer µ of E, called equilibrium distribution, always exists. Specifically, we have: Proposition 4.6. If B is compact, there exists a probability measure µ ∈ P(B) that minimizes the energy. That is, µ solves the minimal-energy problem E(µ) = γ, where . / γ := min E(µ) : µ ∈ P(B) . (4.16) We shall also need the following uniqueness result (cf. Theorem 9.1. in [33]):

$

Proposition 4.7. Let µ' be a probability measure on a compact set B such that U µ = γ ' , constant, $ µ' -a.e. on B and U µ ≤ γ ' everywhere in Rn . Then µ = µ' .

(4.17)

The equilibrium distribution µ of a compact set B has a number of distinguished properties. First, µ is carried by ∂B,

(4.18)

(in the sense that µ(B \ ∂B) = 0). This implies that γ can, in fact, be obtained by minimizing ' ' Eb (µ) := − E(X − Y ) dµ(X)dµ(Y ), (4.19) ∂B

∂B

< ⊂ B such over all Borel probability measures µ on ∂B. Second, there exists an exceptional set B that U µ (X) ≤ γ, U µ (X) = γ,

(4.20)

∀ X ∈ Rn ,

< ∀ X ∈ B\B,

(4.21)

< ⊂ B, recall the notion where γ is as in (4.16). To describe the properties of the exceptional set B of capacity introduced earlier. We have: and

< ⊆ ∂B and Cap (B) < = 0, B

if B ⊆ Rn , n ≥ 3, is compact and X0 ∈ B is such that ∃ C, R > 0 with Cap ({X ∈ B : |X − X0 | ≥ r} ≥ Crn−2 ,

See Theorem 7.1 and Theorem 10.1 in [33]. 24

∀ r ∈ (0, R)

(4.22)

5

< (4.23) =⇒ X0 ∈ B\B.

Proposition 4.8. Let B := Ω, where Ω ⊆ Rn , n ≥ 3, is a bounded, connected open set which satisfies an interior corkscrew condition. Then U µ (X) = γ for every X ∈ B. < is empty. Indeed, if X0 ∈ B, < then X0 ∈ ∂Ω by (4.22) in which case Proof. We claim that B the interior corkscrew condition implies that there exist X∗ ∈ Ω and R > 0, M > 1 such that B(X∗ , r/M ) ⊆ {X ∈ B : |X − X0 | ≥ r} for every r ∈ (0, R). In concert with (4.13)-(4.14) this then implies Cap ({X ∈ B : |X − X0 | ≥ r} ≥ Crn−2 for every r ∈ (0, R) which, in turn, forces < by (4.23). This contradicts the fact that X0 ∈ B < and shows that B < is empty. With this X0 ∈ B\B in had, the desired conclusion follows from (4.21). !

We are now in a position to state and prove the potential theoretic result which is most significant for the goals we have in mind.

Theorem 4.9. Let Ω ⊆ Rn , n ≥ 3, be a bounded, connected open set satisfying a two-sided corkscrew condition and whose boundary is Ahlfors regular. Then the equilibrium distribution on Ω is constant, in the sense that µ = λ Hn−1 7 ∂Ω

for some λ > 0,

(4.24)

if and only of S 1 is constant on Ω. Proof. Set σ := Hn−1 7 ∂Ω. In one direction, if dµ = λ dσ for some constant λ > 0, then we may write λ (S 1)(X) = U µ (X) = γ for every X ∈ Ω, by (4.11) and Proposition 4.8. This shows that S 1 is constant in Ω. In the opposite direction, if S 1 is constant on Ω, take λ := 1/σ(∂Ω) > 0 so that λ σ ∈ P(∂Ω). For the type of domains specified in the statement of the theorem, it has been proved in [10] that the operator 1,pn/(n−1)

S : Lp (∂Ω, dσ) −→ Wloc

(Rn ),

(4.25)

1,q is well-defined and bounded for every p ∈ (1, ∞), where Wloc (Rn ), 1 < q < ∞, denotes the local p version of the usual scale of L -based Sobolev spaces of order one in Rn . From this and standard embedding results it follows that

S 1 is nonnegative and continuous in Rn , harmonic in Rn \ ∂Ω and, given that n ≥ 3, decays at infinity.

