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1, and this extends harmonically to R3 \ {o}. Another way to write the last formula IS
(2.4)
IBI- 1 l l x - yl-l dx
= Iyl-l,
Iyl > 1
where IBI denotes the volume of B( = 471"/3). Thus, the uniform mass distribution on B produces the same external gravitational field as a point mass at O. Yet another interpretation of (2.4) is (2.5)
IBI- 1 l
v(x)dx
=
v(O)
whenever v is a finite linear combination ofthe functions x 1---+ Ix-yl-l with Iyl > 1. It is fairly easy to show that these are dense in Ll(B), the integrable harmonic functions in B, so (2.5) holds for v E Ll(B). This is, of course, just the Gauss mean value theorem for harmonic functions; the point is that modulo an approximation theorem (whose proof we won't give here) it is equivalent to the formula (2.4) for the potential of XB outside B. Now, (2.5) is the simplest case of what can be called a "quadrature identity" i.e. one which reduces an integration (in the present instance, of a harmonic function) to a finite number of point evaluations (classically, such formulae, like that of "Gaussian quadrature" were one-dimensional, and of the form
for polynomals f of a certain degree). In 1907 C. Neumann (whom, I conjecture, was the author of the question studied in [Herglotz, 1914]) discovered a remarkable case of "balayage inwards": the uniform mass distribution on a certain oval-shaped lamina in R2 has the same logarithmic potential outside the lamina as that generated by two point masses inside the lamina. Although he (and Herglotz who shortly thereafter greatly generalized the result) did not interpret this result as a quadrature formula, one may do so, obtaining for Neumann's oval
(2.6)
110 udxdy = (Inl/2) [u(-a,o)+u(a,o)]
2.3 QUADRATURE DOMAINS
for all harmonicintegrable u on 0 (which contains the points (±a,
15
0».
It is easy to see that (2.6) is equivalent to
(2.7)
flo
fdxdy =
(101/2) (f(-a)+f(a»)
for all f E L~(O), the analytic integrable functions on O. This can in turn be formulated in terms of "balayage inwards" for what could be called the Cauchy potential of the uniform lamina 0, namely C«() := ffo(x + iy - ()-l dxdy which satisfies ac/a, = -1!"Xo. We see that (especially in 2 dimensions) there are many equivalent concepts to describe the same phenomenon (of "balayage inwards"). Although, for historical reasons, we have thus far mostly used potentialtheoretic formulations, the reformulation in terms of "quadrature identities" (which is of much later date, arising in [Davis, 1965] and, independently [Aharonov and Shapiro, 1974, 1976]) will usually be preferred by us in the remainder of these lectures. These identities are a simple and natural point of departure for function-theoretic studies, and moreover by varying the setting (e.g. varying the classes of "test functions" allowed in the quadrature identity, allowing more general functionals than point evaluations, etc.) one arrives at new problems. The fruitfulness of this point of view has shown itself in recent works of Gustafsson and Sakai, even when the object of study is classical potential theory and balayage. For an overview see [Gustafsson, 1989]. Remark. The idea to study analytic continuation of potentials into the region occupied by the charges in terms of the solution of a Cauchy problem for the Laplace operator, as in § 2.1, goes back at least as far as the paper "Bemerkung zur Potentialtheorie" by Erhard Schmidt in Math. Annalen 68 (1910) pp. 107-118, where also references to earlier works of Bruns and Stahl are given.
Chapter 3 Examples of "Quadrature Identities" 3.1 About the terminology. It was known already at the time of Newton that "quadrature", i.e. the evaluation of integrals, could sometimes be effected, approximately or exactly, by replacing f(t)dt with a sum of the type 2:;:1 cif(tj) where a ~ t1 < t2 < ... < tm ~ b and the Cj are real constants. For example
J:
(3.1)
3
ill
f(t) dt = f( -1) + 4f(0)
+ f(l)
is valid when f is any cubic polynomial as well as for many other functions (e.g. all odd continuous functions on [-1,1]). Moreover, for smooth f the difference between the left and right hand sides can be bounded by a constant times the supremum of If(4)(t)1 on [-1,1]. This formula can be scaled to an arbitrary interval; if we do so for a number of consecutive equal "small" intervals and add the resulting equations we get "Simpson's rule". The "quadrature identities" we speak of in these lectures have formal similarities with (3.1), but, also these differences: the domain of integration will be an open set in Rn for some n ~ 2; and the family of f for which the formula is to be exact will be some class of harmonic, or analytic functions. Moreover, the point of view is different: we are not here concerned with quadrature identities (q.i.) as a method ofnumerically integrating harmonic (or other) functions over the domains involved. In the next chapter we'll give a formal definition of q.i. (It will turn out that there are several definitions, according as to which subclass of the harmonic functions is to satisfy the q.i. and also whether we care to allow functionals more general than "point evaluations" into the "right hand side" of the formula. It will, in any case, turn out that only rather special domains admit any q.i. at all, and those will be called quadrature domains in the wide sense, or q.d.w.s.). In the present chapter we want only to motivate these definitions, and the theory in the next chapter,
17
3.2 EXAMPLES
by giving some examples. As already discussed in Chapter 2, these examples could also be presented in terms of "balayage inwards" . 3.2 Examples. "The grand-daddy of them all" is the mean value formula, which we have already spoken of: (3.1) where B = B( XO ; R) is a ball centered at the point XO E Rn, and IBI is its volume. The formula holds for all u harmonic and integrable over B(u E Ll(B». It is less well known that for any distinct points xl, ... , xm in Rn there is a bounded connected open set 0 containing all the x j and positive numbers Cj such that
(3.2)
i
(}
u dx =
t
Cj
u(x j ),
Vu E
Ll (0).
j=l
Without the requirement that 0 be connected this would be trivial, one could take 0 = UB( x j ; Rj) where the Rj are so small that these balls are mutually disjoint. The existence of a connected 0 satisfying (3.2) is, however, nontrivial. It follows from results in [Gustafsson, 1981] and [Sakai, 1983], based on variational inequalities (those authors worked in R2, but their methods apply as well in Rn). When n ~ 3 one does not know any nontrivial such domain (Le. with m ~ 2) explicitly. For n = 2 however, one can give explicit constructions (starting with the oval discovered by C. Neumann, mentioned above, and to be considered in a moment). Moreover one knows that all such domains are bounded by algebraic curves which have further remarkable properties (see below). For n ~ 3 one does not even have much qualitative information (e.g. concerning topological structure, or regularity ofthe boundary) concerning 0 that satisfy (3.2).
In the remainder of this chapter we'll work in R2. One construction of q.d. is based on the following two propositions, due to Philip Davis ([Davis, 1974, Ch. 14]; see also [Aharonov and Shapiro, 1976, Thm.1]).
18
3. EXAMPLES OF "QUADRATURE IDENTITIES"
PROPOSITION 3.1. Let n be a bounded plane domain with smooth boundary. Suppose there exists a function S analytic and single-valued in n except for simple poles at Zl, ... , Zm with residues at, ... , am respectively, and continuously extendable to each point ( of an where
it satisfies S(O
= C.
f
(3.3) for every
Then
10 IE L!(n)
Ido-=7r'fajl(zj) j=l
(analytic integrable functions on n).
REMARKS: By "smooth boundary" we mean here so regular that (a) Stokes' theorem can be applied and (b) functions analytic on a neighborhood of n are dense in L!(n). Since this chapter is mainly heuristic and we don't aim at maximum generality, we won't here enter into a discussion of conditions for (b); the reader can think of I in (3.3) as being analytic on a neighborhood of n. In case n is a Jordan domain and the aj are real, (3.3) gives
(3.4)
i
udo-
= 7r
t..
aj u(Zj)
for harmonic functions u which are the real parts of IE L!(n). Since these are dense in Ll(n), (3.4) then holds for u E Ll (n); in this manner we can get q.d. for harmonic functions also from Prop. 3.1, so in the remainder of the chapter we'll only deal explicitly with q.d. for analytic functions. PROPOSITION 3.2. Let 'P be a rational function without poles in 0 (where D denotes the open unit disk) and injective in D. Then, there is a function S analytic on n := 'P(D) except for poles, continuously extendable to each point ( E an and satisfying S( 0 = C there. Moreover, if WI, ••• ,Wm are the distinct poles of 'P, the poles of S in n are at the points Zj := 'P(wj) where wj = l/wj, and the pole of S at Zj has the same order as that of 'P at Wj. (Note that Wj = 00 is allowed). In particular, if the poles of 'P are simple, n satisfies the hypotheses of Prop. 3.1, and the q.i. (3.3) holds. PROOF OF PROP. 3.1: By a form of Stokes' theorem, for on a neighborhood of n
10 I do-
= (2i)-t
t I(O(
d(
I
analytic
3.2
EXAMPLES
19
where r is the positively oriented boundary of O. The last integral is (by residues)
which is (3.3). REMARK: We could of course have allowed S to have multiple poles, and then there would be terms on the right side of (3.3) involving derivatives of f at the Z j. PROOF OF PROP. 3.2: We can define a function S from 0 to the extended complex numbers by
(3.5)
S(IP(w») = 1P*(l/w),
wE 0
where 1,0* denotes the (rational) function defined by 1,0*( w) = IP( w).
S is analytic in 0 except at points Z = lP(t) where t is a pole of 1P*(l/w). Since the poles of 1,0* are {Wj}J!=ll the poles of S are at {1P(Wi)}J!=1 and it's clear that the pole of S at lP(wi) has the same order as that of 1,0 at Wj. Finally, for' E ao we have, = lP(w) with Iwl = 1 so, from (3.5), SeC) = S(IP(w» = 1P*(l/w) = 1P*(w) = lP(w) = (, completing the proof. Combining Prop. 3.1 and Prop. 3.2 gives COROLLARY 3.3. Let 1,0 be as in Prop. 3.2, and suppose all its poles are simple. Then, 0 = 1,0(0) is a q.d. in the sense that (3.3) holds for all f E L!(O). In (3.3), Zj = lP(wi) where {wt, ... ,wm } are the poles of 1,0, wi := Wj -1 and the aj are complex numbers depending
only on 0 (not on I). Let us illustrate this by constructing a non-trivial q.d. We start from the familiar fact that .,p(w) = (1/2)[w + (l/w)] conformally maps {iwi > 1} onto the domain C\[-l,l]. The image of the circle {iwi = R} where R> 1 is the ellipse ER whose parametric equations are
ER : x = (1/2)(R + R- 1 ) cos t, y = (1/2)(R - R- 1) sin t. Hence .,p(Rw) maps {lwl > 1} conformally onto the exterior of this ellipse. If N R is the curve obtained by inversion of E R with respect to o (i.e. by the map Z 1-+ l/z), .,p(Rw)-l conformally maps {Iwl > 1}
20
3. EXAMPLES OF "QUADRATURE IDENTITIES"
onto the interior OR of NR and so, finally .,p(R/w)-1 maps 0 onto OR, i.e.
