Sharp theorems on traces in analytic spaces in tube domains over symmetric cones Текст научной статьи по специальности «Математика»

УДК 519.21

Sharp Theorems on Traces in Analytic Spaces in Tube Domains over Symmetric Cones

Romi F. Shamoyan*

Department of Mathematics, Bryansk State Technical University, Bryansk, 241050,

Russia

Elena Povprits^

Department of Mathematics, Bryansk State University, Novozibkov, 243020, Russia

Received 06.08.2013, received in revised form 06.09.2013, accepted 06.10.2013 New sharp estimates of Traces in Bergman-type spaces of analytic functions on tube domains over symmetric cones are obtained. These are first results of this type for tube domains over symmetric cones.

Keywords: trace estimates, tube domains, analytic functions, Bergman-type spaces.

1. Introduction and preliminaries

In this note we obtain new sharp estimates for Traces in Bergman type spaces of analytic spaces in tube domains over symmetric cones. This line of investigation can be considered as a continuation of our previous papers on Traces in analytic function spaces [1,3] and [2,4-6] where similar results were obtained but only in bounded domains in higher dimension. We remark that in this note for the first time in literature we consider this known problem related with Trace estimates in spaces of analytic functions in unbounded domains in Cn, namely in tube domains over symmetric cones. The first section contains required preliminaries on analysis on symmetric cones. Our new sharp results are contained in the second section of this note. The Whitney type decomposition of tubular domain based on properties of so-called Bergman balls (see [7-9]) serves as important tool for almost all our proofs as in previous papers on this problem in other domains [2-4]. In one dimensional tubular domain which is upperhalfspace C+ (see [7]) our theorems are not new and they were obtained recently in [6]. Moreover arguments in proofs are parallel to those we have in one dimension or polyball in Cn The base of proof is again the so-called Bergman reproducing formula, but in tubular domain over symmetric cone(see, for example, [7-9] for this integral representation). We shortly remind the history of Diagonal map (or Trace) problem. After the appearance of [10] various papers appeared where arguments which can be seen in [10] were extended, changed and modified in various directions in one and higher dimension (see, for example, [11-15] and [1,3,4] and also various references there).

In particular in mentioned papers various new sharp results on traces for analytic function spaces in higher dimension (unit polyballball ) were obtained. New results for large scales of analytic Qp type spaces in polyball were proved(see [6]).

* [email protected]

[email protected] © Siberian Federal University. All rights reserved

Later several new sharp results for harmonic functions of several variables in the unit ball and upperhalfplane of Euclidean space were also obtained (see, for example, [1] and references there).

Probably for the first time in literature these type problems connected with diagonal map in analytic spaces appeared before in [10].

In this book this problem was formulated and certain cases connected with spaces of analytic functions in the unit disk were considered.

These results and various other results were mentioned and proved much later in [14]. Some interesting applications of diagonal map can be seen in [14,16] where other problems around this topic can be found also. The goal of this note to develop further some ideas from our recent mentioned papers and present a new sharp theorems in tube domain over symmetric cones. In upper halfplane of complex plane C+ which is a tube domain in one dimension such results already were obtained previously by author [2], so this problem in tube appears naturally. For formulation of our results we will need various standard definitions from the theory of tube domains over symmetric cones( see [7,17-19]). In this section we also mention some vital facts which will be heavily used in proofs of our assertions (see, for example, also for parallel assertion in other domains [2-5]).

Let Tq = V + ii be the tube domain over an irreducible symmetric cone l in the complex-ification VC of an n-dimensional Euclidean space V. Following the notation of [17] and [7] we denote the rank of the cone l by r and by A the determinant function on V. Letting V = Rl, we have as an example of a symmetric cone on Rl the forward light cone An defined for n ^ 3 by

An = {y e Rl : y?-----vl > 0, y\ > 0}.

Light cones have rank 2. The determinant function in this case is given by the Lorentz form

A(y) = y?-----vl.

(see, for example , [7]).