(4.26)

Thus, the maximum principle applies and gives that Sλ ≤ γ in Rn . With the help of Proposition 4.7 we may then conclude that µ = λ σ, finishing the proof. ! Recall the boundary Hardy spaces H±2 (∂Ω) and note that 1 ∈ H+2 (∂Ω), where 1 denotes the constant function on ∂Ω. Below we show that whether the equilibrium distribution of Ω is a constant hinges on whether H−2 (∂Ω) is contained in L20 (∂Ω, dσ) ⊗ C"n . Proposition 4.10. Let Ω ⊆ Rn be a bounded, connected, two-sided NTA domains with an Ahlfors regular boundary. Then 0 1⊥ 1 ∈ H−2 (∂Ω) ⇐⇒ the equilibrium distribution of Ω is a constant,

in the sense of (4.24).

25

(4.27)

" # Proof. If 1 ⊥ H−2 (∂Ω) = Im − 12 I + C : L2 (∂Ω, dσ) ⊗ C"n , it follows that (− 12 I + C ∗ )1 = 0 and,

further, ( 12 I + C )ν = 0 by (2.21). Next, recall (2.37) and set u := Cν ∈ H 2 (Ω). It follows that u|∂Ω = ( 21 I + C )ν = 0 hence u = C(u|∂Ω ) = 0 in Ω, by (2.41). In summary, the above reasoning shows that DS 1 = Cν = 0 in Ω, where D is as in (2.11). As a consequence, |∇S 1| = |DS 1| = 0 in Ω which proves that S 1 is constant in Ω. At this stage, the fact that the equilibrium distribution of Ω is a constant is a direct consequence of Theorem 4.9. This justifies the left-to-right implication in (4.27). In fact, the converse implication is also valid since all steps above are reversible. ! Let Ω be a bounded, open subset of Rn . The following conjecture has been formulated by P. Gruber: the equilibrium distribution of Ω is a constant (in the sense of (4.24)) if and only if Ω is a ball.

(4.28)

Let us highlight a connection between Conjecture 4.3 and Gruber’s conjecture (4.28). Specifically, we have: Proposition 4.11. In the class of bounded, connected open sets in Rn , n ≥ 3, satisfying a two-sided corkscrew condition and whose boundaries are Ahlfors regular, Gruber’s conjecture =⇒ Conjecture 4.3.

(4.29)

Proof. As already noted earlier, it is clear that if Ω is a ball then K = K ∗ . In view of Theorem 4.9 it suffices to show that if Ω ⊂ Rn is as in the statement of the proposition and has the property that K = K ∗ , then S 1 = constant in Ω. This, however, follows from Lemma 4.5 and the fact that K1 = 21 . ! Work of L.E. Payne and G.A. Philippin in [28], [27] shows that Gruber’s conjecture has a positive answer in the class of starlike C 2,ε -domains in Rn , n ≥ 3. Subsequent work by W. Reichel in [29], [30], has shown that Gruber’s conjecture holds in the class of C 2,ε -domains. On the other hand, E. Martensen has shown in [24] that this is also the case for piecewise smooth domains in R2 . Finally, in [25], O. Mendez and W. Reichel have shown that Gruber’s conjecture is valid both in the class of convex domains in Rn , n ≥ 3, as well as the class of two-dimensional Lipschitz domains. We shall further strengthen the latter result in Corollary 4.14 in § 4.2. For future purposes, here we only wish to record the higher-dimensional version of the main result in [25] as follows: Theorem 4.12. Let Ω ⊂ Rn , n ≥ 3, be a bounded convex domain with the property that S1 = constant on ∂Ω.

(4.30)

Then Ω is a ball.

4.2

The two dimensional case

Here we continue the discussion initiated in the previous subsection, by focusing on the two dimensional setting. The main theorem in this subsection is contained in the following perturbation result, involving the classical Cauchy operator (2.31). Theorem 4.13. Assume that Ω ⊂ R2 is a bounded, connected UR domain satisfying ∂Ω = ∂Ω. Then for every δ > 0 there exists ε > 0, which depends on δ and the geometrical characteristics 26

of Ω (more specifically, the diameter and the constants implicit in the definition of UR domains), with the property that +C ∗ 1 − 12 +L2 (∂Ω,dσ)⊗C < ε =⇒

inf +χΩ − χD +L2 (R2 ) < δ.

D disk

(4.31)

Furthermore, in the class of domains as in the first part of the statement, the following holds: K ∗ 1 = constant on ∂Ω ⇐⇒ Ω is a disk.

(4.32)

Proof. Inspection of (4.31) reveals that C = K + iQ where, with τ := −iν denoting the unit tangent along ∂Ω, ' )ζ − z, τ (ζ)* 1 f (ζ) dσ(ζ), Qf (z) := lim |ζ − z|2 ε→0+ 2π

(4.33)

z ∈ ∂Ω.

