Some application of the Bochner-Martinelli integral Текст научной статьи по специальности «Математика»

УДК 517.55

Some Application of the Bochner-Martinelli Integral

Alexander M. Kytmanov*

Institute of Mathematics, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041, Russia

Received 10.08.2010, received in revised form 10.09.2010, accepted 20.10.2010

The Bochner-Martinelli formula gives the connection between complex and harmonic analysis in Cn. This becomes especially apparent in the solution of the д-Neumann problem: any function that is orthogonal to the holomorphic functions is the д-normal derivative of a harmonic function.

Keywords: Bochner-Martinelli formula, holomorphic function, holomorphic extension, functions with one-dimensional property of holomorphic continuation.

Introduction

The Bochner-Martinelli integral representation for holomorphic functions of several complex variables appeared in the works of Martinelli (1938) and Bochner (1943). It was the first essentially multidimensional representation in which the integration takes place over the whole boundary of the domain. This integral representation has a universal kernel (not depending on the form of the domain), like the Cauchy kernel in C1. In 1957 Lu Qikeng and Zhong Tongde considered the boundary values of the Bochner-Martinelli integral.

However, in Cn when n > 1, the Bochner-Martinelli kernel is harmonic, but not holomorphic. For a long time, this circumstance prevented the wide application of the Bochner-Martinelli integral in multidimensional complex analysis.

Interest in the Bochner-Martinelli representation grew in the 1970's in connection with the increased attention to integral methods in multidimensional complex analysis. Moreover, it turned out that the very general Cauchy-Fantappie representation found by Leray is easily obtained from the Bochner-Martinelli representation (Khenkin). Koppelman's representation for exterior differential forms, which has the Bochner-Martinelli representation as a special case, appeared at the same time.

The Cauchy-Fantappie and Koppelman representations found significant applications in multidimensional complex analysis: constructing good integral representations for holomorphic functions, an explicit solution of the d-equation and estimates of this solution, uniform approximation of holomorphic functions on compact sets, etc.

In sum, one may say that the Bochner-Martinelli formula gives the connection between complex and harmonic analysis in Cn. This becomes especially apparent in the solution of the d-Neumann problem: any function that is orthogonal to the holomorphic functions is the d-normal derivative of a harmonic function.

1. The Bochner-Martinelli Integral

We consider n-dimensional complex space Cn with variables z = (z1, ...,zn). If z and w are points in Cn, then we write (z,w) = z1w1 + ■ ■ ■ + znwn, and |z| = \J(z,z), where z =

* kytmanov@lan.krasu.ru © Siberian Federal University. All rights reserved

(z1,...,zn). The topology in Cn is given by the metric (z, w) ^ |z — w|. If z G Cn, then Re z = (Re zi,..., Re zn) G Rn, where we write Re zj = Xj, and Imz = (Imz1?..., Imzn) with Imzj = yj; that is, zj = Xj + iyj for j = 1,... ,n. Thus Cn ^ R2n. The orientation of Cn is determined by the coordinate order (xi,..., xn, yi,..., yn). Accordingly, the volume form dv is given by dv = dx1 A • • • A dxn A dy1 A • • • A dyn = dx A dy = (i/2)ndz A dz = (—i/2)ndz A dz.

As usual, a function f on an open set U c Cn belongs to the space Ck (U) if f is k times continuously differentiable in U. If M is a closed set in Cn, then f belongs to Ck(M) when f extends to some neighborhood U of M as a function of class Ck (U). We will also consider the space Cr (U) (or Cr(M)) when r > 0 is not necessarily an integer. A function f belongs to Cr(U) if it lies in the class CM(U) (where [r] is the integral part of r), and all its derivatives of order [r] satisfy a Holder condition on U with exponent r — [r].

The space O(U) consists of those functions f that are holomorphic on the open set U; when M is a closed set, O(M) consists of those functions f that are holomorphic in some neighborhood of M (a different neighborhood for each function). A function f belongs to A(U) if f is holomorphic in U and continuous on the closure U (that is, f G O(U) nC(U)).

