CC BY
7
Вестник КРАУНЦ. Физ.-мат. науки. 2019. Т. 26. № 1. C. 28-45. ISSN 2079-6641
DOI: 10.26117/2079-6641-2019-26-1-28-45 MSC 32A07, 432A10, 32A07
ON DECOMPOSITION THEOREMS OF MULTIFUNCTIONAL BERGMAN TYPE SPACES IN
SOME DOMAINS IN Cn
R. F. Shamoyan
Department of Mathematics, Bryansk State Technical University, Bryansk 241050,
Russia E-mail: [email protected]
We present some extensions of well-known one functional results on atomic decompositions in classical Bergman spaces obtained earlier by various authors in some new multifunctional Bergman type spaces in various domains in higher dimension.
Keywords: Bergman type spaces, analytic functions, decomposition theorems.
© Shamoyan R. F., 2019
УДК 517.55+517.33
О ТЕОРЕМАХ ДЕКОМПОЗИЦИИ В МУЛЬТИФУНКЦИОНАЛЬНЫХ ПРОСТРАНСТВАХ
БЕРГМАНА
Р. Ф. Шамоян
Брянский государственный технический университет, 241050, г. Брянск, Россия E-mail: [email protected]
Мы приводим некоторые новые теоремы о декомпозиции аналитических функций из мультифункциональных пространств Бергмана, обобщающие ранее известные результаты подобного типа для обычных пространств Бергмана в различных областях.
Ключевые слова: аналитические функции, пространства Бергмана мультифунк-циональные пространства, теоремы декомпозиции.
© Шамоян Р.Ф., 2019
1. Introduction, preliminaries and main results
The intention of this paper to provide complete analogues of our recent results (see [4],[5])on atomic decompositions in new multifunctional Bergman spaces in the unit ball and bounded pseudoconvex domains in some new multifunctional Bergman spaces in tubular domains over symmetric cones and in some related domains.
The problem of atomic decompositions of Bergman and other spaces in one functional case was considered in various domains in one and higher dimension by various authors (see for example [1]-[3],[14-16],[7],[10],[12] and various references there).It is well-known that these theorems have numerous applications in various problems of complex function theory in one and higher dimension. Note that our results in particular are heavily based on certain new theorems from [7] and [6] on onefunctional Bergman spaces in tubular domains over symmetric cones and in Siegel domains of second type(direct generalizations of bounded symmetric domains). As in mentioned cases of the unit ball and bounded strictly pseudoconvex domains (see[4] and [5]) adding a simple new integral condition (which vanish in case of one functional space) we obtain a new atomic decomposition theorem for analytic multifunctional Bergman spaces in these domains. These assertions can be considered at the same time as direct extensions of previously known results concerning one functional Bergman spaces. To formulate our theorems we need to introduce a group of definitions and notations taken from [7], [6]. Further we also note our results partially are also valid in so-called bounded minimal homogeneous domains in Cn, based on recent results of S.Yamaji (see [8-9] and various references there).These can be done using same methods of proof which we present below.The only tool of our rather transparent proof is so-called Forelly-Rudin type estimate which is available in all these domains(see [7],[8],[6],[4],[5]) and a well-known uniform estimate from below of a norm of Bergman space which is also available in various domains in higher dimension. Note more precisely one part of all our assertions below in various domains and spaces on them is a direct simple corollary of an argument related with ordinary induction and an uniform estimate we just mentioned.The other part uses only Holder"s inequality applied twice and the standard Forelly-Rudin estimate. This can be seen after simple very careful analysis of the proof of the unit ball and tube cases below. First we provide some known assertions on atomic decomposition for one functional Bergman spaces , then we provide direct multifunctional generalizations of these assertions. First we discuss the simpler case of the unit ball then pass same arguments to other domains. We consider Bergman spaces on bounded symmetric domains, tubular domains over symmetric cones and then even more general Siegel domains of second type in Cn and new multifunctional Bergman spaces on them. Since proofs of all assertions in various domains are very similar we will omit some details below leaving them to interested reader. Let Bn or B be the unit ball in Cn , let further H (Bn) be the space of all analytic functions in Bn(or B).
We define the Bergman class in the unit ball in a usual way.
(Apa) (Bn) = {f e H (Bn) : ^ | f (z) |p (1 - |z|)a dv (z) < ,
0 < p < <*>, a > -1, where dv is a normalized Lebeques measure on Bn (we will also use this notation for Lebegues measure in other domains). We formulate a well-known theorem on atomic decomposition of usual Bergman spaces Apa (Bn).
We denote positive constants in this paper as usual by c,c1,c2,... or by ca,cp.
Theorem A. (see, for example, [1,3,14])
Let a > — 1, f e H (Bn)o 0 < p < <*>, let, b > b0, b0 = b0 (p,n, a). Let f e Apa (Bn). Then there exist a sequence {a/} in Bn, such that
—n—1—p
1 — I a/~"
f (z) = I j ' -,z e Bn (si)
j=i l1 — (z, aj))
Where the series converges in the norm topology of (Ap) (Bn) and I|cj1p < and where a > p0, p0 = ao (p, n, m) and the reverse is also true if (s1) holds then f e Ap (Bn) for same values of parameters.