(4.34)

ζ∈∂Ω

|z−ζ|>ε

Let us also note that Q∗ = ∂τ S,

(4.35)

with S as in (4.3), and ∂τ the directional derivative along τ . As a consequence, C ∗ = K ∗ + i∂τ S

(4.36)

+(− 12 I + K ∗ )1+2L2 (∂Ω,dσ) + +∂τ S1+2L2 (∂Ω,dσ) = +C ∗ 1 − 12 +2L2 (∂Ω,dσ)⊗C .

(4.37)

and

Next, a direct calculation shows that for every harmonic function u in R2 \ Ω there holds " # div 21 X|∇u(X)|2 − )X, ∇u(X)*∇u(X) = 0, X ∈ R2 \ Ω. (4.38)

Hence, assuming that N (∇u) ∈ L2 (∂Ω, dσ), Theorem 2.4 and (2.18) give ' # " 2 1 )X, ν(X)*|∇u(X)| − )X, ∇u(X)*∂ u(X) dσ(X) 0 = ν 2 ∂(BR \Ω)

=

'

∂BR

−

'

"

∂Ω

2 1 2 R|∇u(X)|

"

# − R−1 |)X, ∇u(X)*|2 dσ(X)

2 1 2 )X, ν(X)*|∇u(X)|

# − )X, ∇u(X)*∂ν u(X) dσ(X),

(4.39)

where BR is the ball of radius R (assumed to be sufficiently large) with center at the origin. Let us specialize the above to the case when u := S 1. From the definition of the single layer we then have the following asymptotic expansion (∇u)(X) =

1 2 1 2π H (∂Ω)X/|X|

+ O(|X|−2 ) as |X| → ∞. 27

(4.40)

Using (4.40) in (4.39) and letting R → ∞ then yields ' " # " #2 2 1 1 1 . )X, ν(X)*|∇u(X)| − )X, ∇u(X)*∂ u(X) dσ(X) = − H (∂Ω) ν 2 4π

(4.41)

∂Ω

Next, consider the tangential derivative ∂τ along ∂Ω. Two properties of this operator, established in [10], are going to be of importance for us here. First, ∂τ w is well defined for any function w with N (∇w) ∈ L2 (∂Ω, dσ) and for which the pointwise nontangential traces (∇w)|∂Ω , w|∂Ω exist σ-a.e. on ∂Ω. Second, if w is as above then (∇w)|∂Ω = [τ · (∇w)|∂Ω ]τ + [ν · (∇w)|∂Ω ]ν.

(4.42)

Returning to the mainstream discussion, these considerations ensure that ∇u = (∂τ u)τ + (∂ν u)ν and |∇u|2 = |∂τ u|2 + |∂ν u|2 σ-a.e. on ∂Ω,

+∂τ u+2L2 (∂Ω,dσ) + +∂ν u − 1+2L2 (∂Ω,dσ) = +C ∗ 1 − 12 +2L2 (∂Ω,dσ) .

(4.43) (4.44)

In particular, assuming that +C ∗ 1 − 12 +2L2 (∂Ω,dσ) ≤ ε, we obtain +∂τ u+L2 (∂Ω,dσ) + +∂ν u − 1+L2 (∂Ω,dσ) ≤ Cε.

(4.45)

To continue, for σ-a.e. X ∈ ∂Ω we write 2 1 2 )X, ν(X)*|∇u(X)|

− )X, ∇u(X)*∂ν u(X) = − 21 )X, ν(X)* + ρ(X)

(4.46)

where the error term ρ(X) is given by 0 1 ρ(X) := − 12 )X, ν(X)* |∂ν u(X)|2 − 1 − )X, τ (X)*∂τ u(X)∂ν u(X). Note that, by (4.45)

'

∂Ω

|ρ(X)| dσ(X) ≤ Cε,

(4.47)

where C > 0 depends only on the geometrical characteristics of Ω. In conjunction with (4.41), this implies ' " " #2 # 1 2 1 1 H (∂Ω) )X, ν(X)*|∇u(X)| − )X, ∇u(X)*∂ u(X) dσ(X) = − ν 4π 2 =

1 2

'

∂Ω

∂Ω

)X, ν(X)* dσ(X) + O(ε) = H2 (Ω) + O(ε),

(4.48)

where the last equality follows from (2.42) with .v (X) := X. In turn, this implies " #2 H1 (∂Ω) − 4πH2 (Ω) ≤ Cε H2 (Ω).