A domain D in Cn has boundary of class Ck (we write dD G Ck) if D = {z : p(z) < 0}, where p is a real-valued function of class Ck on some neighborhood of the closure of D, and the differential dp = 0 on dD. If k =1, then we say that D is a domain with smooth boundary. We will call the function p a defining function for the domain D. The orientation of the boundary dD is induced by the orientation of D.

By a domain with piecewise-smooth boundary dD we will understand a smooth polyhedron, that is, a domain of the form D = {z : pj(z) < 0, j = 1,..., m}, where the real-valued functions Pj are class C1 in some neighborhood of the closure D, and for every set of distinct indices j,..., js we have dpj1 A • • • A dpjs =0 on the set {z : pj1 (z) = • • • = pjs (z) =0}. It is well known that Stokes's formula holds for such domains D and surfaces dD.

We denote the ball of radius e > 0 with center at the point z G Cn by B(z, e), and we denote its boundary by S(z,e) (that is, S(z, e) = dB(z, e)).

Consider the exterior differential form U(Z, z) of type (n, n — 1) given by

where dZ[k] = d^i A • • • A dZk_i A d^k+i A • • • A dZ„. The form U(Z, z) clearly has coefficients that are harmonic in Cn \ {z}, and it is closed with respect to C.

Let g(C, z) be the fundamental solution to the Laplace equation:

U K.zH^ ¿I-»'-1 f-zF «A

«K.z)

ln - z|2 for n = 1.

(n - 2)! 1

(2ni)n ^ |C - z|2n-2

for n > 1,

2

for n = 1 .

Then

U (C, z) = ]T(-1)k-i -fdCik] A dC = (-1)n-1dc g A £ dC[k] A dC [k],

where the operator d = ^ (dZk) k=1

We will write the Laplace operator A in the following form:

d _1 ( d . d \ d d _ d

If Zk _ Xk + iyk, then — _ - ---i — , and - - .

dQk 2 \dxk dyk) d-k dZk

Theorem 1 (Bochner [3]; Martinelli [12]). If D is a bounded domain in Cn with piecewise-smooth boundary, and f is a holomorphic function in D of class C(D). Then

I f(«^_ {r f I /D (1)

dD K

Formula (1) was obtained by Martinelli, and then by Bochner independently and by different methods. It is the first integral representation for holomorphic functions in Cn in which the integration is carried out over the whole boundary of the domain. This formula is by now classical and has found a place in many textbooks on multidimensional complex analysis.

Formula (1) reduces to Cauchy's formula when n _ 1, but in contrast to Cauchy's formula, the kernel in (1) is not holomorphic (in i and Z) when n > 1. By splitting the kernel U(Z,z) into real and imaginary parts, it is easy to show that

f f (Z)U(Z,i)

dD

is the sum of a double-layer potential and a tangential derivative of a single-layer potential; consequently, the Bochner-Martinelli integral inherits some of the properties of the Cauchy integral and some of the properties of the double-layer potential. It differs from the Cauchy integral in not being a holomorphic function, and it differs from the double-layer potential in having somewhat worse boundary behavior. At the same time, it establishes a connection between harmonic and holomorphic functions in Cn when n > 1.

Let D be a bounded domain with piecewise-smooth boundary, and let f be a function in C 1(D). Denote

Mf (i)_ j f (Z )U (Z,i), i /3D.

dD

We shall write M+f (z) for i / D and M-f (i) for i / DD.

Function Mf (z) is harmonic function for z / dD and Mf (z) ^ 0 as |z| ^ to.

Theorem 2. Under these conditions function M+f has a continuous extension on D, function M- f has a continuous extension on Cn \ D, and

M +f (z) - M-f (z)_ f (z), z / dD. (2)

Formula (2) is a simplest jump formula for the Bochner-Martinelli integral. There are exist many jump theorems for different classes of functions: for Holder functions [11], etc (see [6, Chapter 1]).