maps 0 conformallyonto OR (which is "C. Neumann's oval"). Observe that cp has its poles at WI = iR, W2 = -iR. Thus the Schwarz function of NR has its poles at Zj = cp(l/wj),j = 1,2 i.e. at the points ±2(R2 - R-2)-li. The residues at these poles can easily be calculated using the parametric representation
of the Schwarz function S of N R. For, in general, if we have a parametric representation Z = A( w), S = B( w) and B has a simple pole at Wo with residue p, S (as a function of z) has a simple pole at Zo := A( wo) with residue pA'(wo), when A is analytic near Wo. In the present case Wo = i( or i), p = R-I and A'(wo) = 2R(R2 + R-2)(R2 - R-2)-2. If we introduce the abbreviations (3.7) for the semi-axes of the ellipse ER we thus obtain: the Schwarz function of N R is analytic in the interior of N R except for simple poles at the points ±(2ab)-I.i, at each of which it has the residue (a- 2 + b- 2)/4. Applying (3.3) with 0 the domain bounded by NR and f = 1 we see the area enclosed by NR is (1l"/2)(a- 2+b- 2). (Note that a 2 _b2 = 1 since the foci of each ellipse ER are at the points Z = ±1). We have thus obtained (in a somewhat different way) the result of C. Neumann: inversion of an ellipse about its center gives a curve (denoted NR above) which bounds a domain admitting a two-point quadrature identity, or in other words, the uniform lamina bounded by N R induces the same logarithmic potential outside N R as two suitably placed point masses (each with half the mass of the lamina). It is easy to check that N R is an algebraic curve of degree 4 whose equation in Cartesian coordinates is
(3.8)
3.3 EXAMPLES, CONTINUED
21
This is one of a class of what Herglotz calls "bicircular curves", and investigates in great detail in the cited work. 3.3 Examples, continued. Implicit in the preceding paragraph is the Schwarz function of the ellipse ER. The most direct way to get it is to substitute into the equation of E R
(3.7) where (a, b are given by (3.7)) x = (z + z)/2 and y = (z - z)/2i and solve the resulting equation for z. Alternatively, we can represent the Schwarz function S(z) of ER parametrically by
Since the expression for z = z(w) in (3.8) maps {Iwl > I} conformally to the exterior of ER we see that S is analytically continuable throughout the exterior of ER except for a pole at 00 where it behaves like S(z) = R-2 z + Co + CIZ-1 + .... Moreover, z(w) is locally univalent except near w = ±R-l where z = ±1. Thus, S(z) is analytically continuable along every path in C\{ -1, I}. It is easy to check that S has algebraic branch points of order 2 at z = -1 and z = 1 (the foci of the ellipse ER)' By direct calculation from (3.7) we see, in fact
(3.9)
S(z) = (2a 2 - l)z - 2ab(z2 _ 1)1/2
where we take the branch of (.)1/2 such that S(z) '" R-2 z at 00. From these formulae we can deduce interesting quadrature formulae for the ellipse. The first (known, in equivalent forms, to MacLaurin and perhaps earlier) is obtained as follows. Letting n denote the interior of the ellipse (3.7), if J is analytic on n,
{ jZ dz = (2i)-1 { J(z)S(z)dz. Jn{ Jda- = (2i)-1 Jan Jan We can shrink the contour of integration, until it "just barely" surrounds the segment [-1,1], and obtain in the limit
(3.10)
22
Since
3. EXAMPLES OF "QUADRATURE IDENTITIES"
n
has area
Inl =
nab we can write this
(3.11) Since the right side is the same for all ellipses n with foci at -1,1 we can state: The mean value of an analytic (or harmonic) function over each elliplle of a confocal family ill the .'!ame. This proposition (a generalization of "Gauss' mean value formula") can be extended to Rn. Another interesting formula, perhaps first noted in [Sakai, 1981] is: If f ill analytic and integrable on D := C\n where n ill an elliplle, then fD fda = o. To prove this, let D(t) be the domain bounded by ER and r t := {Izl = t} where t is large. We shall assume f extends smoothly to aD. Also, the hypotheses imply fez) = O(lzl- 3 ) for large Izi. Hence, { fda
= (
JD
fda
+ o( 1)
JD(t)
and {
= (2i)-1
fda
JD(t)
= (2i)-1
(
f(z)zdz
JaD(t)
(
JER
f(z)S(z)dz
+ O(t-l)
where S is the Schwarz function of ER. By Cauchy's theorem, we can move the contour in the last integral outwards (since S has no finite singularity outside E R ), replacing it by r t and then we see this integral is O(t-l). We conclude, letting t -+ 00, that fD fda = o. (The result also holds for f integrable and harmonic, and in this form is valid also in Rn (see [Friedman and Sakai, 1986] and references there). ) 3.4 Example of a q.d. with singular boundary point. Whenever n is a bounded domain whose boundary consists of finitely many nonsingular analytic Jordan curves, we can obtain a kind of q.i. for fn f dO' (with f analytic in n) by writing this as (2i)-1 fan fz dz, replacing z on each component of an by the corresponding Schwarz function, and then using Cauchy's theorem to move the integration contours inwards to n, thus replacing the integral of f over n by line integrals over some curves in the interior of n.
3.5 AN EXAMPLE OF "BALAYAGE INWARDS"
23
It may happen that this procedure even works when an is analytic but has isolated singular points. In this case the Schwarz function will not be regular in a full neighborhood of an but, may be analytic and bounded in a one-sided neighborhood of an, near each singular point. Here is an example. The function z = ep(w) := w + (w 2 /2) is analytic and univalent in {Iwl $ I}, it maps this disk conformally on a Jordan domain n (so-called cardioitl). The Schwarz function S of an is given parametrically by Z
= w + (w 2 /2),
S = w- 1
+ (w- 2 /2).
Clearly this uniquely defines a branch of S(z) that is holomorphic in except for a double pole at z = o. Since epl( -1) = 0, the map w 1-+ z is not injective on a full neighborhood of w = -1, hence S has a branch point (of order 2) at z = -1/2, the singular boundary point of n. We'll see in the next chapter that this behaviour is typical: if n is a bounded simply connected domain and there is a function analytic on n n {z : dist (z, an) < e} for some e > 0 and continuously extendible to an where it coincides with z, then n is the union of non-singular analytic arcs and at most a finite number of singular points, all of which are inward-pointing cusps, as in the example of the cardioid.
n
3.5 An example of "balayage inwards". We have seen how, starting from rational conformal maps of the unit disk, we can construct domains n satisfying q.i. of type (3.4). Now, in (3.4) some aj can well be negative (we leave it to the reader to construct an explicit example). Hence, if we do "balayage inwards" of the uniform mass (or better, charge) distribution on n we can sweep the charges ultimately to the finite set {Zl, ... , zm}. However, this cannot be done keeping the charges positive: they will be positive at the beginning, but beyond a certain point negative charges will be required. This supplies an example that was promised in Chapter 2.
Chapter 4 Quadrature Domains: Basic Properties, I 4.1 Notations,etc. Let us review some notations that we repeatedly use. By 0 we shall always denote an open set in Rn, and LP(O) denotes the usual Lebesgue space with respect to Lebesgue measure dx. By Lt(O) and L~ (0) we denote respectively the subspaces of LP(O) consisting ofharmonic and analytic functions (the latter only in the case n = 2). We shall occasionally also invoke the Sobolev space Wm,P(O) of distributions 'II. in 0 such that {)a u E LP(O) for all multi-indices a with lal :5 m, and its subspace W;a'p(O) which is the closure in Wm,P(O) of COC(O) (the infinitely differentiable functions on Rn whose support is a compact subset of 0). We use the notations of [Adams, 1975] to which we refer for more details of these spaces. Although we consistently use the same multi-index notation in Rn as in the modem theory of distributions and p.d.e. we sometimes depart from this in R2 and use traditional "complex variable" notations: z = x + iy rather than x = (Xl,X2) for a point of R2 ~ C. We'll also write du for area measure in C. Finally, XA always denotes the characteristic function of a set A, and E (or En) shall denote the usual "fundamental solution" for the Laplace operator in Rn. By a quadrature domain in the wide sense (q.d.w.s) we mean an open connected set 0 C Rn such that there exists a distribution JI. with compact support in 0 for which the quadrature identity (q.i.). (4.1)
In
udx = (JI.,u)
holds for all 'II. E Ll (0). We'll say 0 is a quadrature domain (q.d.) and (4.1) a quadrature identity (q.i.) if (4.1) holds for some distribution whose support is a finite set. Lots of variation is possible in these definitions, since we may e.g. wish to require (4.1) to hold only for some subclass of Ll(O) (such as L~ (0), which is a subclass when 0 is bounded). It can be shown for bounded domains 0 subjected to certain regularity hypotheses that
4.2 QUADRATURE DOMAINS AND THE SCHWARZ POTENTIAL
25
L'r (n) is dense in L1 (n), so the choice of Lt (n) as the class of "test functions" u in (4.1), for any 1 ~ P ~ 00, leads to the same class of domains. Also, when n = 2 we will frequently want to require that (4.1) hold for u in L! (n) rather than Ll(n). Usually, /J will be a real measure, and n fairly regular, and then it is easy to see that this distinction is irrelevant since sums of functions in L! (n) and their complex conjugates are dense in L1 (n). By and large the questions that interest us in these lectures are those in which this kind of distinction does not make any difference. Both Gustafsson and Sakai usually work with a definition of q.d. which requires /J to be a bounded measure on n but not necessarily of compact support (so that every domain is trivially a q.d.). As an example of a nontrivial q.d. in this broader sense we have a triangle in the plane. It is not hard to show e.g. if K is the subset of a triangle n consisting of an interior point, and lines joining it to the three vertices, that there is a bounded measure /J supported on K such that (4.1) holds. Again, if we allow unbounded domains n the requirement of compact support is rather restrictive since e.g. a solid circular cylinder in R3 admits an identity (4.1) where /J is a measure supported on the axis of the cylinder. However, for brevity, we won't deal much with those situations in these notes. If in (4.1) (with n = 3, say) n is bounded and we take u(x) = Ix - yl-l where y is a point of R3\n, we see that the Newtonian potential of the uniform mass distribution on n extends harmonically "inwards across an to R3\ supp /J. Thus, as we have observed several times, the concept of q.i. is very closely related to that of "balayage inwards" . 4.2 Quadrature domains and the Schwarz potential. THEOREM 4.1. H n is a bounded q.d.w.s. (so that (4.1) holds) there is a distribution V on Rn satisfying
(4.2)
~V
= Xu - /J
(4.3) 2
I
where U E Wo ,p (n) (Vp' < (0) and supp wEn. REMARKS: The proof we'll give is not the most elementary, but has the advantage that the method employed can be adapted to unbounded
26
4. QUADRATURE DOMAINS: BASIC PROPERTIES, I
domains, although we won't go into that here. Note that from (4.2) ~V = Xo on a neighborhood of an. By well-known results on local regularity of solutions of elliptic p.d.e. this implies that V E Wj!{ (D) for some neighborhood D of an, for every p' with 1 < p' < 00, and the first-order partial derivatives of 11 are in Wj~{ (D). By the Sobolev embedding theorem these first derivatives are therefore Holdercontinuous with Holder exponent A, for each A < 1. In view of (4.3) we then have: V and grad V vanish on an. Before proving the theorem, we will outline a proof of the weaker result where, instead of (4.3), V is only shown to vanish on RN\n, since that is much more elementary. Practically speaking, when n is the interior of its closure, this weaker result is as useful as the stronger one. To this end, let E denote the standard "fundamental solution" for the Laplace operator. For y ¢ n, we may take '11.( x) = E( x - y) in (4.1), which then reads: E * (Xo - J.t) = 0 on Rn\n. Denoting by V the convolution on the left side of this equation, we have (4.2) and the vanishing of V off n, as claimed. (To get sharper information on V we could choose u(x) = E(x - y) in (4.1), with yEan; this is the method that mainly has been employed so far, and is perfectly adequate for bounded n, but as we said, we'll give an alternative proof of Theorem 4.1.) The proof of Theorem 4.1 is based on several lemmas. LEMMA
1
4.2. [Havin, 1968]. Let n be a bounded open set in Rn and = p/(p - 1). Suppose FE Lpl (n) satisfies
< p < 00, p'
(4.4)
J
uFdx = 0,
V'll. E
Lt (n).
Then, there is a distribution U E W;,pl (n) satisfying ~U = F. PROOF: To say'll. E Lt(n) is equivalent (in view of "Weyl's lemma") to saying'll. E LP (n) and J u~
0,
Vx E B\{O}.
This requires a calculation, which we'll do later. From (4.11) and (4.13),
(4.16)
6.(V - Vo)
= Xn
- XB.