(4.49)

+χΩ − χDε +L2 (R2 ) < Cε1/4 ,

(4.50)

We can now appeal to the following isoperimetric stability result, given as Theorem 1.1 in [23]. Given (4.49), there exists a disk Dε such that

28

where C depends only on the geometrical characteristics of Ω. From this, (4.31) readily follows, completing the proof of the first part of the theorem. As for (4.32), first note that if K ∗ 1 is a constant function in ∂Ω, then Lemma 4.5 gives that K ∗ 1 = 12 and S 1 is constant in Ω. As shown in [10], the latter condition forces ∂τ S1 = 0 on ∂Ω. In concert with (4.36), this gives that C ∗ 1 = 12 on ∂Ω. Having shown this, (4.31) then gives that Ω coincides a.e. with a disk, proving (4.32). Since we are also assuming that ∂Ω = ∂Ω, elementary topological considerations then show that Ω must actually be a disk. ! We conclude this section with a few corollaries of Theorem 4.13 (and its proof), starting with: Corollary 4.14. Assume that Ω ⊂ R2 is a connected, bounded UR domain satisfying ∂Ω = ∂Ω. Then S 1 = constant in Ω ⇐⇒ S1 = constant on ∂Ω ⇐⇒ K = K ∗ ⇐⇒ K ∗ 1 = constant on ∂Ω ⇐⇒ K ∗ 1 =

1 2

on ∂Ω ⇐⇒ Ω is a disk.

(4.51)

In particular, Conjecture 4.2 and Conjecture 4.3 are both valid in the class of connected, bounded UR domains Ω ⊂ R2 for which ∂Ω = ∂Ω. Proof. If Ω is a disk, it is easy to see that K = K ∗ , so in this case K ∗ 1 = K1 = 21 . Everything else now follows from Theorem 4.13 and Lemma 4.5. ! Next we show that the equivalence ∇S 1 = 0 in Ω ⇔ Ω is a disk is stable under L∞ perturbations. Corollary 4.15. Suppose Ω ⊂ R2 is a bounded, connected UR domain with the property that ∂Ω = ∂Ω. Then for every δ > 0 there exists ε > 0, which depends on δ and the geometrical characteristics of Ω, for which +∇S 1+L∞ (Ω) < ε =⇒

inf +χΩ − χD +L2 (R2 ) < δ.

D disk

(4.52)

Proof. Note that (4.52) entails +(− 21 I + K ∗ )1+L∞ (∂Ω,dσ) = +∂ν S 1+L∞ (∂Ω,dσ) ≤ +∇S 1+L∞ (Ω) and +∂τ S1+L∞ (∂Ω,dσ) = +)τ, ∇S 1*+L∞ (∂Ω,dσ) ≤ +∇S 1+L∞ (Ω) . In combination with (4.37) these imply that +C ∗ 1 − 12 +L2 (∂Ω,dσ) is small if +∇S 1+L∞ (Ω) is small, so the desired conclusion follows from Theorem 4.13. ! Corollary 4.16. For every δ > 0 there exists ε > 0 for which ( Ω is a δ-SKT domain and ∗ $ % < ε =⇒ +C − C + 2 L L (∂Ω,dσ)⊗C inf D disk +χΩ − χD +L2 (R2 ) < δ.

(4.53)

in the class of all connected, bounded domains Ω ⊂ R2 which satisfy a two-sided local John condition and have Ahlfors regular boundaries, with a certain fixed common bound on the geometrical constants involved in describing these characteristics. Proof. Since C 1 = 21 , +C − C ∗ + $

% < ε implies that +C ∗ 1 − 1 + 2 2 L (∂Ω,dσ)⊗C < ε. Granted

L L2 (∂Ω,dσ)⊗C

this, Theorem 2.2 and Theorem 2.3 give that Ω is a δ-SKT domain for δ = δ(ε) > 0 with δ(ε) → 0 as ε → 0+ . Now, (4.53) follows from this and Theorem 4.13. ! 29

4.3

Higher dimensional results

The nature of the results in this subsection is that certain analytical conditions (involving singular integral operators) do in fact determine the shape of the underlying domain. Our first major result in this regard is as follows. Theorem 4.17. Let Ω ⊂ Rn be a UR domain which satisfies ∂Ω = ∂Ω. Then C = C ∗ ⇐⇒ Ω is either a ball, the complement of a (closed) ball, or a half-space.

(4.54)

Proof. To get started, (2.20) gives C = C∗

⇐⇒ ν(X) # (X − Y ) = −(X − Y ) # ν(Y ) for σ-a.e. X, Y ∈ ∂Ω.

