Iterates of the Bochner-Martinelli integral operator in a ball Текст научной статьи по специальности «Математика»

УДК 517.55

Iterates of the Bochner-Martinelli Integral Operator in a Ball

Alexander M.Kytmanov* Simona G.Myslivets^

Institute of Mathematics, Siberian Federal University, av. Svobodny 79, Krasnoyarsk, 660041,

Russia

Received 10.01.2009, received in revised form 20.03.2009, accepted 29.04.2009

In the present paper we prove the convergence of iterates of the integral Bochner-Martinelli operator in a ball in various spaces: the infinitely-smooth functions, the analytic functions and the spaces conjugate to them, the distributions and the analytic functionals. We give a description of a spectrum of this operator in these spaces as well as the space Lp.

Keywords: Bochner-Martinelli integral operator, iterates.

Iterates of the Bochner-Martinelli integral operator were considered by A.V.Romanov and his result was applied to research the solvability of d-equation. For a ball B in Cn and for the space L2(dB) they have been considered in [1], while for any domain D and spaces W2(D) they have been considered in [2]. Since that, there was no further advancement in generalization of these results to other classes of spaces. Moreover, E.L.Shtraube has shown (se example in [3]) that in any domain it is impossible to expect the convergence of these iterates in spaces Wf for any s.

In the present paper one proves the convergence of iterates of the Bochner-Martinelli integral operator in a ball in various spaces: infinitly-smooth functions, analytic functions and the spaces conjugate to them, distributions and analytic functionals. We give a the description of the spectrum of this operator in these spaces and in the space Lp.

1. Let B = {z G Cn : |z| < 1} be the unit ball in Cn, n > 1 and let S = dB = {z G Cn : |z| = 1} be its boundary. Denote by W|(B) the Sobolev space, s G N. We remind that this space consists of functions f G L2(B) such that the derivatives daf of the order s belong to the space L2(B), where

dMIf

daf

dz I1

.dza

. dza2n

and a = (ai,. .., a2n),

a i + ... + a2n. If

and

(f,g)c2{B) = J f • Z dv в

llf ll£2(B) = \J(f, f )C2(B)

are the scalar product and the norm in £2(B), where dv is the element of volume in Cn, then the scalar product and the norm in space W%(B) are given by the formulas

(f, 9)Ws,(B)

II а У <s

(daf,da g)C2 (B)

*e-mail adress: kytmanov@lan.krasu.ru te-mail adress: simona@lan.krasu.ru © Siberian Federal University. All rights reserved

a

and _

11/IIw|(b) = \J(f,f )W|(b}-

Consider the space W2s+A(S) for 0 < A < 1. It consists of functions / £ W2(S) such that the integral

V- Idaf (z) - daf (Z)|2 d d

|Z - z|2n+2A-1 daCdaz,

S S ||a||=s

converges, where da is an element of the surface S.

We will use the following properties of these spaces (see [4]):

s-1

1) The restriction of the function f £ W2(B), s ^ 1, to S belongs to the space W2 2 (S), and the operator of restriction is continuous.

2) If we denote by G2(B) the subspace of harmonic functions in the space W2 (B), then the

s-1

operator of restriction from G2(B) to W2 2 is a linear topological isomorphism. Observe that the folowing decomposition holds:

W2(B) = GS(B) ©Ns(B),

where the space N2s(B) consists of the functions in W|(B) that are equal to zero on S.

3) Embedding theorems imply that there exists the compact continuous embedding W| (S) C

Ck (S) where s > n + k - ^.

For f £ W2s(B), s £ N, we consider the Bochner-Martinelli formula for smooth functions

f (z) = J f (Z )U (Z, z) -J df A U (Z, z), z £ B, (1)

S B

where

U (z, z) = (n-ini y (-i)k-1 (Z:k - f2k) dc[k] A dZ

vs' y (2ni)n k==l IZ - z|2n

is the Bochner-Martinelli kernel and dZ = dZi A.. .AdZn, dZ[k] = dZ1 A. . .AdZk-1 AdZk+1 A. . .AdZ, df_

k = 1 dZk _ Then I = M + T, where I is the identical operator in Wf (B). Since the operator d is aj bounded operator from W22(B) to W2-1(B), the operator T is the bounded operator from W22(B) to W2 (B) (see for example, [4]). Thus M is also bounded in W2(B).