Suppose n > 1, and D _ {z : p(z) < 0} is a bounded domain in Cn with boundary of class C1,

___ n dp i

where p is defining function. If F / C 1(D), then denote dnF _ J2 §i~pk, where pk _ —— .

dnF is d-normal derivative of function F.

Now consider the homogeneous d-Neumann problem

(dnF _ 0 on dD,

\AF _0 in D. ( )

It is clear that holomorphic functions F satisfy (3). We show that the converse is also true.

Theorem 3 (Aronov, Kytmanov [2]). Let F be a harmonic function in D. The following conditions are equivalent:

1. dnF = 0 on 3D;

2. M +F = F in D; _

3. M-F = 0 in Cn \ D.

Theorem 4 (Folland, Kohn [5]; Aronov, Kytmanov [2]). Let F be a harmonic function in D of class CX(D). The following conditions are equivalent:

1. dnF = 0 on 3D;

2. M +F = F in D; _

3. M-F = 0 in Cn \ D;

4. F is holomorphic in D.

Theorem 5 (Kytmanov [6]). If M + f is holomorphic in D, f G Cx(dD), and dD G C1 is connected, then the boundary value of M + f coincides with f.

It is clear that Theorem 5 is not true when n =1. Also, it is not true if dD is not connected: it suffices to set f =1 on one connected component of dD and f = 0 on the remaining components.

Theorem 4 is proven for continuous functions F by Kytmanov and Aizenberg [7], for integrable functions F by Romanov [13].

2. Functions with the Property of One-Dimensional Holomorphic Continuation along Complex Lines

Consider complex lines lz,b of the form: lzb = {Z : Zk = zk + tbk, k = 1,..., n, t G C}. The point z G Cn and the point b G CPn-1 (b is defined to within of multiplication on a complex number A = 0).

We write the Bochner-Martinelli kernel U(Z, z) in variables t and b. We have |Z — z|2 = |t|2|b|2.

n

Then dZ = dZ1 A • • • A dZn = (b1dt + tdb1) A • • • A (bndt + tdbn) = tn-1 £ ( — 1)j-1bjdt A db[j],

j=i

since db = db1 A • • • A dbn = 0 in CPn-1.

nn

In exactly the same way ( — 1)k-1(Zfc — z,k)dZ[k] = tn-1 ( —1)j—1bjdb[j]. From here we

k=1 j=1

have

Lemma 1. The Bochner-Martinelli kernel in variables t and b has the form

U(Z, z) = dt A A(b),

where

, ^-1 £ ( —1)j-1bjdb[j] ^ ( —1)j-1bjdb[j]

A(b)=(n — 1)!( —1)n 1 j=1 j=1

2n

(2ni)n |b|

Let D be a bounded domain in Cn with smooth boundary. Give the following definition (Stout, 1977). The function f G C(dD) has a property of one-dimensional holomorphic continuation along complex lines if for any complex lines lZjb (meeting D) there exists a function Fz,b with the following properties:

a) Fz,b GC(D n lz,b);

b) Fz,5 = f on the set dD n lz,b;

c) Fz b is holomorphic in interior (with respect to topology of lz,b) points of the set D n lz,b.

Theorem 6 (Stout [14]). If dD G C2 and a function f G C(dD) has a property of one-dimensional holomorphic continuation along all complex lines then f has a holomorphic extension into D as a function of several complex variables.

Proof. Consider the integral

M-f (z) = j f (z)U(Z,z), z / D.

dD

The Fubini theorem, Lemma 1 and the conditions of the theorem imply

M-f(z)= j X(b) j f(z) J = 0.

t

8Dnlz,b

Applying Theorem 4 for continuous functions we have that M+f gives holomorphic extension of f into D. □

Problem 1 (Stout, 1991). Which families L of the complex lines are sufficient for holomorphic extension?