(4.55)

Now, if Ω = B(Z0 , R), for each X, Y ∈ ∂Ω we have ν(X) = (X − Z0 )/R, ν(Y ) = (Y − Z0 )/R, and since (X − Z0 ) # (X − Y ) = (X − Z0 ) # [(X − Z0 ) − (Y − Z0 )]

= −R2 − (X − Z0 ) # (Y − Z0 ) = −(X − Y ) # (Y − X0 ),

(4.56)

we obtain ν(X) # (X − Y ) = −(X − Y ) # ν(Y ) for every X, Y ∈ ∂Ω. Hence, C = C ∗ . Likewise, the last equality in (4.55) checks out if Ω is the complement of a (closed) ball, or a half-space (in which case ∂Ω is a (n − 1)-plane). This proves the left-to-right implication in (4.54). The crux of the matter is, of course, the opposite implication, to which we now turn. To this end, assume that the domain Ω is as in the statement of the theorem and, in addition, satisfies ν(X) # (X − Y ) = −(X − Y ) # ν(Y ) for σ-a.e. X, Y ∈ ∂Ω. From this and (2.4), it follows that ν(X) =

X −Y X −Y # ν(Y ) # , |X − Y | |X − Y |

for σ-a.e. X, Y ∈ ∂Ω.

(4.57)

Fixing Y , this shows that, after eventually modifying ν on a set of σ-measure zero, ν ∈ C 0 (∂Ω). With this in hand and relying on the results proved in [11], we may deduce that Ω is a strongly C 1 -domain.

(4.58)

Next, we record a general identity to the effect that a # b # a = |a|2 b − 2)a, b*a,

∀ a, b ∈ Rn .

(4.59)

Indeed, for every a, b ∈ Rn formula (2.8) allows us to write a # b = −b # a − 2)a, b* which further implies that a # b # a = −b # a2 − )a, b*a = |a|2 b − 2)a, b*a, proving (4.59). When used in the context of (4.57), formula (4.59) yields : X −Y ; X −Y , ν(Y ) , |X − Y | |X − Y |

(4.60)

ν(0) = −en = (0, ..., 0, −1) ∈ Rn .

(4.61)

ν(X) = ν(Y ) − 2

for all X, Y ∈ ∂Ω. To continue, after a translation and a rotation, it can be assumed that 0 ∈ ∂Ω

and

30

In this setting, let ϕ : {x' ∈ Rn−1 : |x' | < r} −→ R,

ϕ(0' ) = 0,

∇' ϕ(0' ) = 0'

(4.62)

(where ∇' denotes the gradient with respect to x' ∈ Rn−1 ) be a C 1 function with whose graph coincides with ∂Ω in a neighborhood of 0 ∈ Rn . More precisely, there exists an open cylinder C := {(x' , xn ) : |x' | < r, |xn | < a} such that C ∩ Ω = C ∩ {(x' , xn ) : xn > ϕ(x' )},

(4.63)

C ∩ ∂Ω = C ∩ {(x' , xn ) : xn = ϕ(x' )}.

(4.64)

Hence, (∇' ϕ(x' ), −1) ν(x' , ϕ(x' )) = = , 1 + |∇' ϕ(x' )|2

if x' is near 0' ∈ Rn−1 .

(4.65)

By making Y = 0 in (4.60) and using the fact that ν(0) = −en we arrive at ν(X) = −en + 2

xn X, |X|2

for X = (x' , xn ) ∈ ∂Ω near 0 ∈ Rn .

(4.66)

In particular, for X = (x1 , ..., xn ) ∈ ∂Ω near 0 ∈ Rn , νj (X) = 2

xj xn |X|2

for 1 ≤ j ≤ n − 1, and νn (X) = −1 + 2

x2n , |X|2

(4.67)

so that further, on account of (4.65) and (4.66), ∂j ϕ(x' ) = −

νj (X) 2xj xn 2xj ϕ(x' ) = , = νn (X) |X|2 − 2x2n |x' |2 − ϕ2 (x' )

1 ≤ j ≤ n − 1,

(4.68)

for each x' near 0' ∈ Rn−1 . Thus, in a neighborhood of 0' ∈ Rn−1 , ∂j ϕ(x' ) =

2xj ϕ(x' ) , − ϕ2 (x' )

|x' |2

1 ≤ j ≤ n − 1.

(4.69)

Consider now the expression ϕ(x' ) |x' |2 + ϕ2 (x' )

for x' @= 0.