Let's denote by V„hi the space of homogeneous harmonic polynomials of degree m in z and of degree I in Z. It is well-known that the space |J Vm,i is dense in L2(S) and all spaces Vm,i have

finite dimensions. Since polynomials from different spaces Pm,i are orthogonal (in L2 (S) and in L2(B)) it follows then each function f £ G2(B) admits a representation of the form

f (z) = £ Pm,i (z), (2)

m,l

where Pm,i £ Pm,i. The series (2) converges uniformly on any compact in B with respect to the norm in the space W2(B) to a function f (convergence in W2(B) follows from usual properties of a complete orthogonal systems in a Hilbert space). By Lemma 5.2 from [3] the following equality holds:

MPm,i = n + m - \ Pm,i. (3)

n + m + i - 1

_ df _

df ^^ d_. We denote by Mf the first integral (in (1)) and by Tf the second integral.

The representation (3) and the properties of the Bochner-Martinelli integral imply the following assertion.

Proposition 1. The operator M : GS(B) —> GS(B) is a bounded self-adjoint operator with i|M||w=(b) = 1■ All rational numbers in the interval (0,1] are eigenvalues of the operator M of infinite multiplicity■ The spectrum M coincides with the interval [0,1].

Proof■ Consider the decomposition of the form (2) for the function f G Gf (B). Using (3), we obtain the decomposition

Mf = E TIm

z—' n + m + I — 1

m,l

It follows that

IMf Il 1(B) = E( nn++mm+l -1, )2UPm,lUl(B) < llf Il2(B).

Since

Mtrf = ¿J n+m-ai — — 1

z—' n + m + l — a.\ — .. . a.2n — 1

E

m,l

it is obvious that

llMdaf |L2(b) < ||daf lllpy

From here and from the norm definition in Wf (B) we have that ||M||wj(b) = 1.

The self-adjointness of the operator M follows from the formula (3) and the decomposition (2). The formula (3) also implies that all rational numbers from an interval (0,1] (and only such numbers) are the eigenvalues of the Bochner-Martinelli operator of infinite multiplicity. □

Proposition 2. Let nO be the projection operator from GS(B) onto the subspace of holomorphic

functions OS(B) C Gf(B)■ Then Mk -► nO in the strong operator topology of space GS(B)

k—

for any s G N.

Proof the proposition follows from the Banach-Steinhaus theorem, the Proposition 1 and the equality (3). □

Let's introduce the Szego operator K in a ball B by the integral

„ / n (n — 1)! ¡' , „, daz Kp(z) = K-7K-ér v(Z ): Z

(2ni)n J ^(1 — (z,C))r

S

where (z, Z) = ziCi + • • • + znCn, and daz is an element of a surface S. The operator K coincides with the projection operator ns0 from the space G2(B) onto the subspace O%(B) for all s. It is known that for any function f £ CX(S) the operator Kf £ O(B) n CX(B) (see, for example,

[5]).

Let's denote by £(S) the space of functions in CX(S) with a topology of the uniform convergence on S with all derivatives.

Theorem 1. For a function f £ £(S) the sequence Mkf converges to Kf in £(S) as k ^ <.

Proof. It suffices to show that f tends to Kf in the spaces Cl (S) for any I £ N. Properties 2), 3) of the spaces W%(B) and the Proposition 2 imply the theorem. □

Denote by £'(S) the space adjoint to £(S) (relatively of a measure da), i.e., the space of distributions on S.

It is known [3], that the restriction of the Bochner-Martinelli kernel on S has the form M(Z, z)da^, where

M= £ Zk Zkzk

2nn k==[ k IZ - z|2n'

Let's denote by

MT (z) = Tc (M (Z,z))

the Bochner-Martinelli transform for the distribution T £ E'(S).

The function MT is harmonic in the ball B and has a finite order of growth near S. Therefore MT determines some distribution on S. We will denote it by [MT]0.

Let's denote by Gf(B) the space of harmonic functions of a finite order of growth nearly S, and let's denote by OF(B) the space of holomorphic functions from GF(B). It is known, that the space GF (B) is isomorphic to space E'(S) ([6]).

Theorem 2. For a distribution T £ E'(S) the sequence [MT]k converges in E'(S) as k ^ to to a distribution in Of(B).

Proof. It is known that the harmonic continuation of a distribution T £ E'(S) is given by the Poisson transform ([6])

PT (z) = Tc (P (Z,z)),

where the Poisson kernel is equal to

P(Z z)=(n - 1)! (1 -|z|2)

; 2n" IZ - z|2«' For any function ^ £ E(S) the value T^> is equal to

T(¿)= lim/ PT(rZ)^(Z) dac. (4)

S

We show that the operator M satisfies the condition

[MT ]o(^) = T (M^).