A family L is sufficient for holomorphic extension if any function f G C(dD) with a property of one-dimensional holomorphic continuation along complex lines lz,b G L has a holomorphic extension into D.

Consider the following family.

Let V be an open set in Cn. We denote

Lv = {lz,b : lz,b n V = 0}.

If dD is connected and V n D = 0 then LV is sufficient family since in this case M- f = 0 in V then M-f = 0 outside D (by uniqueness theorem for harmonic functions).

Theorem 7 (Agranovskii, Semenov [1]). Let dD be connected and an open set V C D. Then LV is sufficient family for a holomorphic extension.

Remind that the smooth (of class CTO) manifold r is generic, if for any point z G r the complex linear span of a tangent space Tz (r) coincides with Cn. Denote Lr the family of complex lines meeting r.

Theorem 8 (Kytmanov, Myslivets [9]). Let r be a germ of generic manifold lying in Cn \ D and let function f G C(dD) have a property of one-dimensional holomorphic continuation along almost all complex lines from Lr. Then there exists the function F G C(D) which is holomorphic in D and coinciding with function f on the boundary dD.

Proof. Consider the Bochner-Martinelli integral with function f:

F (z) = j f (Z )U (Z,z), z /dD,

dD

where U(Z,z) is the Bochner-Martinelli kernel.

Lemma 2. If for the point z G Cn \ D and for all complex lines meeting the point z function f have a property of one-dimensional holomorphic continuation, then F(z) = 0 and all derivatives of order a = (ai,. .., an)

daF dHlF

aza(z) = dza ■■■ dz^ (z) = 0, z G r,

where ||a|| = ai + ... + an.

Let us r be the germ of generic manifold in b Cn \ D, i.e. exists some open set W, in which r is smooth generic manifold of class CIf function f have a property of one-dimensional holomorphic continuation along complex lines from Lr, the by Lemma 2 the Bochner-Martinelli integral and all it derivatives in z vanish on r:

F |r =0

= 0 for all multiindeceis a. (4)

r

dza

The generic manifolds r by local byholomorphic map one can be transforms to the form

= h1(z1, . . . ,zk,U1, . . . ,Um),

r : < ............................................................................................(5)

> = hm(z1, . . . , zk ,U1, . . . , Mm),

where k + m = n, zj = Xj + ¿yj, j = 1,..., k, ws = us + ivs, s = 1,..., m. Moreover real-valued vector-function h = (h1,..., hm) belongs to the class Cin a neibourhood W of the point 0 and holds the conditions

hP<°> = °- Sp <"> = dhp <"> = dus <0> = 0- j-P =1--m. s = 1.....k-

Since under byholomorphic map the derivatives in holomorphic variables transform in derivatives in holomorphic variables the condition (4) one can rewrite in the form

F |r =0, °

= 0 for all multiindeces a, (6)

r

Lemma 3. If real analytic function F, given in neibourhood W of the set r satisfies the conditions (6), then it vanishs in W.

We show that Bochner-Martinelli integral vanishs in the neighbourhood W. Since it real analytic function and the complement Cn \ D is connected, then F(z) = 0 b Cn \ D. Applying Theorem 5 we get that function F is holomorphic in D and its boundary values coincides with f on dD. □

3. Singular Bochner-Martinelli Integral

We identify Cn with R2n under the complex structure zj = Xj + «xn+j, for j = 1,..., n. We will consider a smooth hypersurface S in Cn \ {0} with a singular point at the origin given by

S = {(^(r)x, r) G R2n : x G X, r G [0, R)}, (7)

where ^ G C 1[0, R) satisfies <^(0) = 0 and ^(r) > 0 for r G (0, R), and the point x = (x1,..., x2n-1) varies over a smooth compact hypersurface X in R2n-1 which does not meet 0.