(4.70)

Note that, for j ∈ {1, . . . , n − 1}, the partial derivative of (4.70) with respect to xj is a fraction with the numerator ∂j ϕ(x' )(|x' |2 + ϕ2 (x' )) − ϕ(x' )(2xj + 2ϕ(x' )∂j ϕ(x' )).

(4.71)

This, however, vanishes identically, thanks to (4.69). Hence, the expression in (4.70) is constant 1 for x' near 0' . Denoting the value of this constant by ± 2R , for some 0 < R ≤ ∞, and solving the ' ensuing equation in the unknown ϕ(x ), we obtain the following possible expressions for ϕ(x' ) for x' near 0' ∈ Rn−1 : = = ϕ+ (x' ) = R − |x' |2 + R2 , ϕ− (x' ) = −R + |x' |2 + R2 , ϕo (x' ) ≡ 0. (4.72) 31

= The solutions ϕ± (x' ) = ±R ∓ |x' |2 + R2 correspond, respectively, to the situation when the boundary of Ω agrees with that of B(±Ren , R), near 0 ∈ Rn . On the other hand, the solution ϕo (x' ) ≡ 0 corresponds to the case when ∂Ω is flat (i.e., is contained in a (n − 1)-plane) near 0 ∈ ∂Ω. In summary, the argument above shows that near each boundary point, ∂Ω is either flat, or agrees with a sphere. Thus, ∂Ω = As ∪ Af , where As is the subset of ∂Ω consisting of points near which ∂Ω agrees with some sphere, and Af is the subset of ∂Ω consisting of points near which ∂Ω is flat. We continue the proof, under the hypothesis that As is nonempty. To fix ideas, assume that X∗ ∈ ∂Ω is such that there exists a ball B ⊂ Rn with the property that ∂Ω and ∂B coincide near X∗ . In this case, consider Σ := ∂Ω ∩ ∂B which is therefore a closed subset of ∂B, with nonempty interior Σ◦ relative to ∂B (viewed as a topological space, with the structure inherited from Rn ). The claim we make at this stage is that ∂(Σ◦ ), considered in ∂B, is empty. Indeed, if X ∈ ∂(Σ◦ ) then X ∈ Σ and X ∈ / Σ◦ . In particular, X ∈ ∂Ω and we can run the same argument as above (this time, for X ∈ ∂Ω in place of 0 ∈ ∂Ω) in order to conclude that there exist an open neighborhood U of X along with either an Euclidean ball B ' , or a (n − 1)-plane π, such that U ∩ B ' = U ∩ ∂Ω, or U ∩ π = U ∩ ∂Ω. Hence, W := (U ∩ ∂B) ∩ Σ◦ is a nonempty open set (relative to the topology of ∂B) with the property that either W ⊆ U ∩ ∂Ω = U ∩ ∂B ' , or W ⊆ U ∩ ∂Ω = U ∩ π. As a consequence, either ∂B ∩ π, or ∂B ∩ ∂B ' , has nonempty interior in ∂B. This rules out the former eventuality and, further, forces B = B ' . In turn, this implies that U ∩ ∂B = U ∩ ∂Ω which shows that X ∈ U ∩ ∂B ⊆ Σ◦ and, further, that X ∈ Σ◦ , which is a contradiction. Thus, Σ◦ is an open, nonempty, boundaryless subset of the connected topological space ∂B. We can therefore conclude that Σ = ∂B, i.e. ∂B ⊆ ∂Ω. As a consequence of the reasoning above, ∂Ω is a union of spheres if As is nonempty. Since, nonetheless, the equality ν(X) # (X − Y ) = −(X − Y ) # ν(Y ) may fail if X, Y belong to different spheres, we may ultimately conclude that ∂Ω is a sphere itself, say ∂Ω = ∂B, for some ball B ⊂ Rn . That actually Ω = B, or Ω = Rn \ B then follows from (4.63) and the following general topological result Let O1 , O2 be two open subsets of Rn with the property that ∂O1 = ∂O2 @= ∅. Then ∀ x ∈ ∂O1 ∃ r > 0 such that B(x, r) ∩ O1 = B(x, r) ∩ O2 ⇒ O1 = O2 .