Since any function from Gf(B) expands into a series in homogeneous harmonic polynomials Pm i converging absolutely in B, it follows that

PT(z)=£ Pm,i(z).

m,i

Using (3) we conclude that

MT (z) = £ ++"+■ 11 Pm,i (z). (5)

n + m + i -1

Indeed, MT(z) £ GF(B) and for any 0 < r < 1 the series

PT(rz)= £ Pm,i(rz)

,i

converges absolutely and uniformly in B. Therefore

n+m n + m + ■ 1

MPT (rz) = £ ++"+ ■ 11 Pm,i(rz).

i—' n -4- m -4-1 —

This implies (5).

Since the function y admits a similar decomposition, it follows that

y(z) = £ Qm,i(z),

where Qm,;(z) are homogeneous harmonic polynomials in B. Therefore

n + m _ 1

-(

n + m + Z — 1

^ , . n + m — 1 ^ . .

My(z) = £ + + . Qm,l(z)

' n. m .4- / —

and hence for 0 < r < 1 we have

——« n —I— mm _ 1 /

MPT(y) = £ n + m . 1 PmM)Qm,i(rZ) dac = PT(My). z—' n + m + Z _1 J

Applying the formula (4) we conclude that

[MT ]o(y) = T (My).

Therefore,

[MT ]k (y) = T (Mk y). (6)

Since by the Theorem 1 Mky -► Ky in f(S), for any y G f(S) there exists the limit

k—

lim [MT]k(y). This limit determines a distribution L G f'(S).

k—

We show that PL G (B). It suffices to prove that L is a CR-distribution. For this it is necessary to show that dTL = 0 where dTL is the value L on functions dT y satisfying the condition

dTy da = dy A d£[Z, m] A dZ|S. Using the formula (6) and the Theorem 1 we obtain

5tL(y) = lim dT[MkT]o(y) = k—

= _ lim [MkT]o(5ty) = _ lim T(Mk(Sty)) = _ lim T(K(<9ty).

k—k—k—

Since the function dTy is orthogonal to the holomorphic functions in L2(S), it follows that K(dTy) = 0. By the Hartogs-Bochner theorem (see, for example, th. 7.1 from [3]) we have that PL gOf(B). □

2. Consider the space A(S) of real-valued analytic functions on S as on a real analytic manifold in Cn ~ R2n. Then any function y G A(S) has holomorphic continuation to some complex neighbourhood U C C2n.

The linear functional T on A(S) is analytic if for each complex neighbourhood U of the sphere S there exists a constant c(U) such that

|T(y)| < c(U)sup |y| (7)

U

for any entire function y on C2n [7]. In this definition we can take the holomorphic functions on U instead of entire functions. We denote by A'(S) the set of analytic functionals on S.

The property (7) can be expressed in terms of the function y and its derivatives on S. Namely we consider the following expression

l(y) = Hm "a"Isups |day|

ail —► w

where a! = a^. ■... ■ a2n!. Then the function — G £(S) is belongs to A(S) if and only if £(—) <

and ——) is the radius r of a polydisc U(Z0,r) with the center at the point Z0 G S to which —

holomorphically continues. The union of all such polydiscs gives a neighbourhood U to which — can be holomorphically continued.

The sequence of functions {—k} converges to — in A(S) as k ^ to if the functions —k holomorphically continue to some complex neighbourhood U and in this neighbourhood uniformly

converge to —. Therefore, if T G A'(S) then T(—k) -► T(—).

k—

We can identify the space A(S) with the space G(B) of functions harmonic in B. Each function — G A(S) can be harmonically continued to B as a function $. This function $ is a function of class Cup to a boundary S [8] and since — is real analytic on S it follows that $ G G(B). The maximum principle shows that the topologies in A(S) and in G(B) coincide.

For a function — G A(S) we consider M—. Since the coefficients of the Bochner-Martinelli kernel are the derivatives of the potential of a simple layer, by Lemma 24.1 in [3] we have that IM— ^ I—. Hence lMk — ^ I—. I.e., the iterates Mk — holomorphically continue to the same neighbourhood U C C2n of the closure of the ball B as the function —.

Theorem 3. For a function — G A(S) the sequence {Mk—} converges to K— at k ^ <x in some neighbourhood U of the closure B.