For instance, X may be a (2n — 2) -dimensional sphere with centre at the origin. In any case we assume that X = {x G R2n-1 : p(x) = 1}, where p is a C1 function on R2n-1 \ {0} with real values, satisfying Vp = 0 on X and p(Ax) = Ahp(x) for all A > 0 with some h > 0.

The origin is a singular point of S, for <'(0) < ro. If <'(0) = 0 then 0 is a conical point of S. In the case <'(0) = 0 the point 0 is a cusp.

Using (7) it is easy to determine a defining function of the smooth part of S. Indeed, write p(x) = p(z', xn), where z' = (z1,..., zn-1). Then z G S \ {0} readily implies

z' Re zn

<(Im zn)' <(Im zn) - 37-

p

and so the homogeneity of p yields S = {z G Cn : Im zn G [0, R), g(z) = 0}, with g(z) = p(z', Re zn) - (f(lm zri))h.

Given an integrable function f with compact support on S, the singular Bochner-Martinelli integral of f is defined by

Ms f (z) = p.v.J f (Z )U (Ç,z)

for z G S, where

U K^^ ^A it

and d( = dZi A ... A dZn, while dZ[j] is the wedge product of all differentials dZi,..., dZn but dZj. In the sequel, we drop the designation "p.v." for short.

The properties of the Bochner-Martinelli singular integral operator on smooth hypersurfaces are well understood. We are aimed at investigating this operator on hypersurfaces with isolated singular points. Since MSf is smooth away from the support of f, one can certainly assume without loss of generality that S is of the form (7).

We first represent U(Z, z) in the local coordinates of S close to a singular point. Set

v (y)

m'(s)

for y G X, and v2n(y, s) = —h

|Vy p|'

|Vy p[

Lemma 4. The restriction of the Bochner-Martinelli kernel to the hypersurface S has the form

TT(r s 1 (y), v2n(y, s)), (cfi(s)y - <p(r)x, S - r)) \\2n-2 j A ( N

U(Z,z) =--¡Tn-< \ |2 , r--(f(s)) da{y) -

<?2n (w(s)y - p(r)xl2 + (s - r)2)n

1 (1VC(y,s), (V(s)y - f(r)x,s - r)) Ms))2n-2jsMy)t

&2n (\f(s)y — p(r)x\2 + (s — r)2)

where da is the area form on X induced by the Lebesgue measure in R2n, a2n the area of the (2n — 1) -dimensional sphere, and ivc = (—vn+\,..., —v2n, ... ,vn).

The vector ivc indicates to what extent the surface S fits to the complex structure of Cn. The only point remaining concerns the surface measure on S which is the subject of the lemma below.

Lemma 5. The surface measure on S induced by the Lebesgue measure in R2n is given by

■if'(s) \ 2

dS = (V(s))2n-^ 1 + h2 (jVpy)2ds da(y),

where da is the area form on X.

Lemma 4 applies in particular to conical hypersurfaces S c Rn, in which case p(r) = r and if'(r) = L

From now on, we restrict our discussion to hypersurfaces Sc Rn with conical points.

Let D be a bounded domain in Cn, with n > 1. The boundary of D is assumed to be of the form Y U (Si U ... U SN), where Y is a smooth hypersurface and each Sv is diffeomorphic to a conical hypersurface S as above. Thus, dD is a smooth hypersurface with a finite number

of conical points. Since the analysis at singular points is local, one can assume without loss of generality that N = 1, i.e., dD = YUS where

S = {z G Cn : z = (rx, r), x G X, r G [0, R)},

cf. (7).

For a function f G Ccomp(dD \ {0}) we define the norm

llfi|l2,y(dD) :=(/ |z|-2Y |f |2 dE) 1/2, (8)

av

where 7 G R. Denote by L2'7(dD) the completion of Ccomp(dD\ {0}) with respect to this norm.

It is clear that the weight factor |z| 2y affects the behaviour of functions in L2'7(dD) merely at the conical point 0.