(4.73)

See [11] for a proof. Hence, Ω is a ball, or the complement of a closed ball if As is nonempty. A very similar reasoning shows that Ω is a half-space when Af is nonempty, thus finishing the proof of the theorem. ! To state our next result, recall the Riesz transforms Rk , 1 ≤ k ≤ n, from (2.23). In !the case when Ω ⊂ Rn is a half-space (so that ∂Ω is a (n − 1)-plane in Rn ), it is well-known that nk=1 Rk2 = −I and Rj Rk = Rk Rj for all j, k ∈ {1, ..., n}. It turns out that the same identities are also valid when Ω is a ball in Rn . Somewhat surprisingly, the validity of the aforementioned identities actually determines the shape of the domain in question. Concretely, we have the following. Theorem 4.18. Assume that Ω ⊂ Rn is a two-sided NTA domain with an Ahlfors regular boundary. Then Ω is a ball, the complement of a (closed) ball, or a half-space ⇐⇒

n & k=1

Rk2 = −I and Rj Rk = Rk Rj 32

∀ j, k ∈ {1, ..., n}.

(4.74)

Proof. From (2.24) it follows that C Mν = − 21 Squaring both sides of this identity then yields C Mν C Mν

=

1 4

n &

n &

(4.75)

Rk Mek .

k=1

Rj Rk Mej Mek .

j,k=1

− 14

n &

= − 14

n &

=

Rk2 +

1 4

k=1

&

Rj Rk Mej Mek

&

(Rj Rk − Rk Rj )Mej 1ek .

1≤j0=k≤n

Rk2 +

1 8

k=1

(4.76)

1≤j ? "' #1/2 2 lim sup sup − |ν − ν∆(x,r) | dσ = 0. (5.5) r→0+

x∈∂Ω

∆(x,r)

We also recall the following definition from [10]. Definition 5.7. Let Ω ⊂ Rn be an open set. This is said to satisfy a local John condition if there exist θ ∈ (0, 1) and R > 0 (required to be ∞ if ∂Ω is unbounded), called the John constants of Ω, with the following significance. For every p ∈ ∂Ω and r ∈ (0, R) one can find pr ∈ B(p, r) ∩ Ω, called John center relative to ∆(p, r) := B(p, r) ∩ ∂Ω, such that B(pr , θr) ⊂ Ω and with the property that for each x ∈ ∆(Q, r) one can find a rectifiable path γx : [0, 1] → Ω, whose length is ≤ θ−1 r and such that γx (0) = x,

γx (1) = pr , and dist (γx (t), ∂Ω) > θ |γx (t) − x|

∀ t ∈ (0, 1].

(5.6)

Finally, Ω is said to satisfy a two-sided local John condition if both Ω and Rn \ Ω satisfy a local John condition. Finally, following G. David and S. Semmes [6] we make the following. Definition 5.8. Call Σ ⊂ Rn uniformly rectifiable provided it is Ahlfors regular and the following holds. There exist ε, M ∈ (0, ∞) (called the UR constants of Σ) such that for each x ∈ Σ, n−1 n−1 R > 0, there is a Lipschitz map ϕ : BR → Rn (where BR is a ball of radius R in Rn−1 ) with Lipschitz constant ≤ M , such that $ % n−1 Hn−1 Σ ∩ BR (x) ∩ ϕ(BR ) ≥ εRn−1 . (5.7) If Σ is compact, this is required only for R ∈ (0, 1].

References [1] I. Chavel, Isoperimetric Inequalities, Cambridge Univ. Press, Cambridge, 2001. [2] R. Coifman, A. McIntosh and Y. Meyer, L’intégrale de Cauchy definit un opérateur borné sur L2 pour les courbes lipschitziennes, Ann. Math., 116 (1982), 361–388. [3] R. Coifman and Y. Meyer, Le théorème de Calder´ on par les ”méthodes de variable réelle”, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 7, A425–A428. [4] G. David, Courbes corde-arc et espaces de Hardy généralisés, Ann. Inst. Fourier (Grenoble), 32 (1982), no. 3, xi, 227–239. [5] G. David and D. Jerison, Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals, Indiana Univ. Math. J., 39 (1990), no. 3, 831–845. [6] G. David and S. Semmes, Singular Integrals and Rectifiable Sets in Rn : Beyond Lipschitz Graphs, Astérisque No. 193, 1991. [7] G. David and S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, AMS Series, 1993.