Let us introduce some convinient notations regarding multi-indicses.

If t = (t1,..., tm), then t* = (tm,... ,t\) and, for a e R, t + a = (t1 + a,... ,tm + a). Also, if t, k e Rm, then t < k means tj < kj for all 1 ^ j ^ m.

We are going to use the following multi-index

go =((j - 1)tH , where (r - 1)d = n - 1.

\ 2 l^j^r 2 r

H(Tq) denotes the space of all holomorphic functions on Tq. We denote m cartesian products of tubes by Tm, Tm = Tq x ... x Tq the space of all analytic function on this new product domain which are analytic by each variable separately will be denoted byH(Tm). In this paper we will be interested on properties of certain analytic subspaces of H(T.m). By m we denote below a natural number bigger than 1. For t e R+ and the associated determinant function A(x) we set

(Tq )=( F e H(Tq ): ||F |U«» = sup |F (x + iy)|AT (y) < <*} , (1)

I x+iyeTn J

( [7] and references there). It can be checked that this is a Banach space.

n

For 1 ^ p, q < and v e R, and v >--1 we denote by Ap,q(Tq) the mixed-norm

r

weighted Bergman space consisting of analytic functions f in Tq that

l|F||a- = (7 (jV IF(x + iy)lPdx)q/P Av(y)^^) <

This is a Banach space. Replacing above simply A by L we will get as usual the corresponding

larger space of all measurable functions in tube over symmetric cone with the same quazinorm

n

(see [17,19]). It is known the Ap,q(Tq) space is nontrivial if and only if v >--1,(see [7,18]).

r

And we will assume this everywhere below. When p = q we write (see [7])

Apq (Tq ) = Ap (Tq ).

This is the classical weighted Bergman space with usual modification when p = to.

Let Tg = Tq x ... x Tq. To define related two Bergman-type spaces Ap(Tg) and A^°(Tg) (v and t can be also vectors) in m-products of tube domains Tg we follow standard procedure which is well-known in case of unit disk and unit ball [2-4,10,14]. Namely we consider analytic F functions F = F(z1,... zm) which are analytic by each zj,j = 1,..., m variable, and where each such variable belongs to Tq tube and define as H(Tg) the space of all such functions. For example we set, for all Zj = Xj + yj, Tj G R, j = 1,..., m, F(z) = F(zi,..., zm), t = (t1, ..., Tm)

A~(Tm) = j F G H(Tq) : ||F = sup |F (x + iy)|AT (y) < to! , (2)

[ x+iyerm J

|F(x + iy)| = |F(xi + iyi, ...,xm + iym)|, where AT (y) is a product of m-onedimensional ATj (yj) functions, j = 1,... ,m. Similarly the

n

Bergman space Ap can be defined on products of tubes for all t = (t1, ..., Tm), Tj >--1, j =

Tr 1, ... , m. It can be shown that all spaces are Banach spaces. Replacing above simply A by L we

will get as usual the corresponding larger space of all measurable functions in products of tubes

over symmetric cone with the same quazinorm.

The (weighted) Bergman projection Pv is the orthogonal projection from the Hilbert space

L"l (Tq) onto its closed subspace A^(Tq) and it is given by the following integral formula (see [7])

Pvf(z) = Cv f Bv(z,w)f (w)dVv(w), (3)

JTn

where

Bv(z,w) = CvA-(v+ n)((z - w)/i) is the Bergman reproducing kernel for

A2 (Tq )

(see [7,17]).

Here we used the notation

dVv (w) = Av-r (v)dudv.

Below and here we use constantly the following notations w = u + iv G Tq and also z = x + iy G Tq .

Hence for any analytic function from A^(Tq) the following reproducing integral formula is valid(see also [7])

f(z) = Cv f Bv(z,w)f(w)dVv(w). (4)

JTn

We will use even more general version of this assertion for AP'q. We denote everywhere below by Cp the Bergman representation constant.

Reproducing formulas are bases of all our proofs as in simpler cases of unit disk and unit polyball. In this case we say simply that the f function allows the Bergman representation via Bergman kernel with v index.