According to dD = Y U S, the norm (8) can be splitted into two seminorms. The first of the two corresponds to integration over dD \ S and controls the behaviour of functions on the smooth part of the boundary. The second seminorm corresponds to integration over S and specifies the behaviour of functions close to the singular points. Under the parametrisation (7), the hypersurface S is identified with the cylinder X x [0, R). In this manner the second seminorm actually stems from the norm

R

-2(7-n+1/2) 1/1|2 ^ 1/2

i2,Y-n+i/2(Xx[0'R)) := I / r ||f (X)

r

on functions f G Ccomp(X x (0, R)), which is clear from Lemma 5.

Introduce a function k(x, y; t) defined for (x, y) G X x X and t > 0 by the equality

k(x, y; t) =

_ 1 ((v(y),V2n(y)), (y - tx, 1 - t)} i (ivc(y), (y - tx, 1 - t)}

^2„ (|y - tx|2 + (1 - t)2)n a2„ (|y - tx|2 + (1 - t)2)n'

Using this kernel, we can rewrite the singular Bochner-Martinelli integral in the form

Ms f (x,r)=y fc(x,y; S)f (y, s) da(y), (9)

X

where (x, r) and (y, s) are identified with z = (rx, r) and Z = (sy, s), respectively. Note that the integral over X is singular, for k(x, y; r/s) has a singularity at y = x provided s = r.

Theorem 9 (Kytmanov, Myslivets [8]). Integral (9) induces a bounded linear operator in L2'7(X x [0, R)) provided 1 - 2n < 7 < 0.

Let f be an arbitrary smooth function on dD which is supported in S and equal to 1 in a neighbourhood of the vertex. Pick a smooth function ^ with a larger compact support in S which is equal to 1 on the support of f. Such a function ^ is called covering for f.

Corollary 1. The singular Bochner-Martinelli integral fMs^ induces a bounded linear operator in L2'7(dD) provided |y| < n - 1/2.

Denote by the Mellin transform defined on functions f (r) on the semi-axis. It is given

by

f = J r^f (r) d^

0

for A G C.

Composing the singular Bochner-Martinelli operator (9) with the Mellin transform yields

c C

r ^

Mr^xMs f (x, r) = J r-lX diJdiJk(xy, S) f (y• s) da(y) =

~-tX ■ ' "ix

r s s

0 0 X

CC CO

= .f dsJUr-* k(x,y; l)v)f (y,s) da(y)

0 X 0

where (x, r) and (y, s) are identified with the points z = (rx, r) and Z = (sy, s) of S, respectively.

In the integral over r G (0, to) we change the variables by r = st, where t runs over (0, to). This gives

C C

-iX

Mr^xMsf (x,r) = J s-lX dS^Jt-lX k(x, y; t) j)f(y,s) da(y)

S s J KJ —t

0 X 0

= J Mt^xk(x,y; t) Ms^xf (y,s) da(y) X

for x G X and A G C. It follows that

Ms f (r) = M-lr a(A)Mr'^xf (r'), (10)

where f (r) := f (x, r) is thought of as a function of r G (0, to) with values in functions of x G X, and a(A) is a family of singular integral operators on X parametrised by A varying on a horizontal line in the complex plane. The action of a(A) is specified by

a(A)f (x) = J Mt^xk(x,y; t) f (y) da(y).

X

The family a(A) is usually referred to as the conormal symbol of the pseudodifferential operator (9) based on the Mellin transform. To evaluate it more explicitly, we denote by Z the unique root of (y — tx,y — tx) + (1 — t)2 =0 in the upper half-plane, i.e.,

1 + (x,y) + V\y — x\2 + \x\2\y\2 — (x,y)2 /,,\

Z = iT\x\2 . (11)

Lemma 6. In the strip 0 < Im A < 2n — 1, the Mellin transform of k(x,y; t) has the form

Mt^xk(x,y; t) =

( — 1)n-i expnA n- (2n — 2 — j)' A . . A . _

= —'—— -77-~.(iA + 1)(»A + 2)... (iA + j — 1) x

(n — 1)! smhnA j=0 j]-(n — 1 — j)!