4.3 SOME APPLICATIONS
31
Now, suppose 0 i= B. We'll derive a contradiction. Consider first the case where Xu = XB a.e. Then from (4.16) and Weyl's lemma V - Vo is harmonic on RR and, vanishing for large lxi, it vanishes identically. From (4.12) and (4.15), Vex) - Vo(x) < 0 at any boundary point x of 0 inside B, hence there are no such boundary points and B C 0 whence, since IBI = 101, B = O. This is a contradiction, therefore Xu - XB is non-zero on a set of positive measure, which implies there are points of 0, and hence of ao not contained in B. Let y be such a point. On some neighborhood D of y, ~ V = Xu so V is nonconstant and subharmonic with V(y) = 0, hence sup VID > o. Therefore, max(V(x)- Vo(x» for x ERR, is positive; let it be attained at xl. We must have V(xl) > 0 because of (4.15). Hence xl E 0 (since V vanishes outside 0). But then, for x on some neighborhood N of xl
so V - Va is subharmonic on N. This is impossible unless XB(X) = 1 a.e. on a neighborhood of xl, i.e. xl E B n 0 and V - Vo is constant near xl, and hence on the closure of the component of RR\(aB u aO) containing xl. Thus, there is a point x 2 E aB u ao where V - Vo attains its maximum. By the reasoning just conducted for the point xl, we deduce that x 2 E B n 0, and have arrived at a contradiction, which finishes the proof. There remains, however, the detail of verifying (4.15). We shall give several proofs. First proof of (4.15). By a simple scaling argument we may assume B = B(O; 1). Vo is uniquely determined on B\{O} as the solution of the Cauchy problem: ~ Vo = 1, with Vo and grad Va vanishing on aBo Let's just do the cases n ~ 3. Since we are looking for Vo with radial symmetry, it is fairly easy to guess the solution,
(4.17)
2nVo(x) =
Ixl 2 + (2/(n - 2»lxI 2 - n
-
n/(n -
2).
This clearly solves the Cauchy problem, and hence is the unique solution. We have thus to verify that F(r) := (n - 2)r2 + 2r 2- n - n is positive for r in (0,1), which is evident since F(l) = 0 and F'(r) < 0 on (0,1).
32
4. QUADRATURE DOMAINS: BASIC PROPERTIES, I
Second proof of (4.15). If En is the standard fundamental solution for .6. in Rn,
Vo
= (XB -IBI· 0) * En = XB * En -IBI· En
so we have to show
(4.18) for 0 < Ixl < 1. Now, the function y ~ En(Y - x) is subharmonic on Rn, so (4.18) follows, indeed for all x, by the sub-mean value property.
A third proof of (4.15) follows from a general proposition, essentially contained in [Sakai, 1982, §14], and which (as we'll see) has other applications: PROPOSITION 4.8. (M. Sakai) Let n be a bounded connected open set in Rn symmetric with respect to a hyperplane H. Suppose J.L is a positive measure whose support is a compact subset of n n H and V E GI(n\ supp J.L) satisfies
(4.19) (4.20)
.6.V = Xo - J.L
V
in
n
and grad V vanish on
an.
Then
(4.21)
Vex) > 0
for x E
n\ sUPPJ.L.
PROOF: Without loss of generality we may take H to be {x n = O}. We employ the notation x = (x'; x n ) where x' := (xt, ... , Xn-l). Let n+ := {x En: Xn > O}. We'll show first
anY ~ 0
(4.22)
on
n+.
Since .6. V = 1 on n+, an V is harmonic in n+, so (4.22) will follow if we show
(4.23)
lim sup (an V)(x) ~ 0, z-y
zEO+
Vy E
an+.
4.3 SOME APPLICATIONS
33
We consider three cases: (i) yEan, Yn ~ 0 (ii) Yn = 0, y' E n\ supp p (iii) Yn = 0, Y' E supp p. In case (i), an V(x) -+ 0 as x -+ y because of (4.20). Next, observe that V# (x) := V( x' j -x n) satisfies the same hypotheses as V, so V#V is harmonic in n and vanishes on an, hence identically in n, so V(x'j -x n ) = V(x'j x n ) and (an V)(x'j 0) = 0 for x' ¢ supp p. Thus
....
(4.23) is verified in case (ii). To handle case (iii), observe first that if V is the extension of V to Rn obtained by defining it as 0 off n, then '"
....
V = (~V) * En where En is a fundamental solution as above. Hence, for x E n+ we have V(x) =
(an V)(x) =
In En(x - z)dz - JEn(x - z)dJ1.(z).
In (an En)(x - z)dz - J(an En)(x - z)dp(z).
The first integral is continuous with respect to x and when Xn = 0 it vanishes because of the odd symmetry of z 1-+ (an En) (x - z) = (an En) (x' - Z'j -zn) with respect to Zn. As to the second integral, a simple computation shows that an En(x - z) > 0 for Xn > 0, Zn = 0 so this integral is ~ 0 for Xn > 0 and (4.23) is verified also in case (iii). Hence (4.22) holds. Moreover, anY < 0 on n+, otherwise, by the (strong) maximum principle an V = 0 on n+ whence, in view of (4.20), V = 0 on n+ contradicting (4.19). Since V = 0 on an and an V < 0 on n+ it is clear that V > 0 on n+. Since V is symmetric about {xn = O}, (4.21) follows, and the proof is finished.
The modified Schwarz potential of the ball B = B( X O j R) is positive on B\{xo}, i.e. (4.15) holds. COROLLARY.
If we examine carefully the proof of theorem 4.7, we see that the only properties of the ball B that were used are (i) B is the interior of its closure, and B has connected complement (ii) B is a q.d. whose m.S.p. is positive (i.e. (4.15)). We required (i) in two ways: (a) to assure that if Rn\B contains points of n, it also contains a boundary point of nj and this conclusion would fail if B had a "hole" in it, (b) to infer from n ::> B and Inl = IBI that there are points of n outside
34
4. QUADRATURE DOMAINS: BASIC PROPERTIES, I
Bj and this would be incorrect if B could be 11\K where K is some closed set of measure zero. Therefore, we can formulate the following more general uniqueness theorem. THEOREM 4.9. [Sakai, 1982] Let 11 be a bounded open set in Rn with connected complement and which is the interior of its closure. H it satisfies the q.i. (4.1) and its modified Schwarz potential (i.e. the function V furnished by Theorem 4.1) is positive on 11\ supp 1', then 11 is the unique bounded open set satisfying (4.1). As an interesting application of this, recall the q.i. for an ellipse, proved in Chapter 3:
f
(4.24)
10
holds for u E
-1
L1 (11)
where 11 is the elliptic lamina
11 = {(x, y) E R2 : (x/a?
(4.25) and a 2
u du = 2ab 11 (1 - x 2)1/2 u( x, O)dx
-
+ (y/b)2 < I}
b2 = 1 (so the ellipse has its foci at {-I, I}).
COROLLARY 4.10. [Sakai, 1982]. H 11 is a bounded open set ::) {-I, I} such that (4.24) holds, for all u E Ll(11) then 11 is given by (4.25). PROOF: By Theorem 4.9 we need only verify that the m.S.p. of the ellipse (4.25) is positive in the ellipse, outside the segment [-1,1], and that is guaranteed by Proposition 4.8. REMARK: This proof extends to ellipsoids in Rn. A q.i. analogous to ( 4.24) holds where n
(4.26)
11 = {x E R
n :
"[)xj/aj?
< I}
j=1
and al (4.27)
~
a2
~
... > an>
In
o.
Namely,
udx = JUdI',
Vu E LU11)
4.3 SOME APPLICATIONS
35
where I' is a certain positive measure supported in the set n-l
K:= {x ERn:
Xn
= 0
and
L (a~ - a~)-l x~ ~ 1}
j=l
the so-called "focal ellipsoid" of O. (For more details, see [Khavinson and Shapiro, 1989a].) Observe that K lies in the symmetry plane {xn = O} of 0, so Prop.4.8 is applicable and shows that the ellipsoid (4.26) is uniquely characterized among bounded open sets by the q.i. (4.27). There is another proof of Cor. -4.10 that is more elementary, based on reasoning from [Gustafsson, 1989]. Namely, suppose 0 is the elliptic lamina (4.25) and D any other bounded open connected set satisfying the same q.i. (4.24) as O. The first part of the argument is general and works in Rn. Let VD, Vo denote the corresponding m.S.p., and reason as in the proof of Theorem 4.7. We find that VD - Vo attains a positive maximum M at some point of Rn. Now, however, we won't assume that Vo > O. Instead, observe that (since ~(VD- Vo) = XD-XO), VD- Vo is subharmonic on D. If it took the value M in D, by the maximum principle it would be = M on all of D, and this is a contradiction since the set (aD)\O is non-empty and at points of this set VD - Vo is zero. Hence, the maximum is taken at a point y in O\D, and 0 = grad (VD - Vo)(y) = - grad Vo(y). Thus, we shall have arrived at a contradiction if we show grad Vo(x)
(4.28)
i= 0, 't/x
E
0\ supp 1'.
We'll only complete the proof in R2. We have to show that the m.S.p. of the ellipse (4.25) has no critical points in 0\ [-1, 1]. In view of the relation between the m.S.p. and the Schwarz function discussed in Chapt.1, we have to show: The equation S(z) = z, where S is the Schwarz function of 00 (0 being given by (4.25)) has no solution in 0\ [-1, 1]). This, however, is immediate: S satisfies the identity
so if S(z)
= z,
then z
= x + iy
satisfies
(x2/a 2) + (y2/b2) = 1
36
4. QUADRATURE DOMAINS: BASIC PROPERTIES, I
=
i.e. only points z on the ellip"e can satisfy S(z) z (indeed, this is true even for the multiple-valued function S after analytic continuation along any path in C\ {-I, I}).
4.4 Subharmonic quadrature domains. If 11 is a q.d.w.s. (i.e. satisfies (4.1» it has a m.S.p. V which may or may not be non-negative on 11\ supp J.I.. Sakai was the first to realize the significance of this distinction, and discovered the important consequences for uniqueness questions when V ~ 0, as we have illustrated in the last Section. He showed that, when J.I. is a measure
(a)
Vex)
~
0
on
11\ supp J.I.
is equivalent to
(b) for all "ubharmonic /unction" u integrable on 11 and continuous on "upp J.I.. That (b) '* ( a) is rather easy, choosing as test function in (b), u(x) = En(x - y) with y E 11\ supp J.I.. In the other direction, one requires a non-trivial approximation theorem which we won't enter into here, and refer to Sakai's book. In view of the implication (a) '* (b), and the results of the last Section, we get e.g.
(2/1r)
11
-1
(1 - x 2 ) u(x, O)dx
110
~ 1111- f
u
dO'
for u subharmonic inside the ellipse 11 of (4.25), i.e. a "ub-mean value property for ellipse", and the analogous inequality in Rn is also valid. A q.d.w.s. for which (a) (or what is the same, (b) )holds shall be called a subharmonic quadrature domain. It is easy to give examples of q.d. that are not subharmonic q.d. For example, as shown in Chap. 3, there is a q.d. whose associated J.I. is a finite sum of point masses. If one of these is negative (as can well happen, even if there are just 2 point masses) the domain can't be a subharmonic q.d. Suppose, e.g. that the q.i. is
4.4 SUBHARMONIC QUADRATURE DOMAINS
for harmonic u, and that el < we should have
o.
37
Then, if 0 were a subharmonic q.d.
(4.29) for continuous subharmonic functions u. But, taking for for
u(z) = { log Iz - zll loge
Iz - zll ~ e Iz - zll ~ e
in (4.29), we get a contradiction for sufficiently small positive e. However, even if 0 is a q.d.w.s. for a positive measure I-' it can fail to be a subharmonic q.d. In this case Theorem 4.9 is not applicable, and indeed 0 may then not be the only q.d. corresponding to 1-'. We give the following simple, instructive example, also due to Sakai. Let 0 be the open unit disk in the complex plane, and 0 the annulus {(5/12) < Izl < (13/12)}. Note that 101 = 101 = 7r. There is a uniquely determined number e such that
(4.30) holds for u E
1
L1 (0),
13/12
5/12
r dr
Indeed, the integral on the left equals
121r
u(re it ) dt =
1
0
13 / 12
(A log r
+ B)r dr
5/12
for some constants A, B (depending on u) that are independent of r, so (4.30) holds if
1
13/12
(4.31)
loge = 2
r log r dr.
5/12
It is easy to compute that (3/4) < e < 1. Let now I-' denote the normalized arc-length measure on {Izl = e} so that (4.32)
38
4. QUADRATURE DOMAINS: BASIC PROPERTIES, I
holds for u E L1 (0). Thus, 0 is a q.d.w.s. with respect to J.I., and so is n. It follows that the m.S.p. of 0 takes negative values, or what is equivalent, the formula
does not hold for all subharmonic integrable functions on OJ these assertions can be checked by simple computations, which we leave to the reader.