Note these assertions have direct copies in simpler cases of analytic function spaces in unit disk, polydisk, unit ball, upperhalfspace C+ and in spaces of harmonic functions in the unit ball or upperhalfspace of the Euclidean space Rn. These classical facts are well-known and can be found, for example, in [14] and in some items from references there.

Above and throughout the paper we write C (sometimes with indexes) to denote positive constants which might be different each time we see them (and even in a chain of inequalities), but is independent of the functions or variables being discussed.

In this paper we will also need a pointwise estimate for the Bergman projection of functions in Lp'q (Th), defined by integral formula ( [7]), when this projection makes sense. Note such estimates in simpler cases of unit disk, unit ball and polydisk are well-known (see [13]).

Let us first recall the following known basic integrability properties for the determinant function, which appeared already above in definitions. Below we denote by As the generalized power function [7,17].

Lemma 1. 1) The integral

' x + iy

My) = /

J Rn

A-

dx

n

converges if and only if a > 2--1. In that case

r

Ja(y) = CaA-a+n/r (y),

a e R, y e Q.

2)Let a e Cr and y e Q. For any multi-indices s and ft and t e Q the function

y ^ Ap (y + t)As(y)

belongs to L1(Q, Andy(y)) if and only if > g0 and K(s + ft) < -g0j. In that case we have

h Ap (y + t)As(y)Jy(y) = Cj.sAs+a (t).

We refer to Corollary 2.18 and Corollary 2.19 of [18] for the proof of the above lemma or [7]. As a corollary of one dimensional version of second estimate and first estimate (see, for example, [19] Theorem 3.9) we obtain the following vital Forelly-Rudin type estimate for Bergman kernel (A) which we will use in proof of our main result [19]

Lemma 2.

f Aß(y)|Ba+ß+n(z,w)ldV(z) < CA-», (5)

JTn

n

ß > -1, a >--1, z = x + iy, w = u + iv (see [19]).

r

We will also need for our proofs the following important fact on integral representations (see [9]).

nn

Lemma 3. Let v >--1, a >--1, then for all functions from the integral representations

of Bergman with Bergman kernel Ba+V(z,w) (with a + v index) is valid.

The following result can be found in [19](section4).

n 11

Lemma 4. For all 1 < p < <x and 1 < q < ro and for all — ^ p\, where--1— = 1 and

r pi p

nn

--1 < v and for all functions f from Af,'q and for all--1 < a the Bergman representation

formula with a index or with the Bergman kernel Ba(z,w) is valid.

We remark this result is a particular case of a more general assertion for analytic mixed norm Ap'q classes (see [19]), which means that our main result below partially admits also some extentions, even to mixed norm spaces which we defined above. We note also that (see [19])

If (x + iy)|Arp+q(y) < cp,q,r,v 11/1|a™, 1 < P,q< - 1. (6)

All the mentioned results together with properties of the Whitney decomposition of tubular domain over symmetric cones based on Bergman balls [7,8] are used heavily during all proofs of our assertions.

Lemma 5 ( [7-9]). Given 6 £ (0; 1) there exists a sequence of points {zj} in Tq called 6-lattice such that calling {Bj} and {Bj} the Bergman balls with center Zj and radius 6 and 6/2 respectively then

A) the balls {Bj} are pairwise disjoint;

B) the balls {Bj} cover Tq with finite overlapping;

C) f As(y)dV(z) x f As(y)dV(z) = CsA2n +s(ImZj);

Jb, (z, ,S) JBj (z, ,S)

s >--1, J = IBg (Zj )| x A^r (Im Zj ),j = 1,...,m,J x A ^ (Im w), w £ Bs (zj).

2n

Lemma 6 ( [7-9]). For any / £ A2(Tq) we have for any ^ >--1

r

/(z)= i B^(z, w)f (w)AM-21 (Imw)dV(w).