((«A + j)A — iAZB)Z-lX-j-i + ( — 1)j-i((tA + j)A — iAZB)Z-lX-j-i

x

where

(1 + \x\2)n (Z — Z)2n-i-j

1 i A = -((v(y),v2n (y)), (y, 1))--(iVc(y), (y, 1)),

a2n a2n

1 i B = a_ ((v(y),V2n (y)), (x, 1))--(iVc(y), (x, 1)).

a2n a2n

We are now in a position to specify the inverse Mellin transform in the representation formula (10).

Theorem 10 (Kytmanov, Myslivets [8,10]). For |y| < n - 1/2 the singular Bochner-Martinelli integral admits the representation

Msf (r) = 2n / r,Aa(A)Mf'^f (r') dA. (12)

Im A=(n-1/2)-7

Proof. By Lemma 6, the conormal symbol a(A) of Ms is well defined and holomorphic in the strip 0 < Im A < 2n - 1. Hence, we may apply the inverse Mellin transform in this strip, as desired. □

Let |y| < n - 1/2. Pick a smooth positive function w on dD\ {0}, such that w = r-Y close to 0. The action of Ms in L2'7(dD) is unitary equivalent to the action of w-1Msw in L2(dD). This latter operator in turn differ from the action of Ms in L2(dD) by a compact operator. Hence, it suffices to study the algebra generated by the operator MS in L2(dD).

Denote by A = A(C(dD), MS) the C* -algebra generated by the Bochner-Martinelli integral MS, its adjoint operator MS, and by all multiplication operators a/ with a G C(dD).

Theorem 11 (Kytmanov, Myslivets [8]). If D is a domain with connected boundary, then the C* -algebra A is irreducible.

Proof. In order to prove the irreducibility of the algebra A it is sufficient to show that A has no nonzero invariant subspaces in L2(dD).

Consider the subalgebra C = {a/ : a G C(dD)} of A generated by all multiplication operators with a G C(dD). Our objective is to show that each invariant subspace of this subalgebra has the form L2 (dD) := {x2f : f G L2(dD)}, where x2 is the characteristic function of some measurable subset a c dD. Indeed, let E be an invariant subspace of the subalgebra C, and let P : L2(dD) ^ E be the orthogonal projection. □

Lemma 7. Suppose A is an arbitrary C* -algebra in the algebra L(H) of all continuous linear operators in a Hilbert space H. If E is an invariant subspace of A and P : H ^ E the orthogonal projection, then P commutes with all operators of the algebra A.

Theorem 12 (Kytmanov, Myslivets [8]). The algebra Kc A, where K is the ideal of compact operators in the Hilbert space L2 (dD).

Corollary 2. The Calkin algebra A = A/K of the algebra A is well defined.

In contrast to the case of smooth S, where the Calkin algebra is isomorphic to the algebra of continuous functions on the cosphere bundle S*S, the algebra A is not commutative under the presence of conical points. Hence the familiar theorem of Gelfand-Naimark is not applicable. The operator MS is no longer essentially selfadjoint if S has point singularities, unless n =1.

This work was supported by grants of AVTsP 2.1.1./4620 and NSh-7347.2010.1

References

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Некоторые приложения интеграла Бохнера-Мартинелли

Александр М. Кытманов

Формула Бохнера-Мартинелли дает связь между комплексным и гармоническим анализом в Cn. Это становится особенно ясным в решении д-задачи Неймана: любая функция, ортогональная голоморфным функциям, является д-нормальной производной гармонической функции.

Ключевые слова: формула Бохнера-Мартинелли, голоморфная функция, голоморфное продолжение, функции с одномерным свойством голоморфного продолжения.