Chapter 5 Quadrature Domains: Basic Properties, II
5.1 Regularity of the boundary. The simplest regularity result is the following two-dimensional one. THEOREM 5.1. Let n be a simply connected plane domain, Zo E an and suppose for some e > 0 there is a function S holomorphic on N := n n De, where De = {z : Iz - zol < e} continuously extendable to N and satisfying S(O = (, V( E annD e • Let cp map the unit disk o of the w-plane confonnally on n. Assume cp extends continuously to points of aD sufIiciently close to w = 1, and cp(l) = zoo Then cp extends analytically to a neighborhood of 1. PROOF: S 0 cp is holomorphic at points of D near 1, and satisfies S(cp(w)) = cp(w) for Iwl = 1, w near 1. Let cp* be defined by cp*(w) = cp(w); it is holomorphic in D and extends to points of aD close to 1, and S(cp(w)) = cp*(l/w) holds on an arc of aD containing 1, so by the theorem of Painleve (cf. Proposition 1.4) the left and right hand functions are analytic continuations of one another across this arc. In particular, cp*(l/w) is analytically continuable inward across this arc, so cp*(w) (and hence cp(w)) are continuable outwards, which proves the theorem. In particular, any q.d.w.s. n which is simply connected has this sort of boundary, in view of Theorem 4.5 and the remarks following it. The same applies (by decomposition) if n is finitely connected. Recently in [Sakai, 1989] it was shown that every q.d.w.s. in the plane is finitely connected, a problem that had been open for some time. In the more particular case where n satisfies (4.6) with JL of finite support, a good deal more can be said. It was shown [Aharonov and Shapiro, 1976, Theorem 3] that an is algebraic, more precisely: there is a non-constant polynomial P(X, Y) with real coefficients, irreducible over the complex field, such that P(x, y) = 0 for x + iy E an. Thus, for example, an annulus, or a triangle, cannot be of this type since their boundaries (while algebraic) are given by polynomials that are reducible. The latter, moreover, is not a q.d.w.s. because of Theorem 5.1.
40
5. QUADRATURE DOMAINS: BASIC PROPERTIES, II
[Gustafsson, 1983] showed moreover that P above must have as leading terms a constant multiple of (X2 +y2)m for some mj moreover an must consist of all the real points (x, y) satisfying P( x, y) = 0 except possibly for finitely many. No relults of thu nature are known in n ~ 3 dimensionl. Valency of the Schwarz function. Let us again consider a bounded q.d. n in R2, for which the associated distribution has finite support, or what comes to the same, there is a function S analytic in n except for m poles (counting multiplicity) and continously extendable to all points of an, where S(C) = (. This function has a whole series of remarkable properties, the deepest of which are given in [Gustafsson, 1983, 1988] and [Sakai, 1988a]. In view of the introductory nature of these notes, I won't go into all those here, but just prove one simpler result from [A vei, 1977]. 5.2
THEOREM 5.2.
in
(a) (b)
With
n
and S as above, the equation S(z) = w has
n m
zeroes, if
m- 1
w E
zeroes, if
C\n
wEn.
PROOF: We give a proof based on the concept of Brouwer degree, following [Shapiro, 1987]. (For the basic topology see e.g. [Lloyd, 1978].)
Let C denote the compactified complex plane (Riemann sphere), and consider the map F: C -+ C defined by
F(z) = {S(Z) z
, ,
Note that F is continuous from C to itself. Since C has no boundary the Brouwer degree deg(F,C,w) is constant for w E C. Now, for w = 00 the value of this degree is -m + 1. Indeed, w = 00 is taken on by F exactly m times (counting multiplicities) in n and (because F is conjugate-meromorphic on n) each of these oo-points contributes -1 to the calculation of deg(F,C,oo). Moreover the only solution of F(z) = 00 with z E C \n is z = 00, and since F is the identity map
5.3 VARIATIONAL PROPERTIES OF Q.D.
on a neighborhood of 00, the contribution of z = +1. We conclude that
(5.1)
deg(F, C, w) = -m + 1,
\/w E
00
41
to the degree is
C.
Now, (a) follows because for wEe \n, F(z) = w has exactly one solution z E C \n (the map being the identity near z, hence orientationpreserving and contributing +1 to the degree) so it must have m solutions in n (each contributing -1). Likewise (b) follows, because for wEn all solutions to F( z) = w must lie in n: each contributes -1, so there are m -1 of them. The proof is complete. With a little more work we can prove a more complete result. Let's call a boundary point Zo of n regular if the conformal map cp in theorem 5.1 satisfies cp'(w o ) i- 0 at the point Wo of aD that maps to zoo (This is equivalent to saying that an is, near Zo, a non-singular analytic Jordan arc as defined in Chapter 1.) The alternative is that cpl(WO) = 0 in which case
cp(w) = Zo
+ c(w -
wo?
+ O(lw -
woll)
near Wo for some nonzero constant c (because cp" ( wo) = 0 would be incompatible with univalence in D) and an has a cusp at Zo, which points into n. We can now state: with the notations of Theorem 5.2, S(z) = w has in n: m - 1 zeroes if w is a regular point of an, and m - 2 zeroes if w is the vertex of a cusp on an. The proofs, which follow from (5.1) in a similar way as ( a), (b) above, are left to the reader. &.3 Variational properties of q.d.
Quadrature domains (of the special type considered in §5.2) have an interesting variational property, as the following result (generalizing one from [Shapiro and Ullemar, 1981]) shows.
n and S be as in §5.2, and assume moreover tbe distinct poles ZI, ••• , Zm of S in n are all simple. Let cp be any confonnal map of a domain D in tbe w-plane onto n, witb cp(Wj) = Zj (j = 1,2, ... ,m). H f is bolomorpbic in D and THEOREM 5.3. Let
(5.2)
f'(wj)
= cp'(Wj)
(j
= 1,2, ... m)
42
5. QUADRATURE DOMAINS: BASIC PROPERTIES, II
then
(5.3) Thus, the area of feD) (counting multiplicity if f is not univalent), f satisfying (5.2), is not less than Inl.
for any
PROOF: Let K>. denote the (Bergman) reproducing element at oX i.e. K>. E L~ (n) and
E
n,
(5.4) for all g E L~ (n). If the q.i. for n is
(5.5)
L ~ t, hdu
cjh(zj),
Vh E L! (ll)
then (specializing to h E L~ (n), and using (5.4) and (5.5)) we get (5.6) Thus, the quadrature property of n is equivalent to an identity involving Bergman kernels, an observation due to [Epstein, 1962]. Let us also denote the kernel K>.(z) by K(z, oX). (Observe that our convention, in defining K(z, oX) is that it is conjugate analytic in oX.) If k( w, t) denotes the Bergman kernel of D, then [Bergman, 1970]:
(5.7) Taking here t
= Wj
and using (5.6), we get m
(5.8)
cp'(w) =
L
Cj
cp'(Wj) -1 . k(w, Wj)
j=1
i.e. the derivative of the mapping function i.'J a linear combination of the kwj(j = 1, ... m). This important relation is due to [Avci, 1977, Theorem 8].
5.3 VARIATIONAL PROPERTIES OF Q.D.
43
Our theorem follows immediately. For, assuming as we may that the right side of (5.3) is finite, in the Hilbert space L~ (D), J' -cp' vanishes at WI, ••• , Wm so it is orthogonal to k W1 ' ••• ,kwm and hence (by (5.8)) to cp' so 1If'1I = IIcp' + (J' - cp')11 ~ IIcp'lI, and the proof is finished. Remarks 1. Originally, Aharonovand I elaborated the theory of quadrature domains in the hope that this would help us to solve certain extremal problems in conformal mapping. Thus far we have not been able to complete this program because of technical difficulties, see [Aharonovand Shapiro, 1973, 1974, 1978]. 2) Avci's result embodied in (5.8) gives another proof that a simply connected q.d. (of the special type considered in this section) is the conformal image of the open disk 0 under a rational map, since the r.k. of 0 is k(w,t) = (1/1I'"){1 - tw)-2 and then cp, as determined from (5.8), is rational. To Avci is also due a converse result: if D is any (not necessarily simply connected) domain with reproducing kernel k(w, t) and some single-valued univalent function cp has a derivative that is a finite linear combination of r.k., m
cp'(w) =
(5.9)
L
aj k(w, Wj)
j=l
for distinct points {Wj} C D, then cp(D) =: h E L~ (n) we have
In hdu Iv h(cp(w))lcp'(wWdu =
n
is a q.d. Indeed, if
= (H,cp')
where H(w) := h(cp(w))cp'(w) is in L~ (D) and (,) denotes the inner product in L~ (D). In view of (5.9) we thus have
(5.10)
1 =f hdu
n
~l
o'j H(Wj)
=
f
Aj h(Zj)
~l
where Aj := o'j cp'(Wj) and Zj := cp(Wj), so n is a q.d. (There is an obvious generalization of these results to q.i. where, in the right side of (5.10), we allow functionals like h'(zj), h"(zj), . .. ). It should be stressed that the essential idea here, to relate r.k. and quadrature identities, is due to Davis (see Chapter 14 of [Davis, 1974]).
44
5. QUADRATURE DOMAINS: BASIC PROPERTIES, II
5.4 Other varieties of quadrature domains. In this section we consider briefly, just to illustrate some possibilities, three other notions of q.d. that have been considered. 5.4.1. In this paragraph, we consider plane domains n with rectifiable boundary to which there is associated a distribution J.L with supp J.l C n such that
f f ds = (J.l,J)
(5.11)
Jan
holds for all analytic functions f in n of some suitable class, where ds denotes arc length measure on an. Again, when J.l is a point mass and n a circular disk we have a classical mean value formula. Domains satisfying (5.11) for J.l of finite support seem first to have been considered in [Aharonov and Shapiro, 1978, Appendix 2]. The simply connected case was studied in [Shapiro and Ullemar, 1981] and the general case in [Gustafsson, 1987]. Two natural choices for the class of test functions f in (5.11) are Hl(n) and El(n) (see [Duren, 1970, Chapter 10] for the definitions). For domains with regular boundaries, these classes coincide. The theory of q.d. of type (5.11) is greatly complicated by a remarkable counter-example due (in a slightly different form) to Keldys and Lavrentieff (see [Privalov, 1966]). Namely, there is a Jordan domain n with rectifiable boundary, and a point Zo E n, such that
(5.12)
f
Jan
f ds =
L(an)· f(zo)
holds for every polynomial f, L(·) denoting length (or Hausdorff 1measure), and n is not a disk, in fact there are such n with L( an) = 27r and arbitrarily small diameter. There is, namely, a univalent function O. In (5.12) we may take in place of I any function continuous on nand analytic in n, since on a Jordan domain such a function is uniformaly approximable by polynomials (Walsh's Theorem, see [Gaier, 1980].) Now, r.p is a homeomorphism of 0 on n so its inverse function 'l/J is a homeomorphism of n on D. Taking I = 'l/J", with k a positive integer, in (5.12) we get
for k = 1,2, ... hence 1r.pI(eit)1 = 1 a.e. Being an outer function, with r.p1(0) > 0, r.p' is the constant 1 so r.p( w) = w and n is the unit disk. One can prove versions of this theorem also for non- Jordan domains. In view of Theorem 5.5 it is natural to ask: Is there a domain n in R3 other than a ball, homeomorphic to a ball and with a "nice" bo'Undary s'Uch that Jan 'U dS = c'U(O) holds for all harmonic functions
47
5.4 OTHER VARIETIES OF Q.D.
u continuous in n P (Here dS is surface measure on an and C is the area of an.) The answer is: no. This was shown very recently in [Shahgolian, 1990] (and analogous result in Rn). 5.4.2 Quadrature domains for L! ". Another class of domains n which leads to interesting function-th~retic problems is that for which
1 f
(5.16)
fdq =
()
Cjf(Zj)
j=1
holds whenever f E L!,8 (n), the subclass of L!(n) consisting of functions with single-valued integrals, i.e. f which are derivatives of single-valued functions in n. If n is simply connected, then of course L!,,, (n) = L! (n) and we get nothing new, but if it is not L!,8 (n) is a proper closed subspace of L! (n). It is known [Aharonov and Shapiro, 1976, Lemma 2.4] that (5.16) for all f E L! ,,(n) implies: There exists a function H analytic in n except fo~ simple poles at {Zj} and continuously extendable to an such that H«()-( is constant on each connected component of an. Using this, we'll prove the following result, which in a more general form was proved in [Sakai, 1972]. The proof has been rediscovered independently by M. Schiffer and by D. Aharonov. THEOREM 5.6. If n is a bounded plane domain containing is a finite union of continua, and (5.17)
then
n
i
f dq =
Zo
and
an
Inl· f(zo),
is a disk centered at zoo
For those n to which the theorem applies this is, of course, an essential strengthening of Epstein's theorem. PROOF OF THEOREM 5.6: We may assume Zo = O. By the remarks immediately preceding the theorem, if r := an consists of components r 1, r 2, ... , r r there is a function H analytic in n except for a simple pole at 0, and complex constants at, ... , a r such that H extends continuously to I' and
H«()=(+aj,
(EI'j
(j=1,2, ... ,r).