JTc,

Lemma 7 (Besov space, [7-9]). Let Dz be the natural extension to the complex plane space Cn of the generalized wave operator Ox on the cone Q Oz = [A( j-jZ)] which is the differential operator of degree r. We define for 1 < p < x as Bp(TQ) the Besov space the sincee of all holomorphic functions / so that

i Ianf (x + iey)IPAnp-^ (y)dxdy < x. JTn n

n J To n

BPp = Ap, v > - 1, 1 < P<pv, Ap c Bp, v > - 1,

„ _ v + 2n - 2 „ v + 2n - 1 1 < P<Pu, P =-n - 1 , P =-n-1-.

r r

n

Lemma 8 ( [7-9]). a) Let v >--1, 1 < p < x. For all F £ Ap we have

□lF(z) = C t Bv+l(z,w)DmF(w)Am(Imw)dVv(w) To

To

m ^ 0, l is large enough;

For l = 0 this is valid, 1 ^ P < R, for any R > 1.

b) Let v> f - 1,a> f - 1. Then for all Aa(Im z)F(z) £ L™ and all m > 0:

F(z) = C t Bv+a(z,w)DmF(w)Am+a(Imw)dVv(w). To

2. On sharp estimates for traces in analytic function spaces in tube domains over symmetric cones

In this paper we restrict ourselves to Q irreducible symmetric cone in the Euclidean vector space Rn of dimension n, endowed with an inner product for which the cone Q is self dual. We denote by Tq = Rn + iQ the corresponding tube domain in Cn.

This section is devoted to formulations and proofs of all main results of this paper. As previously in case of analytic functions in unit disk, polydisk, unit ball and upperhalfspace C+ and in case of spaces of harmonic functions in Euclidean space [1,3,4] the role of the Bergman representation formula is crucial in these issues and our proofs are heavily based on it and some lemmas we provided above and they are parallel to cases we considered before [2-4].

As it is known a variant of Bergman representation formula is available also in Bergmantype analytic function spaces in tubular domains over symmetric connes and this known fact (see [7,17-19]), which is crucial also in various other problems in analytic function spaces in tubular domains (see [7] and various references there) is used also in our proofs below.

Theorem 1. Let f G Ap(Tq), 1 < p < v G Rn, Vj > v0 for fixed v0 = v0(p, n, r, m), for all

m n n

j = 1,... ,m. Then f (z,..., z) G Aps, where s = (vj--) + 2 —(m — 1) with related estimates

j=i r r

for norms. And for all n ^ pi, where pi is a conjugate of p the reverse is also true. For each g function g G Ap(Tq) there is an F function, F (z,..., z) = g(z), F G Ap (Tq). Let in addition

m

fe f) (zi, ...,Zm)= Cp f (w) H - W)/i)dVp (w),

J To _1

^ i J (w^i^"t((zj

n j=i n

mt = fi +--, zj G Tq,j = 1,. .. ,m.

r

Then the following asertions hold for all fi so that fi > fi0 for some fixed large enough positive

number p0. The Tp Bergman-type integral operator (expanded Bergman projection) maps Aps (Tq) to Ap(Tm), v = (vi,.. .,Vm),Vj > vo,j = 1,..., m.

Proof. We have using A), B), C) of lemma 5 the following chain of estimates for family Eg (zj) of Bergman balls, zj = Xj + iyj

I = \F(z,..., z)\pAj=1 To

(y)dydx = (Bj = Bs(zj)) = Bj(zj, 5)