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[8] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. [9] H. Federer, Geometric Measure Theory, reprint of the 1969 edition, Springer-Verlag, 1996. [10] S. Hofmann, M. Mitrea and M. Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains, preprint (2007). [11] S. Hofmann, M. Mitrea and M. Taylor, Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains, J. Geom. Anal., 17 (2007), no. 4, 593–647. [12] A. Hurwitz, Sur le problème des isopérimétres, C. R. Acad. Sci. Paris, 132 (1901), 401–403. ´ [13] A. Hurwitz, Sur quelque applications géométrique des séries Fourier, Ann. Sci. Ecole Norm. Sup., (3) 19 (1902), 357–408. [14] D.S. Jerison and C.E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math., 46 (1982), no. 1, 80–147. [15] C.E. Kenig and T. Toro, Free boundary regularity for harmonic measures and Poisson kernels, Ann. of Math., 150 (1999), no. 2, 369–454. [16] C.E. Kenig and T. Toro, Poisson kernel characterization of Reifenberg flat chord arc domains, Ann. ´ Sci. Ecole Norm. Sup., (4) 36 (2003), no. 3, 323–401. [17] N. Kerzman, Singular integrals in complex analysis, pp. 3–41 in “Harmonic Analysis in Euclidean Spaces”, Proc. Sympos. Pure Math., XXXV, Part 2, Amer. Math. Soc., Providence, R.I., 1979. [18] N. Kerzman and E.M. Stein, The Cauchy kernel, the Szeg¨ o kernel, and the Riemann mapping function, Math. Ann., 236 (1978), no. 1, 85–93. [19] J. Král and D. Medkov´ a, On the Neumann-Poincaré operator, Czechoslovak Math. J., 48(123) (1998), no. 4, 653–668. [20] L. Lanzani and E.M. Stein, Szeg¨ o and Bergman projections on non-smooth planar domains, J. Geom. Anal., 14 (2004), no. 1, 63–86. [21] B. Lawson and M. Michelson, Spin Geometry, Princeton Univ. Press, Princeton, N.J., 1989. [22] M. Lim, Symmetry of a boundary integral operator and a characterization of a ball, Proc. Amer. Math. Soc., 45 (2) (2001), 537–543. [23] F. Maggi, Some methods for studying stability in isoperimetric type problems, Bull. Amer. Math. Soc., 45 (2008), 367–408. [24] E. Martensen, Eine Integralgleichung f¨ ur die logarithmische Gleichgewichtsbelegung und die Kr¨ ummung der Randkurve eines ebenen Gebiets, Bericht u ¨ber die Wissenschaftliche Jahrestagung der GAMM, Z. Angew. Math. Mech., 72 (1992), no. 6, T596–T599. [25] O. Mendez and W. Reichel, Electrostatic characterization of spheres, Forum Math., 12 (2000), no. 2, 223–245. [26] M. Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces, Lecture Notes in Mathematics, Vol. 1575, Springer-Verlag, Berlin, 1994. [27] L.E. Payne and G.A. Philippin, On some maximum principles involving harmonic functions and their derivatives, SIAM J. Math. Anal., 10 (1979), no. 1, 96–104.

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[28] G.A. Philippin, On a free boundary problem in electrostatics, Math. Methods Appl. Sci., 12 (1990), no. 5, 387–392. [29] W. Reichel, Radial symmetry for elliptic boundary-value problems on exterior domains, Arch. Rational Mech. Anal., 137 (1997), no. 4, 381–394. [30] W. Reichel, Radial symmetry for an electrostatic, a capillarity and some fully nonlinear overdetermined problems on exterior domains, Z. Anal. Anwendungen, 15 (1996), no. 3, 619–635. [31] S. Semmes, Chord-arc surfaces with small constant. I, Adv. Math., 85 (1991), no. 2, 198–223. [32] P. Van Lancker, The Kerzman-Stein theorem on the sphere, Complex Variables Theory Appl., 45 (2001), no. 1, 73–99. [33] J. Wermer, Potential Theory, Lecture Notes in Mathematics, Vol. 408, Springer-Verlag, Berlin-New York, 1974. [34] W. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. [35] M. Zinsmeister, Domaines de Lavrentiev, Publications Mathématiques d’Orsay, 85-3. Université de Paris-Sud, Département de Mathématiques, Orsay, 1985.

————————————– Steve Hofmann Department of Mathematics University of Missouri Columbia, MO 65211, USA e-mail: [email protected]

Emilio Marmolejo-Olea Instituto de Matemáticas Unidad Cuernavaca Universidad Nacional Autónoma de México A.P. 273-3 ADMON 3 62251 Cuernavaca, Mor. México e-mail: [email protected]

Marius Mitrea Department of Mathematics University of Missouri Columbia, MO 65211, USA e-mail: [email protected]

Salvador P´ erez-Esteva Instituto de Matemáticas Unidad Cuernavaca Universidad Nacional Autónoma de México A.P. 273-3 ADMON 3 62251 Cuernavaca, Mor. México e-mail: [email protected]

Michael Taylor Mathematics Department University of North Carolina Chapel Hill, NC 27599, USA e-mail: [email protected]

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