48
5. QUADRATURE DOMAINS: BASIC PROPERTIES, II
Fix now a complex number ..\,1..\1 = 1, and let G.\(z) := 1H(z) + ..\z. Then G.\ is analytic in n except for a simple pole at 0 and for ( E rj, G.\(() lies on the line 1aj +R. Thus, K.\:= G.\(r) is the union of r horizontal segments so C\K.\ is connected (where C is the Riemann sphere) and the Brouwer degree of the map G.\ : n -+ C is constant on C\K.\. The value of this constant is 1 since G.\ takes the value 00 exactly once and so G.\ is a conformal map of n on C\K.\ (this result can of course also be obtained from the argument principle). Consequently, G~(z) is never 0 for zEn, i.e. 1H'(z) + ..\ =/: O. Since ..\ is an arbitrary number of modulus 1 we conclude: IH'(z)1 =/: 1 for zEn. It follows by continuity (since H has a pole at 0) that IH'(z)1 > 1 for zEn. Also, the regularity theorem 5.1 applies to the present situation to show that each r j is a non-singular analytic curve, except for, possibly, a finite exceptional set (of cusps pointing into n) and by a calculation done previously H' (() = T( () 2 at each regular boundary point (. Putting all this together we see that 1/ H' is analytic in n, with a double zero at 0 and no other zeroes. Moreover, it is of absolute value less than 1, and its boundary values are of modulus 1. Hence this function has a single-valued analytic square root F. We have F(n) c 0 (open unit disk), moreover the cluster set of F at each point of an is on aD. Again, by degree theory, the Brouwer degree of the map F : n -+ 0 is constant on 0, and this constant is 1 since F has precisely one zero in n. Consequently F maps n conformally on 0, from which it follows that n is simply connected. Theorem 4.6 now shows n is a disk centered at 0, and the theorem is proved. REMARK: The method employed does not seem to yield anything analogous for (5.16) with m > 1. However, there are some further results for quadrature domains with resped to L!,6 in [Gustafsson, 1983].
an.
5.4.3 Support of J.I. meets As remarked earlier, Sakai and Gustafsson usually work with a definition of q.d. in which J.I. is a measure whose support is allowed to meet Let's give just one example of such a domain, to see what kind of questions can arise. Let n be a triangle, whose boundary consists of oriented line segments The Schwarz function Sj of rj is of the form Sj(z) = ajz + bj for certain complex constants aj, bj. If f is analytic on n, we have
an.
rt, r 2, r3'
(5.18)
L
fda- = (2i)-1
3
~
i
3
j
fidz = (2i)-1
~
i
j
J(z)Sj(z)dz.
5.4 OTHER VARIETIES OF Q.D.
49
Now, since Sj is analytic everywhere, we can by Cauchy's theorem replace the path of integration r j by any other smooth arc with the same endpoints, and lying in O. For example, denoting the vertices of o by Zl, Z2, Z3 (so that r 3 is the segment from Zl to Z2, etc.) and letting Zo be any point of 0, let Cj be a smooth arc in 0 joining Zo to Zj such that C l , C 2 , C 3 are mutually disjoint except for their common endpoint Zoo Then r3 can be replaced in (5.18) by the path C l uC2 (with correct orientation), and similary for r 2, rl. Then (5.18) takes the form (5.19) where J.L is a certain complex measure living on the closure of U~=l Cj (in fact it is of the form (Ajz + Bj)dz along Cj, for suitable complex constants Aj, B j). So we have a great multiplicity of quadrature formula (5.19). However, one of these is distinguished: if Zo is the point where the angle-bisectors of the triangle 0 meet, and Cj is the line segment joining Zo to zj, then the measure J.L appearing in (5.19) turns out to be real. Gustafsson has shown (unpublished) that, subject to certain topological restrictions (so that supp J.L isn't "too massive"), this is the only case of a real measure satisfying (5.19). He has also generalized this to polyhedra in Rn (the role of angle bisectors then being played by hyperplanes bisecting the dihedral angles formed by pairs of adjacent faces). As another interesting example of an identity (5.19) where J.L measure whose support meets ao we have (5.20)
f f du =
10
c
IS
a
f f ds lao
where ds is arc-length measure, c = 101· L(aO)-l, and (5.20) is to hold for, say, all analytic functions f in 0 continuously extendable to O. The higher-dimensional analog is (5.21)
f
10
udx = c
f
lao
udS
for harmonic functions u, where dS is surface measure and c is the volume of 0 divided by the (n - 1 )-dimensional measure of a~. One
50
5. QUADRATURE DOMAINS: BASIC PROPERTIES, II
solution for n is a ball. If n is assumed homeomorphic to a ball, with sufficiently regular boundary, in (5.21) then it is known that n must be a ball. This follows from an argument of [Kosmodem'yanskii, 1981] applied to a theorem of [Serrin, 1971]. For further discussion see [Khavinson, 1987]. 5.5 Existence of q.d. Let us give a few remarks concerning methods to construct, or at least prove existence of, quadrature domains. In the plane, we can get simply connected q.d. by mapping the unit disk conformally by a function
2, since as is well known, there are relatively few conformal maps in these spaces. In [Khavinson and Shapiro, 1989b] this is discussed in more detail. We shall return to this shortly. First, however, let's prove Prop. 6.1. PROOF OF PROP. 6.1: To simplify the proof we'll accept (from the p.d.e. literature) that our hypotheses imply that all 8 0 1.£, lal ~ 2, extend continuously to a neighborhood N of (7, that is independent of u. Now, for ZED, Z near (7, R/7(z) lies in N\D (by Prop. 1.2) and -(uoR/7) is harmonic in N\D and has the same boundary values on (7 as u(z) (namely, 0) so it is a good candidate to be a harmonic extension of u. The crucial thing is to verify (A) The first partial derivatives of 1.£ and of -(1.£ oR/7) match on (7. Indeed, assuming this, and letting 1.£# denote the harmonic extension of 1.£ into DuN, 1.£# and -(1.£ 0 R/7) are harmonic on N\D and have
54
6. SCHWARZIAN REFLECTION, REVISITED
the same "Cauchy data" on the boundary arc tT, i.e. the functions and their gradients coincide on tT and so, by Holmgren's uniqueness theorem, are identical. So, we have only to check (A). It is convenient notationally to write
R tT : (x,y)
f-+
(P(x,y),Q(x,y»
where P, Q are real and P + iQ = RtT(x + iy), and to let '1.£2 := ~;. We have to show that (with P z := 8P/8x, etc.)
hold on
tT,
0= 8z (u(x,y)
+ u(P,Q»)
=
'1.£1
+ '1.£1 Pz + '1.£2 Qz
0= 8y (u(X,y)
+ u(P, Q»)
=
'1.£2
+ '1.£1 Py + '1.£2 Qy
'1.£1
:= ~:,
i.e. that
(6.2) (6.3)
Py '1.£1
+ (1 + Qy)U2 =
0
hold on tT. Now, on tT the functions '1.£, P - x, Q - y all vanish, hence their gradients are proportional and we have (6.4) (6.5) on 0'. But since R is anti-conformal we have identically P z = -Qy, Py = Qz' Using these relations in (6.4), (6.5) we obtain (6.2), (6.3) and this proves (A), and hence Prop. 6.l. REMARK: We could reduce the input of "elliptic regularity theory" in the above proof to: '1.£ and its first partial derivatives extend continuously to 0'. In this case, the above calculations show that the function obtained by piecing together '1.£ and -('1.£ 0 R tT ) is in C 1 (N) for some neighborhood N of 0', and harmonic on N\O'. The proof can then be concluded by a "removable singularities" lemma analogous to Prop. 1.4. This is, again, a general result so we'll state it in Rn.
6.2 STUDY'S INTERPRETATION OF SCHWARZIAN REFLECTION
55
CI(B), where B is an open ball in RR, and harmonic on B\r, where r is a non-singular CI hypersurface. PROPOSITION 6.2. Let u bein
Then u is harmonic in B. This proposition can be proved in the same way as Prop. 1,4, so we'll just sketch the argument. The crucial thing is that the analog of Lemma 1.5 holds in the setting of Prop. 6.2. This is because a function in CI(n) which, together with its first partial derivatives, extends continuously up to a portion r of an which is a nonsingular CI hypersurface (indeed, even the "Lipschitz graph" regularity of r suffices) can be Cl_ extended across r, [Smith, 1971, Ch. 17]. Thus, by the technique we employed in proving Lemma 1.5 (since each :~u. is in C(B)nCl(B\r» 1 the distributional derivatives aiajU can be computed "piecewise" to be the locally integrable functions 8!2;~i on B\r. Since ~u = 0 on B\r, the distributional Laplacian of u in B vanishes, and then Weyl's lemma gives the harmonicity of u in B. Although Prop. 6.2 is very well known, and often useful, it is not easy to find it stated in the literature. One finds in [Carleson, 1967] results of similar character, but apparently not this one. 6.2 Study's interpretation of Schwarzian reflection. An elegant geometric interpretation of anti-conformal reflection was sketched in [Study, 1907], and we give an account of this, with more details, in the present section (see also [Khavinson and Shapiro, 1989b]). The key idea is that anti-conformal reflection, and Schwarzian reflection of harmonic functions vanishing on an analytic curve (as given by Prop. 6.1) have simple geometric interpretations when we look at the picture in C2 rather than in C ~ R2. This is precisely analogous to the situation in classical geometry, where geometrical properties of "real" figures become more simple and unified when we extend those figures from R2 to C2. To give a familiar illustration: consider two circles in the plane, neither of which encloses the other. Then there are points from which the tangents to the two circles have equal lengths. The locus of such points is immediately seen to be the line passing through the points of intersection of the circle, if they intersect. But what if they do not? The answer is, if we consider the coordinates in the Cartesian equations of the circles to be complex rather than real numbers, there are still two (finite) intersections in C2, and the (complex) line joining them still meets R2 in a line, which is still the desired locus ("radical axis" of the circle pair).
56
6. SCHWARZIAN REFLECTION, REVISITED
Since this section is included mainly for its aesthetic appeal, we introduce some simplifying assumptions that alter nothing essential in the discussion. Thus, suppose we have a real algebraic curve r in the plane, described by the equation cp(x, y) = where cp is a polynomial with real coefficients, and irreducible over C. To ensure r is nontrivial we'll assume cp(O, 0) = and grad cp(O,O) does not vanish. Let now u be a real-valued harmonic function in some "sufficiently large" neighborhood of (0,0), say in DR : {x 2 + y2 < R2}. Then it is well-known that u extends analytically to a function in the ball 1 2 B of radius 2- / . R about in C2, where it is holomorphic and satisfies the partial differential equation
°
°
u
°
(6.6) where X, Y denote complex variables: X = x + ix', Y = y + iy' with x,y,x',y' real. There is a holomorphic function fez) in DR whose real part is u/2, so u(x, y) = f(x
+ iy) + f(x + iy) =
where g, defined by g(z) =
(6.7)
fez)
u(X, Y) = f(X
f(x
+ iy) + g(x -
iy)
is holomorphic in DR. We thus have
+ iY) + g(X -
iY)
for (X, Y) E B, where f,g are holomorphic in DR. For the next part of the discussion we require a basic geometric figure in C2, well known in classical projective geometry. Let (Xl, Yt) be a point of C2 and consider the pair of (complex!) lines through (Xl, Y 1 ) defined by (6.8)
Rl := {(X, Y) E C 2
:
(6.9)
B1 := {(X, Y) E C 2
:
X
+ iY =
Xl
+ iYd
X - iY = Xl - iYd.