= \F(z,..,z)\pA>=1

j=iJ Bj By lemma 5

E(Vj - n ) + 2 n (m-i) pAj=1 (y)dydx < cV sup \F(z,..,z)

j=iV

AT (y)dydx

I = \F(z,..., z)\pAi=1

(y)dydx = (Bj = Bs (zj )) = Bj (zj, 5) =

To

= £/ \F(z,...,z)\pAj=1

j=i] B

By lemma 5

(y)dydx ^ C^^ ( sup \F(z,..., z

j=i

zeBj

AT (y)dydx

'Bj

OO OO

<

< CiY, '"Y sup \F(zi,..,zm)\p |AV1 + r (Imz i)J... |AVm+r (Imz m)

jl =i jm=i ZiEBj1

zm ^ Bjm O TO „ „

< ^Y ... \F(zi,..,zm )\pAV1-n (Im zi)....AVm-n (Im zm )x

j1=i jm=i (Bji )' (Bjm, )'

p p m

xdxi...dxmdyi...dym = ... \F(zi,..., zm)\p Y]_ AVj-r (Imzj)dzi...dzm;

■J To J To j = i

s

B

p

We used the fact that for 1 < p < Bj = Bg(zj) = Bj(zj, S), j = 1,..., m, S > 0

C f

|F(zj)|p < C

f |F (w)|pdudv

Ib> ) A2n (v)

|Bg (

j B^ (zj )

|F(w)|pdudv, Zj G Tq;

(7)

(see [7-9]) m-times by each variable zi,..., zm and item (7) of lemma 5 at last step. It remains to show the second part of this theorem. Let

ï> [f (zi,...,zm)j = CJ f (w^A

' Tn

>—ti j

j=1

dVp(w), mt = (fi +—) ; zj G Tq, j = 1,..., m;

/ \ n n

£ is large enough. Then Tpf (z,...,z) = f (z); z G Tn, f G A?; - < pi; s >--1, 1 < p < +<»

V / r r

by lemma 4. We show now that Tp acts from (A?) to (Ap), if £ is large enough.

This will finish the proof of our theorem.

For this we will use lemma 2.

We have the following chain of estimates for Tp integral operator (expanded Bergman projector).

First we have using Holder's inequality twice and lemma 2 y1 + y2 = t; —I--= 1; £ >

p Pi

— - 1; Y2 > (2 — - 1 + fi)

r V r /

1

pi m

; Y2 <

min Vj - — + (fi + —) ( — ) + 1

r pi m

'Tn

If (w)^|A-

j=i

= C

|f (w^dVp (w)

|dVg(w) ) < C(I • J) dVa (w)

p p 1

n |AY1 P j=i

n |AY2P1 j=i

, zj G Tq, j = 1,..., m.

p

|

|

Zo —w

z o — w

p

J < C m/ -dViH_) ^ ^(H A—T (Im zj ) ), zj G Tq, j = 1,..., m;

n

Pi \ n £ +2 — 1 n

for m ( — t = (Y2p1m) — £--; y2 > -r-; £ >--1 and hence we have now using

ypj r pim r

this the following estimate

f f m

Jt ... Jt |(Tp f)(zi,...,zm)|^AV1 — r (Im zi)) ...(AVm — r (Im zm)) jQ dxj dyj < C\\f \\*AV

Zj = Xj + ¿yj, j = 1, ..., m.

The last estimate follows again directly from inequality of lemma 2 and Fubini's theorem and some calculations based on estimate

AV1 — n T (Im z1 )

A

Vm — ^—T

(Im zm) n dxj dyj j=1

'Tn JTn

n |A™ |

j=1

< C3

Av (Im w)

, w G Tq;

n m n n n n

where Vj---t > -1; j = 1,..., m; v = £ (vj--) + 2—(m — 1) +---ft; —v >--1 and ft

r j = i r r r r

is large enough.

The proof of theorem 1 is now complete. □

A complete analogue of this theorem is true also for p = to case. This second theorem follows directly from Bergman reproducing formula provided above for Af and some elementary

estimates like Holder inequality for m functions and estimate (5).

n

Theorem 2. Let f G Af (T^), v G Rn, vj > - — 1, for all j = 1,..., m. Then f (z,..., z) G Af,

m

where s = Y1 vj. And the reverse is also true — for each g function g G Af(TQ) there is an F

j=i

function, F(z,..., z) = g(z), F G Af°(Tm). Let in addition

/m

f (w^A-i((zj — w)/i)dVp (w), -h j=i

n

mt = ft +—, zj G Tq, j = 1,..., m. Then the following assertions hold for all ft, ft > ft0 for some r

fixed large enough positive number ft0.

