Ann. Univ. Ferrara - Sez. VII - Sc. Mat.Vol. XLVII, 169-175 (2001)

Branch Points, Fourier Integrals and Pompeiu Problem.

FAUSTO SEGALA (*) - MATTEO TRIOSSI (**)

SUNTO - Si prova che se p è un polinomio e la mappa h(w)4kp(w) è univalente sul discounitario D del piano complesso, allora V4h(D) ha la proprietà di Pompeiu.

ABSTRACT - Let h be the square root of a polynomial and assume that h is univalent onthe unitary disk of the complex plane. Then the set V4h(D) has the Pompeiuproperty.

1. – Introduction.

In 1929 D. Pompeiu posed the following very hard problem. Let V be a sub-set of R 2 and fC 0 (R 2 ).

If

s(V)

f dx40 ( rigid motion s(1.1)

it is true that ff0?When the answer to this question is positive, V is called a Pompeiu

set (or V has the Pompeiu property). Many contributions towards thesolution of the problem were given since 1972-73 following the fundamentalsideas of the papers of Zalcman [3] and Brown-Schreiber-Taylor [1]. Ina recent paper of Garofalo-Segala [2], the Pompeiu property of a bounded

(*) Indirizzo dell’autore: Università di Ferrara, Italy.(**) Indirizzo dell’autore: Universidad Carlos III, Getafe, Spain.

FAUSTO SEGALA - MATTEO TRIOSSI170

simply connected open set V4h(D) is studied by analyzing the Riemannmap h (D is the unitary disk).

In this work we give a further contribution in that direction. Precisely weprove the following theorem.

THEOREM. Let p(w) be a polynomial and assume that h(w)4kp(w) isunivalent on D . Then V4h(D) is a Pompeiu set.

2. – Preliminaires.

From now on V will be a bounded simply connected open subset of theplane. By the celebrated theorem of Brown-Schreiber-Taylor [1], V is a Pom-peiu set iff xׯV (the Fourier-Laplace transform of the characteristic function of¯V) is not identically zero on every «circle» z 1

21z 224aD0 in C 2 . In [2] it is

shown that xׯV is identically zero on a «circle» of radius ka iff the function

rO ¯V

e rz2a

rS(z) dz(2.1)

is identically zero for every rD0, where S(z) is the Schwarz function of V .In this work we will prove that the Fourier integral (2.1) is not identically

zero, where V is under the hypotheses of Theorem, by analyzing its asymptot-ic expansion for rKQ .

We recall that the Schwarz function S(z) is given by

S(z)4 h u 1

h 21 (z)v .

If we denote by T the set of the singularities of h (obviously outside D), the set

of the singularities of S(z) is given by S4h g 1

Th (see picture below).

T1T S∂Ω

Figure 1

BRANCH POINTS, FOURIER INTEGRALS ETC. 171

3. – Polynomials of degree one.

We begin by proving theorem when h(w)4kw2a , with NaND1. By theinvariance of the Pompeiu property under rigid motions and dilations, we canassume

h(w)4o11w

221 .

We choose the branch kz4kre iw/2 , 2pEwGp and then we cut C on

11 w

242t , tF0, that is on T4 [2Q , 22]. It follows 1

T4 k2 1

2, 0l ,

S4 k k3

221, 0l . In this simple case we can give the explicit construction of

S(z). In fact we have

S(z)4o111

4z 218z21(3.1)

and we make the cut on 11 1

4z 218z42t , that is on k k3

221, 0l.

We must analyze the Fourier integral

J(r)4 ¯V

e rz2 (a/r) S(z) dz

with S(z) given by (3.1).We change the integration along ¯V with the integration along the path of

the picture below:

Gδ

Figure 2

FAUSTO SEGALA - MATTEO TRIOSSI172

Let d be the radius of the circle C . Denote by s(z) the jump of S and take intoaccount that

s (z)Ai

k2NzN, zK0 .(3.2)

If we require rkdKQ , for rKQ , we have a

rS(z)K0 on [2A , 2d] and

then e 2(a/r)S(z)A12 a

rS(z). Therefore we can write

J(r)AC

e rz2 (a/r) S(z) dz1a

r

2A

2d

e rz s(z) dz .

Now we make some calculations about the second Fourier integral. From (3.2)it follows

a

r

2A

2d

e rz s (z) dzAia

k2r

2A

2d

e rz dz

kNzN4

4ia

k2r

d

A

e 2rz dz

kz4

ia

k2r 3/2

dr

Ar

e 2u du

ku.