We have thus through the given point a "red" line Rl with direction numbers (1, i) and a "blue" Bl with direction numbers (1, -i). In classical geometry these lines often occur (usually drawn from points of R2) where they are variously called "minimal", "isotropic", "the null
6.2 STUDY'S INTERPRETATION OF SCHWARZIAN REFLECTION
57
circle (X - X l )2 + (Y - Yi)2 = 0", or "the lines joining (Xl, Yd with the circular points at infinity" (cf. [Struik, 1953]). We may also remark that these lines have a completely different role as the complex bicharacteristics of the differential operator appearing in (6.6) ("complexified Laplacian"). Now, choose a second point (X3, Y 3 ) distinct from (Xt,Yd and consider the analogous lines R 3,B3 through it. Then Rl meets B3 in a (finite) point (X2' Y 2 ) and Bl meets R3 in a (finite) point (X4' 1'4). These four points are then the vertices of a quadrilateral which, to have a convenient name, we'll call a Study quadrilateral. It is clear that the vertices numbered 2 and 4 are given by:
(6.10) (6.11) Assuming that the whole quadrilateral lies in B we then get, using (6.7)
+ U(X3' Y3) = I(Xl + iYd + g(Xl - iYl ) +I(X3 + iY3) + g(X3 - iY3) = I(X2 + iY2) + g(X4 - iY4) +I(X4 + iY4) + g(X2 - iY2) = U(X2' 1'2) + U(X4' Y4). U(Xl' Yl
)
Thus, the extension to C2 of a harmonic function satisfies the functional equation (6.12) where (Xj,Y;) (j = 1,2,3,4) are the successive vertices of a Study quadrilateral contained in the domain of holomorphy of Suppose now that u happens to vanish on r. Then (because
I} of F equals the number of w-points in {Izl < I}, hence is bounded by a number independent of w, so 00 is a regular point or pole of F. But F is not rational, having infinitely many zeroes. The proof is now complete.
Chapter 7 Projectors From L2(80) to H2(80) 7.1 Introduction. This and the next chapter are devoted to topics in one-variable complex analysis where the Schwarz function finds applications. 0 shall denote a bounded domain of finite connectivity in C whose boundary r is everywhere smooth (at least, to start with; a Holder-continuous unit tangent vector certainly suffices for our needs, but we won't discuss here the question of minimal smoothness assumptions). 7.2 The Hilbert operator of a plane domain. Let F denote a Holder-continuous complex-valued function defined on r, and consider the "integral of Cauchy type" (7.1)
{211"i)-1
J{~
-
z)-1 F{()d(.
This defines a pair of analytic functions, one (denoted li{Z» in 0 and one (denoted le{z» in Oe := (:\0. Note that Oe may have several components. It is well known (see, e.g. [Muskhelishvili, 1968]) that both Ii and Ie extend continuously to r. Note also that le{oo) = O. Moreover, for (0 E r the limits (7.2) and (7.3) exist and satisfy the "Plemelj-Sokhotski relations"
+ (211"i)-1 P.V.
i {( i«-
(7.4)
Fi{(O) = (1/2) F«(o)
(7.5)
Fe«o) = -(1/2)F«0) + (211"i)-1 P.V.
(0)-1 F(Od(
(0)-1 F(Od(
64
7. PROJECTORS FROM L'(80) TO H'(80)
where P. V. denotes "Cauchy principal value", i.e. lim e-O
f
J1'-'I>e
... d(.
Therefore
(7.6) We can thus state PROPOSITION 7.1. With the above-stated regularity assumptions concerning r and F, F can be expressed as Fi - Fe, where Fi, Fe are m C(r), and: (a) Fi is the boundary value of a function holomorphic in n. (b) Fe is the boundary value of a function holomorphic in ne and vanishing at 00. Moreover, Fi and Fe are the unique elements of C(r) satisfying (a), (b) and F = Fi - Fe.
We have only to check the uniqueness and, in view of linearity it suffices to consider F = O. But, if Fi - Fe = 0, then the functions in ni, ne whose boundary values are Fi, Fe respectively combine to form a single entire function which, vanishing at 00, is O. Hence Fi = Fe = O. From the theory of singular integrals it is known that the principal value integral in (7.4) has the same qualitative behaviour as the Hilbert transform, e.g. it maps LP(r; ds) -+ LP(r; ds) for 1 < p < 00 and preserves the class of functions Holder-continuous of order A > 0 (for references see [Garcia-Cuerva and Rubio de Francia, 1985, p.225]). In particular the maps taking F 1--+ Fi and F 1--+ Fe defined by (7.4), (7.5) initially for smooth F, extend continuously to L2(r) and we define in this way bounded linear maps from L2(r; ds) to the closure (in L2(r; ds» of the smooth functions which are boundary values of functions holomorphic in n, and analogously to the closure of the smooth functions which are boundary values of functions holomorphic in ne and vanishing at 00. We denote these closures by H2(r) and H;(r) respectively, so that H2(r) is just the (boundary values of) the Hardy space H2 associated to n, [Duren, 1970]. We have thus the direct sum decomposition (writing henceforth L2(r), or L2 for L2(r; ds» :
(7.7)
7.2 HILBERT OPERATOR OF A PLANE DOMAIN
65
and correspondingly, a contin'Uo'U3, linear idempotent map (projector) from L2(r) --+ H2(r) which we shall denote by H, and call the Hilbert operator (or projector). Note that it is, in general, an "oblique" projector, i.e. not self-adjoint or, in other words, the decomposition (7.7) is not orthogonal. The orthogonal (or Szego) projector from L2(r) --+ H2(r) shall be denoted by S. In [Kerzman and Stein, 1978] the relation between H and S was studied. Although their main program concerned several complex variables, they also studied the one-variable case, and specifically, how to express S in terms of H. The idea is that H can, in principle, be computed directly from its singular integral representation (because the "Cauchy kernel" is directly available), whereas S cannot; conceived as a singular integral operator, it is based on the Szego kernel which is not known a priori. This is an important theme in function theory: to compute, or estimate "deep" entities (like Szego or Bergman kernels, Riemann maps, harmonic measures, solutions of boundary value problems) in terms of "naive" ones that are easily computed such as Cauchy integrals, potentials, or geometric quantities like lengths, angles, areas. The classical Neumann-Poincare problem was of just this character, to compute the solution of Dirichlet's problem in terms of double-layer potentials (which one "just writes down"). PROPOSITION 7.2. [Kerzman and Stein, 1978]. The Szego projector can be expressed in terms of the Hilbert projector by
(7.8)
S = H [I + (H - H*)]-1
where I denotes the identity operator on L2(r). PROOF:
(7.9)
(7.10)
We have the immediate relations
HS=S,
SH* = S,
SH=H
H*S = H*.
Hence S(H - H*) = H - S, whence Sri + (H - H*)] = H. Since the bracketed operator has its spectrum on the line {Re A = I} it is invertible, and that gives (7.8).
66
7. PROJECTORS FROM L2(8n) TO H2(8n)
Now, H is expressible in tenns of a singular integral and so, therefore, is its adjoint H*. Kerzman and Stein made the important observation that in the integral kernel representing the operator H - H* the "singular" parts cancel and we are left with an integral operator having a smooth kernel (in particular, H - H* is compact), and that makes possible in principle effective modes of attack for computing numerically the inverse operator in (7.8). We refer for details of this to their paper as well as [Kerzman and Trummer, 1986], and shall instead follow up another of their results. THEOREM 7.3. [Kerzman and Stein, 1978]. H H = 5, i.e. if H self-adjoint, n is a circular disk.
1S
The converse is evident: when n is a disk H2(r) and H:(r) are mutually orthogonal. Actually, in the cited paper very strong regularity hypotheses are imposed: n is assumed simply connected, and r is COO. Our main purpose in this chapter is to prove the following generalizations. THEOREM 7.4. H, for some Zo E orthogonal to H2(r), n is a disk.
n,
the function z
1--+
(z - zo)-l is
This implies Theorem 7.3 because if H is self-adjoint, every function in H:(r) (in particular (z - zO)-l, restricted to r) is orthogonal to
H2(r). THEOREM 7.5. H H - H* is of finite rank,
n is a
disk.
The proofs will be given in Section 7.4. First, however, I want to discuss some related matters. 7.3 Relation to the Neumann-Poincare problem. The "Hilbert operator" of a domain, as defined in §7.2 is closely related to the Neumann-Poinca.re-Fredholm solution of the Dirichlet problem in tenns of a double-layer potential. If F is a (for the moment) real-valued smooth function on r, its double-layer potential is defined for z E C\r by
(7.11) where a/an denotes differentiation in the direction of the outer nonnal. This defines a pair of harmonic functions Ui, U e in n, ne respectively.
7.3 RELATION TO THE NEUMANN-POINCARE PROBLEM
67
It's easy to see that the above integral equals
Re (211'i)-1
i «-
Z)-1 F«)d(.
Thus, Ui(Z) has boundary values on r given by Re (HF). Consider a smooth complex-valued function F + iG on reF, G real). The boundary value (from n) of its double-layer potential is, denoting by C the operator of complex conjugation: DOW
Re (HF)
+ iRe (HG)
=(1/2)[HF + CHF + iHG + i(CHG)] =(1/2)(H + CHC) (F + iG). Hence, for any complex-valued smooth function on r the boundary values from n of its double-layer potential are obtained by applying to it the operator (1/2)(H + CHC). Thus, an L2(r) version of the Dirichlet problem is: given F on r, solve (7.12)
(H
+ CHC)G =
2F.
The double-layer potential of G is then harmonic in n and has boundary values F. The essence of Fredholm's solution of the Dirichlet problem (which we are here, for convenience, looking at in the context of L2(r) rather than the usual c(r), but that is not really essential) is then that (7.13)
H+CHC=I+K
where K is a compact operator (so that the Fredholm-Riesz theory applies to I + K). When n = 0, it is easy to check that H+CHC maps each FE c(r) to F + (211')-1 Jo27r F(eit)dt, so the K in (7.13) is just the rank one operator of orthogonal projection onto the constants. Conversely: THEOREM
7.6. If the rank of H
+ CHC -
I is finite,
n
is a disk.
The special case of rank one was also obtained by D. Khavinson (unpublished). Theorem 7.6 will be proved in the next Section. Before
68
7. PROJECTORS FROM L2(80) TO H2(80)
turning to it, we wish to remark that the solution to the Dirichlet problem in two dimensions, embodied in (7.12), doesn't require explicit reference to potentials, only to the operator H (or what is the same, to Cauchy's integral). This gives a rather neat way to "package" Fredholm's solution of the Dirichlet problem. Of course, one must first show (under suitable hypotheses on S1) that K in (7.13) is compact, and this is not trivial, and we'll assume this result (of Fredholm) here. A rather simple computation shows that, if r o, rt, ... r m denote the components of r (with ro outermost) the kernel of H + CHC consists of functions equal to 0 on ro and to some complex constant Cj on each rj(j = 1,2, ... ,m). In particular H + CHC (which is a Fredholm operator of index 0) is injective when r = ro (i.e., S1 is simply connected) and hence surjective. Thus, every FE L2(r) is the sum of a function in H2(r), and the complex conjugate of one, so it is the boundary value of a harmonic function in S1. H m ~ 1, Fredholm-Riesz theory tells us H + CHC has codimension m. If we choose a point Zj inside the contour rj (j = 1, ... , m) the functions {log Iz - zjlli=1.2 .... m are linearly independent modulo ker (H + CHC), so every function in L2(r) can be expressed as a function of the form Ei=l a j log 1Z - Z j 1 and one in the range of H + CHC, leading also in this case to the solution of Dirichlet's problem.