The Tp Bergman-type integral operator (expanded Bergman projection) maps Af(TQ) to

nm

Af(Tm), v =(vi,...,vm), vj >r — 1, j = 1, ..., m, s = £ vj.

r j=i

Proof. Note using the obvious property of AT (y) function we have one part of theorem since we have obviously that

sup |f (z,..., z)|[AT(y)] < sup ... sup |f(zi,...,zm)|[AT1 (Imzi)]...x

zeTn zieTh zmeTn

n

x[ATm (Im zm)]; Ti + ... + Tm = t > 0, Tj >r — 1,j = 1,..., m, z = x + iy.

Let us show the reverse implication for this theorem. For this we have to use heavily two lemmas which we formulated above, namely lemma 3 and lemma 2.

m

First we have that if f G Af (Tq); s = £ vj and if

j=i

Tpf) (zi,...,zm) = cJ f (w^A

'Th

j=i

dVp(w), mt = ft +—; ft > fto, zj G Tq

then (Tp/) (z,..., z) = f (z), z G Tq by lemma 3 for all ft, ft > fto, zj G Tq, j = 1,..., Then we have that by Holder's inequality for m functions and (5), ft1 = ft — s

|( Tp f) (zi,...,zm)|[Avi (Im zi )...Avm (Im zm)] < Cp ||f ||A«» x

HA"

/Th j=i

dVpi (w)

I[Avj (Im zj)

j=i

^ Cp

A

'Thj=i

—t

dVpi (w)

I]Avj (Im zj)

j=i

z„- — w

tj

m.

zo — w

tj

x

< jUV JTn Ap+n (j™) )

]jAVj (Im zj) Lj=i

< ro.

n

for all 3 > 30 and Vj >--1,w = u + iv,j = 1,..., m.

r

Hence we have

m

l( Tp f) (zi,...,zm )in AVj (Im Zj) < Cp\\f \\A~ ,Zj G TQ,j = 1,..,m.

j=i

The proof of theorem 2 is complete. □

Let

LP (TQ) = {f — loc. integrable in Tq :

\\f\\l? = f if (z)lP fl ATj-n/r(Vj)dxjdyj < ro}. h j=i

To define the next space of functions we remind the reader that the family of Bergman balls Bg (z) forms an r-lattice in tubular domain Tq ( [7-9]). We denote by Bgm(Z) standard m-cartesian product of such Bergman balls in Cm, Z = (z1,..., zm). By (Tq) we denote all f analytic functions in T m, so that

m

/ if (zi,...,Zm)|PTT ASj (Vj )dxj dy3

JbT(Z) j=i

belongs to Liri. Tm (TQ), where sj = Vj--for all j = 1,... ,m, and where 1 < p < ro,

n n r

Vj >--1, Tj >--1, for all j = 1,... ,m. Note in polyball complete analogues of these classes

were considered in [3] and the complete description of Traces of these spaces were also given. We obtain below a complete analogue of that result in case of tubular domain over symmetric cone.

Theorem 3. Let Vj > v0 and Tj > t0 for some fixed positive numbers v0 = v0(p,n,r,m) and t0 = t0(p,n,r,m), 1 < p < ro. Let f G Kp T(TQ) then f(z,...,z) belongs to AsS(Tq), mn

s = Y1 (Vj + Tj) + 2( — )(2m — 1) and for every f function f G Ap there is an F function j=i r

n 11

F G Kp , so that F (z,...,z) = f (z) for all — < pi, —\--= 1. Let in addition

' r p pi

/m

f (w^A-i((zj — w)/i)dVp (w), -h j=i

n

mt = 3 \—, Zj G Tq , j = 1,... ,m. Then the following assertions holds for all 3, so that 3 > 30 r

for some fixed large enough positive number 30. The Tp Bergman type integral operator (expanded Bergman projection) maps AsS(Tq) to K% T(Tq), v = (vi, ..., Vm), t = (Ti:..., Tn), Vj > v0, Tj > To, j = 1, ... , m.