Proof. We consider a neighbourhood U to which all functions Mk— have a holomorphic continuation. We continue — as a harmonic function in G(B) and expand it into a series in homogeneous harmonic polynomials Pm,i, i.e.,

—(z,z) = Y, Pm,i(z,z). (8)

m,l

This series converges uniformly and absolutely in some ball Br D B, r > 1 (§4, ch. 11 [9]). By the Kiselman theorem a series of the form Pm,i (z,w) absolutely and uniformly converges to

m,l

the holomorphic continuation — of the function — in some neighbourhood U C C2n of a ball Br. We can chose this neighbourhood to be the ball Li (the symmetric domain of the fourth type) [10]. Since

m>k ( n + m - 1 \kr>

Mk— = E ^——H Pm'i>

+ m + I — 1/

m,l

it follows that the holomorphic continuation Mk — in U satisfies the estimate

|MV | lpml (z,w)l

m,l

on any compact from U.

Thus the sequence {Mk—} is uniformly bounded on any compact lying in U. Hence the sequence {Mk—} converges uniformly in some neighbourhood compactly lying in U. The limit of this sequence is the function

K—(z) = E Pm,0 (z).

m

In particular, a function K— G O(B). □

Theorem 4. For the analytic functional T G A'(S) the sequence [MkT]o converges weakly in A'(S) to the analytic functional [KT]o at k ^ to.

The proof repeats the proof of Theorem 2 with E(S) replaced by A(S) and E'(S) by A'(S). 3. Let function y £ Lp(S), p > 1. The harmonic continuation of the function y in a ball B is given by Poisson integral

Py(z) = | y(Z)P(Z,z) .

S

It is known that (ch. 7, §4 [11])

sup / |Py(rZ)|pda < to

0<r<1J S

and the function Py(rZ) -> y(Z) with respect to the norm of the space Lp(S) and almost

ri1

everywhere.

Let's consider the decomposition of the function Py into homogeneous harmonic polynomials of the form (2):

Py(Z)=£ (Z).

This series converges absolutely and uniformly inside the ball B. For the function — £ (S), with —\— = 1 the decomposition P— has the form

p q

P-(Z ) = £ Qm,i(Z).

Then the pairing of the functions y and — is given by the formula

(y,V0 = / y(Z)-(Z)dac = lim| Py(rZ)P-(rZ) dac =

S S

= r1lmi /X) Pm.l (rZ)Qm,l(rZ) daC = limX Pm.'(Z)Qm,i(C)

S

li^V(Pm,i ,Qm,i)r2(m+I). (9)

r — 1

Thus the sum of the series

m,

in the Abel-Poisson sense equals (y,

Let's denote by grp(B) the space of functions y which are harmonic in B and such that the condition

sup / |y(rZ< œ

<r<i 7

0<r<1

S

is satisfied. It is known that the normal boundary values of the function y £ gP(B) determine the function from Lp(S) [12]. Moreover, the Poisson integral determines an isometry of the spaces Lp(S) and gp(B).

Proposition 3. The operator M is a bounded linear operator from Lp(S) to Lp(S), its spectrum coincides with the interval [0,1] and for any y £ Lp(S), — £ (S) the following equality holds:

(My, -0) = (y, M-). (10)

Proof. It is known [13] (and also [3]) that M— G gP(B) ~ Lp(S) if — G LP(S). Moreover, this operator is bounded because it is the singular integral operator satisfying a cancelation condition (see Th. 3, ch. 2 [11]). These conditions are checked in [14]. Equality (10) evidently follows from equality (9).

Since the homogeneous polynomials Pm,i G gP(B) for all m,l and p > 1, it follows

/ n + m/ — 1 \

from equality (3) that Pm i are the eigenfunctions and the numbers (---) are the

\n + m + l — 1J

eigenvalues of the operator M. We show that the operator M in Lp(S) has no other eigenvalues. Let's assume that for some function — G gP(B)

M— = X—.

Using equality (10) we conclude that

(M—,Pm,i) = ( — , MPm,i),

hence,

X(—,Pm,i) = n + m j-T (—,Pm,i ).

n + m + l — 1

n + m/ — 1

If X = --- for all m,l then the scalar product (—,Pmi) = 0 for all m,l. Since the

n + m + l — 1

linear combinations of functions Pm, i are dense in Lp(S), it follows that — = 0. □

The authors were supported by RFBR, grants 08-01-00844 and 08-01-90250; by Rosobra-zovanie, grant №2.1.1/4620.

References

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