7.4 Proofs of the preceding theorems. We now turn to the proofs of the preceding theorems, in which the Schwarz function plays an essential role. Perhaps the simplest proof is that of Theorem 7.6, so we'll start there. PROOF OF THEOREM 7.6: To bring out the idea in its simplest form, suppose first that S1 is simply connected and K := H + CHC - I has rank one. (Note that since K does not annihilate constant functions its rank is always ~ 1.) Multiplying on the left by H in H +C H C = 1+ K gives HCHC = HK, hence HCHC has rank ~ 1, and so = 1. For any f E H2(r), HCHC] = Hf. Now, the function ] = a + hz, for suitable choice of a,b in C (not both 0), annihilates HCHC, so a+hz E ker H, which means there is a function 9 in H2(S1 e ) satisfying g(z) = a + hz for z E r. Hence b :I 0, otherwise 9 = a on r and hence in S1 e , so g( 00) = a :I 0, a contradiction. Now g(z) = O(lzl- 1) near 00, so g(z)(a+bz) =: h(z) is holomorphic in S1 e and bounded, and regular at 00. Its boundary values on r are la + bzl 2 , so 1m h is harmonic in S1e and zero on r = aOe whence it
7.4 PROOFS OF THE PRECEDING THEOREMS
69
°
is == and h is constant. Hence la + bzl 2 = C for z E r where C > 0, and so r must be a subset of the circle {Iz + ab-11 = c1/ 2 ·lbl- 1}. For topological reasons, r must be the whole circle, so 0 is a disk and the proof is finished in this case. H 0 were not simply connected, Oe would consist of several components and the above argument would only show that h is constant in each component, so h(z) = Cj for z E rj (j = 0,1, ... m) where r 0 is the outermost component of r and r 1, ... , r m (where we may suppose m ~ 1) are the remaining components. Thus ro is a whole circle, and each rj is an arc of a circle concentric with roo (Possibly that of least radius, say r m, is the whole circle; for topological reasons rj, with 1 ~ j ~ m - 1 are proper arcs.) Now, the presence of a boundary component that is a proper arc could be excluded by simply saying, we demanded more regularity than that (i.e. continuously turning tangent); but there is no need to resort to such low tactics, we can exclude this case as follows. Fix any fo E H2(r), then reasoning as before with a + bfo instead of a + bz, we conclude that a + bfo(z) is, for z E r, the boundary value of a function holomorphic in Oe. Let Zl, Z2 be distinct points on the same component of a~, satisfying IZll < 1, IZ21 < 1, and let fo be a branch of [(z - zJ)/(z - Z2)P/3. This is in H2(O). There is no loss of generality in assuming ro is {Izl = 1}. For zEro, fo(z) = [(1 - zlz)/(1 - Z2Z)]l/3 and this function is not holomorphic in Oe, so we have a contradiction to the assumption that ro is not all of a~. Finally, let's suppose rank K = k ~ 1 (and hence, rank HCHC ~ k). Consider n:::;=oCjzj : Co, ... ,Ck E C}. By linear algebra there is a choice of {Cj}, not all 0, such that the E lies in the kernel of H (reasoning as above), so with this choice of {Cj}, c.p := E;=o Cj zj satisfies: ~Ir is a boundary value of H;(r). Moreover Cj i= for at least one j ~ 1 (as before). Now we play the same game with linear combinations of the functions z, z~, ... zc.pk and conclude: there are complex constants ao, ... ,ak (not all zero) such that z(ao + al ~ + " .ak~k)lr is a boundary value of H;(r). Thus, in view of what we know about c.p, zlr is the boundary value of a function meromorphic in Oe. Call this function S(z). Now, S can have no poles in Oe. Because, if S has a pole at Zo E Oe, then E~=l CjSj must have a pole there. But this function coincides on r with ' J) =
(8.3)
1. Tn f,
>. E C.
Henceforth we'll usually suppress the subscript n when there is no danger of confusion and write simply P or T. From (8.3) we see that T2 i3 complex-linear. The following are easily verified, where denotes the inner product in L2(n), i.e. < f,g > = f'9 dO' :
In
J
(8.4)
< f, Tg > =
(8.5)
=
(8.6)
< CF,G >=< CG,F >
f gdO'
(f,g E L~
(n»)
(f'9EL~(O») (F,G E L 2 (n»)
< T2 f,g >=< PCPC f,g > = < CPC f,g >=< Cg,PCf > = < PCg,PCf >, so
Consequently,
(by (8.6»
Hence T2 is a linear, positive (i.e. non-negative; hence in particular self-adjoint) operator on L!(n) of norm ~ 1. The norm is certainly 1 if n is of finite area since then constant functions belong to L!(n). T2 has also a geometrical interpretation. Since Q := C PC is the orthogonal projector from L2(n) to CL!(n) (the conjugate-analytic functions), PQP is a self-adjoint operator on L2(n) having L!(n) as an invariant subspace, and its restriction to this invariant subspace is T2. Thus the spectral properties of T2 reflect the geometry of the pair ofsubspaces L!(!l) and CL!(n) within L2(n). Here we shall only give a few results. For the purposes of this chapter we'll say 00 is smooth at one of its points Zo if, for some e > 0, r(zo; e) := (00) n D(zo; e) is a Jordan arc with a Holder-continuous oriented unit tangent vector (where D(zoi e) denotes {z : Iz - zol < e}), and n n D(zo; e) lies to one side of r( Zo; e).
76
8. THE FRIEDRICHS OPERATOR
THEOREM 8.1. [Friedrichs, 1937]. H 0 is bounded and ao is smooth at each of its points, To is compact. That is, if {In} C L~(O) and In -+ 0 weakly, then liTo In II -+ o.
We'll deduce this later from some more general results, based on [Shapiro, 1981]. Since that paper is rather inaccessible, we'll give the details.
8.3 Weak· limits of sequences in L!(O). In this section we consider the isometric embeddings (8.8)
where 0 denotes the closure of 0, and M(O) is the Banach space of bounded complex measures on 0, conceived as the dual of the Banach space C(O) of continuous complex-valued functions on o. In this manner L! inherits from M(O) a weak* topology which, for sequences, takes the form (for notational reasons, we denote by I du the image of IE L!(O) in M(O) under the imbedding (8.8»: In du -+ I du
(8.9) {:=:}
(weak*)
10 In u du 10 lu du, for all u -+
E C(O).
When no confusion is to be feared, we express (8.9) simply by saying In -+ I (weak*). Let now 0 be any bounded open set, r = a~, and define r reg to be the set of points of r in whose neighborhood r is smooth in the sense defined in §8.2. Observe that r reg is relatively open, and so the set rirr := r\rreg, the set of "irregular boundary points of 0" is compact. THEOREM 8.2. Let Un} C L!(O),
J..I.
E M(O), where 0 is any
bounded open set, and suppose (8.10)
Then, there exists
In du -+ dJ..l.
IE L!(O) and v
(weak*). E
M(O) such that
(8.11)
dJ..l. = Idu +dv
(8.12)
suppv C
rirr.
8.3 WEAK· LIMITS OF SEQUENCES IN L~(O) COROLLARY
77
1. H rjrr is empty, the unit ball of L~(n) is weak* closed
in M(n). REMARK l: If T denotes the circle, the analogous proposition that the unit ball of the Hardy space HI (T) is weak* closed in M(T) is equivalent to the F. and M. Riesz theorem [Duren, 1970, p.4l]. REMARK 2: If an is smooth except for a "corner" at Zo of angle a 1= 7T' or 27T', then it is easy to show there exist Un} C L~(n) such that {In dO'} tends weak* to 6(zo)' Suppose, for simplicity, that Zo = 0 and n is contained in the wedge Wa := {z : 0 < (J < a} (where z = re i8 , and n n 0 = Wa n D. Let g(z) = (z - .\)-m where .\ E C\W a and m ~ 3 are chosen so that 9 E L~(Wa) and fw", gdO' 1= O. It's easy to see that this is possible if a is not 7T' or 27T' (note that if a = 7T', i.e. Wa is a half-plane, fw", 9 dO' vanishes for all 9 E L~(Wa». If now
cp E c(TI) we have, writing In(z) = n 2 g(nz), that In E L~(n) and
{ I/n(z)ldO' ::; {
in
iw",
I/n(z)ldO' = {
iw",
Ig(z)ldO'
and
which says that Un dO'} tends in the weak* topology of M(n) to a non-zero constant times a 6-mass at O. COROLLARY
2. H nrr is empty, the following are equivalent for Un}, I
in L~(n) : (i) In -+ I (weak*) (ii) supn II In II < 00 and In(z)
-+
I(z) lor each zEn.
PROOF: That (i) => (ii), for any n, is easy and left to the reader. Suppose now (ii) holds. It is enough to show that any subsequence of {In} has, in turn, a subsequence which converges (weak*) to f. But, any subsequence of {In} has a further subsequence which converges weak* to some p. E M(n), and by Corollary 1 this p. may be identified with an element of L!(n), dp. = gdO' for some 9 E L!(n). Now, this subsequence converges pointwise to g(z), and so 9 = I, which completes the proof.
78
8. THE FRIEDRICHS OPERATOR
More generally, for any bounded open for {In}, I in L!(n) when supn II In II
n
the following are equivalent
< 00 :
(a) In(z) -+ I(z), all zEn (b) 10 Incpdu -+ Io/cpdu for all cp E C(n) vanishing on rirr. The proof is nearly identical with that of Corollary 2, hence omitted. To prove Theorem 8.2, we require a lemma.
LEMMA 8.3. Let u E Cl(n) and u(C) = 0 for all , E an. Assume moreover that u vanishes on a neighborhood of rirr. Then ~ E A(n), where (8.13)
A(n) := {u E Loo(n) :
In ul du
= 0,
all f E L!(n)}
PROOF: The hypotheses imply that u satisfies a Lipschitz condition
on
n.
(8.14) Extend u to all of R2 by taking it to be 0 outside n. Then, it is easy to see that the extended function (still denoted by u) satisfies (8.14) for all z}, Z2 in R2. In particular, u belongs to the Sobolev space W 1 ,P(R2) for all p < 00. Since also supp u c n and u vanishes on a neighborhood of the irregular boundary points, it is easy to deduce that u E W~,p(n). Henceforth we shall take p = 3. Thus, there is a sequence {CPn} in cOX'(n) such that CPn -+ U in the norm of Wl,3(n). n -+ ~ in L3(n), and so for all 9 E L3/2(n) This implies
8li
(8.15)
lim n-+oo
f
10
g(z) a:,n du vZ
= f g(z): duo 10
vZ
Choosing, in particular, g(z) = (z-a)-l where a is any point in C\n, we have 9 E L:/2(n) and, since I g8!Jin du = 0 for each n, (8.15) tells us that I 9 ~ du = O. Hence I ~ f du = 0 for every rational function f with simple poles, all outside of n, and since [Bers, 1965] these functions are dense in L!(n), and ~ E Loo(n), this last relation
8.3 WEAK· LIMITS OF SEQUENCES IN Ll(O)
79
extends to all IE L!(n), that is to say, ~ E A(n) and the lemma is proven.
In
In
8.2: By hypothesis In r.p du --+ r.p dJ1. for all r.p E C(n). In particular, choose here r.p = ~ where .,p E C8"(n). Then, since I In~ du = 0 for all n, we get that I ~dJ1. = 0 for all such .,p. Let us now write J1. = J1.i + v where J1.i is the restriction of J1. to n, and supp v c an. Then I ~dJ1.i = o. Considering J1.i as a distribution in n, this says = o. By "Weyl's lemma" this implies that the distribution J1.i is identified, in the canonical way, with a holomorphic function I on n. Since J1.i is a bounded measure, we have I E L!(n). Hence we have In du --+ I du + dv (weak*). Thus, v E M(an) and we have 9n du --+ dv (weak*) where 9n := In - / E L!(n). To complete the proof, we have to show supp v c rjrr. Let (0 denote any point of rreg. It follows from the definition of regular point that there is a function v E Cl(~), where ~ is some neighborhood of (0 in R2, such that v(e) = 0 for all ( Ern ~ and grad vko =f:. o. We may take v real, so the last relation is equivalent to ~ ko =f:. o. Multiplying v by a "cut-off function" in coo(R2), with support in ~ and equal to one on a neighborhood of (0, we may assume that v is defined and of class CIon all of R2 and vanishes on all of r and on a neighborhood of rjrr. Finally, let P denote any polynomial in x, y with real coefficients, and set u = Pv. By the lemma, ~ E A(n). Hence PROOF OF THEOREM
W
Since supp vCr and v vanishes on r, the first integral on the right vanishes and we have I P ~ dv = o. This holding for all polynomials P, the measure ~ dv vanishes. Since ~ =f:. 0 on a certain neighborhood of (0, supp v cannot meet this neigborhood. Since (0 was any point of r reg, supp v n r reg = 0 and Theorem 8.2 is proven. Implicit in the above is an approximation theorem which may be of some interest (the case n = unit disc was proved by [Reich, 1976] using Fourier series): Let Q be any bounded open set, and let C#(r) denote the set 0/ functions in C(r) which vanish on rjrr. Then, the restrictions to r of functions in C(Q) n A(Q) are dense in C#(r).