At this point we choose dr41. Then

a

r

2A

2d

e rz s(z) dzAiak2

r 3/2

1

Q

e 2u 2du .

Now we analyze the integral over C .

C

e rz2 (a/r) S(z) dzAi

r

2p

p

exp ye iw2a

2 k2kre 2iw/2z e iw dw4

4ia

2 k2r 3/2

2p

p

exp (e iw ) e iw/2 dw4ia

k2r 3/2

0

Q

exp (2e 2y ) e 2y/2 dy4

4ia

k2r 3/2

0

1e 2u

kudu4

iak2

r 3/2

0

1

e 2u 2du .

BRANCH POINTS, FOURIER INTEGRALS ETC. 173

Summing up, we conclude that

J(r)Aiakp

k2r 23/2 , rK1Q .

4. – Polynomials of general degree.

Let p(w) be a polynomial with even degree N42p12 with N distinctroots. Then the Riemann surface of kp(w) is a sphere with p handles, i.e. asurface of genus p (picture 3), since the roots x1 , R , xN are ramification pointsand we make p cuts along p lines joining p couples of these roots (see figure 3).

Figure 3

When N42p11, the Riemann surface is of the same type, if Q is regard-ed as a ramification point. Therefore in the case N even, we may choose a

branch of kp(w) which is holomorphic outside N

2cuts and which has a pole of

order N

2at Q . In the case N odd we may choose a branch of kp(w) holomor-

phic outside N11

2cuts, with a cut joining a finite root with Q . In the picture 4

we illustrate the set T of the singularities of h(w).We give also a picture which illustrates the corresponding set S of the sin-

gularities of the Schwarz function.A cut T relative to h(w) on the plane w induces a cut S relative to S(z) on theplane z . In general it is extremely complicated to find the principal contribu-tion to

g

e rz2 (a/r) S(z) dz where g is a contour around S .

To solve the problem we explain our idea.

Let A , B the extreme points of T and h g 1

Ah , h g 1

Bh the corresponding

FAUSTO SEGALA - MATTEO TRIOSSI174

D D

∞

N oddN even

Figure 4

points of S . We denote by L the segment line joining h g 1

Ah and h g 1

Bh and

choose the cut T in the following way:

T41

h 21 (L).

Then the corresponding set S is given by S4L! So we can assume that thecuts relative to S(z) are segments line. In the case N42p even, we have toconsider two types of Fourier integrals:

H(r)4g

e rz2 (a/r) S(z) dz(4.1)

N oddN even

Figure 5

BRANCH POINTS, FOURIER INTEGRALS ETC. 175

where g is a contour around the cut L ,

K(r)4C

e rz2 (a/r) S(z) dz(4.2)

where C is a circle around zero and S(z) has a pole of order p in 0 .The integral (4.2) is studied in Garofalo-Segala [2] by using the method of

the steepest descent and its principal contribution for rKQ is given by

const . r2

2

p11 exp gCrp21

p11 hfor some constant C .

In order to find the principal contribution to H(r), we observe that L is farfrom zero and therefore

H(r)Aa

r

L

e rz s(z) dz

where s is the jump of S(z) over L .We write L4]A1 tB , t [0 , 1 ]( with Re BD0 and make some calcula-

tions:

H(r)Aa

re rA

1/2

1

e rtBk12t dt4a

re r(A1B)

0

1/2 e 2rBu kuduAakp

2B 3/2

1

r 3/2e r(A1B) .

In the case N odd, we have to integrate along a segment line whose zero is anextreme point. By using the invariance of the Pompeiu property under rota-tions, we may assume L4 [2z 0 , 0 ] with z 0D0. Then we integrate along thepath of the picture 2 and the radius d of the circle is forced by the conditionrd N/2KQ .

Then the calculations are of the same type of those developed in section 3.The proof of theorem is complete.

R E F E R E N C E S

[1] L. BROWN - B. M. SCHREIBER - B. A. TAYLOR; Spectral synthesis and the Pompeiuproblem, Ann. Inst. Fourier, 23 (1973), pp. 125-154.

[2] N. GAROFALO - F. SEGALA, Univalent functions and Pompeiu problem, Trans.Amer. Math. Soc., 346 (1) (1994), pp. 137-146.

[3] L. ZALCMAN, Analyticity and the Pompeiu problem, Arch. Rat. Anal. Mech., 47(1972), pp. 237-254.

Pervenuto in Redazione il 27 marzo 2001.