Proof. Using arguments from the proof of the theorem 1 and applying lemma 5 and (7) we have the following chain of estimates

P oo

/ |F(z,..., z)lPAri (y)dydx = IF(z,..., z)lpAri (y)dydx <

J To. j=i JBj

< C^i siBp |F (z,...,z)| p)(j Ari (y)dydx J <

j=i

oo

< sup |F(zi,...,zm )|pAr2 (Im zi)...Ar^ (Im zm) <

jl = l jm = 1 ZieB31

zm G B j

oo oo „ „

^ Ci ... J . J IF(zi,...,zm)IPAr3 (Imzi)...Ar™ (Imzm)dxi...dxmdyi...dym <

jl=1 jm=iBji (z) BJm (z)

m m \

< c/ . . I ... |F(wi,...,Wm)|^ AVj (Imwjm duidvA x

J To J To \j B(zi) JB(zm) = = J

UATj (zj)d'Xjdy

Lj=i

, Wj = Uj + ivj, j = 1,..., m

where ri = = (vj + Tj) +--(2m — 1), r2 = Vj + Tj +--, rJ3 = Vj + Tj +--, j = 1,..., m.

To get the reverse we again have to modify a little the proof of theorem 1. Namely we use the following additional estimate to get the result (see [19,20])

f dVp(w) ^ C(ImWj)p+n R>n ( e

1 —--<-/ /—x ; ft>— 1,t > o,zj e ih,Wj e in. (8)

j aM j-^) A*

Bj (wj)

We omit easy technical details. We have for x = [Y2pim — ft — n ) ( )

V r J \pim J

m

If (w)IPdVp (w) n A-x Im zj |Tpf(zi,...,zm)|p < C\ -m-^-,ft>- — 1,

JTo n |AY1P( zj-w )|

j=i 1

n ^ i 1\ ft + — 1

Yi + Y2 = - + ft — , Y2 >

\r ) \mJ pim

ft is large enough, ft > ft0.

Using (8) and then (5) m times we have after some calculations finally the estimate

\\Tpf Ikv < c\\f ||Ap.

The proof of theorem 3 is complete. □

As we see proofs are based heavily on properties of r-lattices based on Bergman balls and related estimates(see [7,8]). Complete analogues of all our assertions in disk, polyball can be found in [2-4,14]. Moreover similar arguments we used in our proofs in polydisk and polyball can be seen in [2,14].

x

Let

A-

{f e H(Tm)

\f (zi,

\P1

AVl-n/r (yi)

dxidyi

Av

1-n/r

1

Pm

(ym) dxmdym

< to}, 1 ^ pj < >--1,j = 1,...,m.

Some assertions of this note can be extended even to AV,q spaces in tubular domain Tq (see for definition the previous section) and to mixed norm spaces p = (pi,... ,pm), v = (vi,..., vm) in products of tubular domains. For complete analogues of these mixed norm spaces in polyball we refer the reader to [6]. These results hovewer are not sharp. Note tubular domains are the most typical examples of general unbounded Siegel domains of second type.For the most typical examples of general bounded Siegel domains of second type the so-called bounded pseudoconvex domains with smooth boundary the complete analogues of all our results are also valid (see [21]).

Some results of this paper based on lemma 7 and 8, can be also easily extended to analitiec Besov Bp spaces on product domains (see [9]).

The second author of this work is particularly supported by the Russian Foundation for Basic Research, grant 13-01-97508.

p1

z

m

References

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Точные теоремы о следах в аналитических пространствах в трубчатых областях над симметрическими конусами

Роми Ф. Шамоян Елена Повприц

В статье предъявлены первые точные результаты о следах пространств Бергмана и типа Бергмана аналитических функций в трубчатых областях над симметрическими конусами.

Ключевые слова: аналитическая функция, трубчатая область, конус.