COROLLARY 3.
80
8. THE FRIEDRICHS OPERATOR
is any measure on r which annihilates those restrictions then, as was shown in the proof of Theorem 8.2, v is supported on rjrr, and hence annihilates elements of C#(r). Hence, by a standard duality argument, the result follows. PROOF: If
V
We can now easily prove Theorem 8.1. Suppose, then, that fl satisfies the hypotheses there and Un} C L!(fl) is a sequence tending weakly to o. We must show
IIT/nll-+ o.
(8.16)
There is no loss of generality to assume T/n =1= o. Define IIT/nll-1 so that 9n E L!(fl), 119nll = 1 and (T/n,9n) = that, using (8.4)
9n = T/n . liT/nil, so
(8.17) Now, hn := by II· lit)
/n 9n
is in L!(n) and (denoting the norm in that space
for some constant C. Moreover, for each Zo
En
so {h n } tend to zero pointwise in n. By Corollary 2 to Theorem 8.2, {hndO'} -+ 0 (weak*) in M(n), hence
so, in view of (8.17), (8.16) holds and Theorem 8.1 is proven. REMARK: It's easy to see that Theorem 8.1 remains true if we allow cusp singularities on an which point into n (i.e. angles of 271").
8.4 The Friedrichs operator, geometry, and the Schwarz function. Under the hypotheses of Theorem 8.1, T2 has a sequence of eigenvalues 1 = AO > Al ;:::: A2 . .. decreasing to zero, and a corresponding
8.4 FRIEDRICHS OPERATOR, GEOMETRY, SCHWARZ FUNCTION
81
sequence of eigenfunctions {CPn(z)}n>O which form an orthonormal basis for L~(n) and are easily seen ~ satisfy the remarkable "double orthogonality" relations (8.18)
(8.19) Clearly ).0 = 1, corresponding to CPo = constant. There may be degeneration in the sense that 0 is an eigenvalue of T2, even of infinite order. In the case of the disk we have in fact rank (T2) = 1 and ).1 = ).2 = ... = o. In general, ).1 is a measure of the "angle" between L! and C L!. It is characterized by
(8.20) where sup
IS
over
As was already indicated, these eigenfunctions enable one to write down solutions of the basic boundary value problems for the biharmonic equation. Friedrichs also studied the essential spectrum of T2 in cases where T2 is not compact. [Norman, 1987] showed that this essential spectrum contains 0 if an is smooth near at least one point. Moreover, extending analysis of Friedrichs, he showed that if an consists of finitely many smooth arcs joined so as to for~ "corners" with interior angles Ql, ••• Q n
then each of the numbers 1sm Qj 12 is in the essential specQj
trum of T2. Thus, the spectral properties of T2 reflect the geometry of n in an interesting way. Norman also investigated the injectivity of T2, and obtained the following result.
n be a bounded domain whose boundary is smooth near two of its points (1, (2. Suppose that an coincides near these points with nonsingular analytic arcs 1'1,1'2 having
THEOREM 8.3. [Norman, 1987]. Let
82
8. THE FRIEDRICHS OPERATOR
Schwarz functions S10 52. H 51 and 52 are analytically continuable along paths in n into some neighborhood where they do not coincide, then T2 is injective. Thus, for example, if an contains two segments of different straight lines, T2 is injective. We won't give the proof here, because it involves technical difficulties concerning boundary behaviour, but shall outline a heuristic argument which indicates the main idea of the proof. So, suppose n satisfies the hypotheses of Theorem 8.3 and that I is a nontrivial function in ker T2. In view of (8.7), T I = 0 and that means, by (8.4) (8.21)
10 I gdu
= 0,
lor all 9 E L!(n).
a
By virtue of Lemma 4.2, or more accurately its analog for the operator, there is a function 11. E W~,2 (n) satisfying Bu/az = I in n. Thus (a/az)(u - zJ) vanishes in n, and so for some 9 holomorphic in n,
(8.22)
u(z) = zl(z) - g(z)
in
n.
If we can deduce that I = 0 we'll now have a contradiction, which would prove the theorem. So far, the analysis has been rigorous, but now we'll argue heuristically. Since 11. "vanishes on an " we get from (8.22) (il I and 9 had boundary values in a suitable sense this would be rigorous, but functions in L~(n) don't in general have boundary values, which is why we use the adjective "heuristic"): z = g(z)/I(z) on an. Thus, if r l is any analytic arc of an its Schwarz function has a meromorphic continuation throughout n given by 9 / I. This continuation is the same for r 2, and our hypotheses now imply the sought-for contradiction. Similar reasoning shows that if an is a single analytic arc, its Schwarz function extends meromorphically throughout n, and n is a quadrature domain. As we'll see in the next section, T2 has finite rank in a q.d., so we get the surprising result: if an is a non-singular analytic Jordan curve, and T2 is not injective, then T2 is of finite rank. Thus, if T2 maps any nontrivial function to 0, it maps almost everything to
O! 8.5 The Friedrichs operator and quadrature domains. In [Shapiro, 1984] the following theorem was proved:
8.5 FRIEDRICHS OPERATOR AND Q.D.
83
8.4. Let n be a planar open set whose boundary consists of IS finite number of continua. Then, the following assertions are equivalent: (i) n is a quadrature domain, in the sense that
THEOREM
for all h E L~(n). Here the non-negative integers numbers Cij and points Zj En do not depend on h.
rj,
complex
(ii) The "Friedrichs operator" To is of finite rank. It is of interest that in a quadrature domain (even one "in the wide sense", in which case it needn't be of finite rank) the Friedrichs operator has a property much stronger than compactness, as given by the following result from [Shapiro, 1987]. The open set n is a q.d.w.s. if and only if there exists a compact set Ken and constant C such that the Friedrichs operator T of n satisfies THEOREM 8.5.
We refer to the cited papers for the proofs of the last two theorems. The last inequality implies exponential decrease of the eigenvalues of
T2. It seems there are numerous problems of interest connected with relating spectral properties of T6 to the geometry of n. These problems can of course be generalized, in the spirit of "Hankel operators", by introducing a "symbol function" cp E LOO(n) into the definition of T, i.e. considering the operator f 1-+ Po MtpC f where Mtp denotes multiplication by cpo Similary, one can introduce Friedrichs-like operators in the context ofthe "boundary spaces" L2 (r; ds) and H2 (r; ds) that Were studied in the last chapter.
Chapter 9 Concluding Remarks 9.1 The Schwarz potential in en. We first encountered the Schwarz potential as attached to a nonsingular real-analytic arc in R2 (a concept which immediately permits generalization to a real-analytic "hypersurface" (i.e. of dimension n -1) in Rn). We have seen that if the arc happens to be a closed curve bounding a lamina of uniform density in R2, the singularities of the Schwarz potential are precisely those encountered by the logarithmic potential of this lamina, considered initially in the exterior of the lamina (where it is harmonic) when it is extended harmonically inwards across the boundary of the lamina; the nature of this extension was the theme of the "Preisaufgabe" that Herglotz studied in his paper which we have often cited. Once we re-interpret this problem (as we have done, and which Herglotz never did) in tenns of a Cauchy problem (for the Laplace equation) we are outside the usual territory of potential theory and in the domain of partial differential equations, especially the question of propagation of singularities for solutions to Cauchy problems. This subject has been extensively studied for (linear) hyperbolic equations in Rn, less so for elliptic equations. Since the solutions of an elliptic Cauchy problem, in case the partial differential equation is linear with real-analytic coefficients, and the initial manifold in Rn as well as the Cauchy data are real-analytic, extends always some distance into en, it is natural from the beginning to place the action in en rather than Rn in the elliptic case, i.e. to study the question in the framework of the holomorphic Cauchy problem, and that is what we shall do in this Section. It is a remarkable fact that the singularities (in Rn) of the Schwarz potential are much better understood when seen from the vantage point of en, as we'll learn shortly. Concerning the holomorphic Cauchy problem in general, there still does not seem to exist an adequate global theory. The Cauchy-Kovalevskaya theorem gives the local existence and uniqueness theory near non-characteristic points of the initial manifold. The local theory of singularities that "arise" at characteristic points of the initial manifold is the subject of the seminal paper [Leray, 1957]. Later, this
9.2 THE ELLIPSE, REVISITED
85
theory was reworked and extended in [Garding, Kotake and Leray, 1964] and other works. As to global results, much is known when the Cauchy data is given on a hyperplane. (e.g [Miyake, 1981], [Persson, 1971], [Johnsson, 1989]) but almost nothing when it is given on a more general (even an algebraic) manifold. For the Laplace operator, there are some rudimentary results in [Khavinson and Shapiro, 1989a] and a fairly complete treatment for data on quadric surfaces in [Johnsson, 1990], where ideas of Leray et al are adapted and extended into a global context. The following discussion is very sketchy, intended only to illustrate some of the ideas that come into play.
9.2 The ellipse, revisited. As an illustration of the "cn viewpoint" (here with n = 2) let us seek to "explain" why the Schwarz potential of the ellipse has singularities at the two foci, and nowhere else in R2. We'll take as our ellipse (9.1)
r:= {(x,y)
E R2: Ax2 +By2
= I}
where A = a- 2 ,B = b- 2 and a > b> o. The foci of r are at ±c, where c = (a 2 - b2 )1/2. Along with r we'll consider its "complexification"
(9.2)
r:= {(X, Y)
E C 2 : AX 2
+ By 2 = I}
where X = x + ix', Y = y + iy' and x, y, x', y' are real. From here on we'll take for granted familiarity with the Cauchy-Kovalevskaya theorem (C-K theorem) and related notions like "characteristic point", etc. (cf. [Garabedian, 1964]). The (modified) Schwarz potential of r is the (unique) solution to the Cauchy problem
r
(9.3)
.6.v = 1 near
(9.4)
v and grad v vanish on
r.
Since there are no characteristic points on r, v exists as a real-analytic function in a neighborhood of r by virtue of the C-K theorem. Hence it extends to a holomorphic function on some neighborhood N (in
v
86
9. CONCLUDING REMARKS
v vex, Y) solves the holomorphic (meaning
e2 )
of r.1t is clear that = here: in e 2 ) Cauchy problem (9.5)
(9.6)
v,
av
ax
and
av
ay
vanish on
t n N.
v
t
Now, is holomorphically extendible along every path on that does not go through a characteristic point. For a complex one-dimensional non-singular variety ("curve") in e 2 defined by cp{X, Y) = 0 where cp is holomorphic near (XO, yO) and vanishes at (XO, yO), the condition for this point to be characteristic for the operator in (9.5) is
(9.7) Hence, the characteristic points on tions
t
are gotten by solving the equa-
(9.8)
(9.9) This shows there are 4 characteristic points on independent choices of sign in
t,
corresponding to
(9.10)
v
t
Now, analytic continuation of along (so that the Cauchy data remains 0, as prescribed by (9.6» must lead to a singularity when we reach a characteristic point. This is a general phenomenon (also in en), and can be seen as follows. Suppose first we are in a neighborhood M of a point of cp(X, Y) = 0 where never simultaneously vanish. (In the present situation, cp = AX2 +By2 -1, but that special and its gradient vanish on it's choice plays no role.) Then since
U,
t:
v
-u.
r,
9.3 PROPAGATION OF SINGULARITIES, AN EXAMPLE
87
easy to see V =