Let To be a tubular domain over symmetric cone, let {z/}be a 8-lattice in To
(see [7], [11]), 8 e (0,1), {z/} e To, H(To) - be the space of all analytic functions on To.
Let further
Ap (Tn) = {f e H(To) :J |f (z)|pAa—n (Imz)dv(z) <
, i f a—r 'To
where a > n — 1,1 < p < ^ be the Bergman space in To, (AJ) be determinant function in To.(or we use below sometimes T\ )(see [7,11]).
We formulate a known theorem on atomic decomposition of (Ap) (To) spaces in To (see [7,11]).( well-known one functional result which has many applications) Theorem B. (see, for example, [7,11])
Let {z/} be a 8-lattice in To , 8 e (0,1), z/ = Xj + iy/, z/ e To, j > 0. Then
IIf Ik -1 |f (zj)|p Av+n (y/). j
// f G Ap, ¿¿en
//
£|Aj|p Av+n (y,) < c ||f Q . (S2)
j=i v
£|Aj|pAv+n (yj) < j=i
then the reverse to (s2) holds, if f e (Ap) then
f (z) = I (A) (Bv (z,z/))(Av+n (yj)
where Bv is Bergman kernel ([7,11]).
In [1] and [5], [10], [14] similar atomic decomposition theorems were obtained (or mentioned) for ap Bergman spaces in bounded symmetric domains, Siegel domains and in bounded pseudoconvex domains with smooth boundary for Ap spaces (one functional Bergman spaces). ( )
In [4], [5] these assertions in analytic space were extended to multifunctional (Ap) spaces in the unit ball and in bounded strongly pseudoconvex domains with smooth boundary. This paper provide such results in other type domains in Cn. To formulate
oo
these results we need some very basic definitions and lemmas on these domains,namely tube domains, Siegel domains and bounded symmetric domains, and on Bergman Aa spaces on them.
Let Q be a bounded symmetric domain in Cn (see [13]). Then Q is uniquely determined by their analytic invariants namely r-rank of Q, a, b, all of them are positive integers. The Bergman reproducing kernel is
K (z, w) = .1 . , z, w E Q, h (z, w)
where h (z, w) is a sum of homogeneous monomials in z and w,
N = a (r - 1)+ b + 2
and the orthogonal projection P of L2 (dV) onto A2 (dV) is given by the well-known formula
(Pf)«=/ -f e L2(dV),z€Q,
Ja h (z, w)N
'Q h (z, w)N
where dV is the normalized volume measure in Q. Let further
a > -1, dVa (z) = Cah (z, z)a dV (z),
where ca is special constant so that dVa (z) has total mass 1 on Q. Let also further
AS (Q) = {f E H (Q) : ^ | f (w) |p (h (w, w)a) dV (w) <
where a > (-1), 1 < p < and where H (Q) is a space of all analytic functions on Q.
The definition of the problem weighted Bergman spaces in classical simplest bounded pseudoconvex domain the unit ball is the following.
Let
l If1(z)|q1 ■ ■ ■ |fm(z)|qm (1 - |z|2) ^ dv (z) < «,,
where
m
£ ak >-1,qj E (0,, j = 1, ■■■,m. k=1
Then can we say that there is a atomic decomposition for each {/j} function, j = 1,..., m?
The answer is true when m = 1 (see [1, 5, 7]). Our goal is to show that when qj = P,j = 1,...,m, p E (0,<*>), the answer is also true, that is each function fj,j = 1, ■ ■ ■,m can be decomposed into atoms, under the following additional new integral condition
f (w ) f (w )= C f f1 (z) ■■■ fm(z) dva (z) (1) f1 (w1) ■■■ fm (wm) CW -n+E«-n+I+a", (1)
(1 -(z,w1)) m ■■■ (1 -(z,wn)) m
where wj e Bn, j = 1, ■ ■ ■,m, a > -1.
The following is our theorem on atomic decomposition for multifunctional weighted Bergman spaces which completely extends the theorem on atomic decomposition of one functional weighted Bergman spaces in the unit ball from [14],[15].
Theorem 1.
Let ak > -1, fk eH(Bn), k = 1,•••,m, me N and 1 <p < ^ or p = 1. Suppose that
max ak + 1
b > n +--
p
Let
r m p i 2\(m-1)(n+1)+im=1 «k
f ■■■, /m)Ag =JB IQ I ft |P (1 - W*) k=1 dv (z) .
If for all zj e B, j = 1, ■ ■ ■, m,
r / \ f ( \ r ( f1 (z) ■ ■ ■ fm (z) dva (z) ( ,
f1 (z1) ■ ■ ■ fm (zm) = Cb -;-x(n+1+a(2)
nm=1 (1 -<z, Zj ))(n+1+a )/m
and (/1, ■ ■ ■, /m)Aa < —, then there exists a sequence (a/) in B such that every function /k can be represented in a form
ft (z) = £ efV /"/ ^b-, k = 1,..., m, (3)
(pb-n— 1—a.fc) 1 I |2\ P
1 - laj1
j=1 (1 -<Z, aj ))
C(k) Cj
< rc, k =
where the series converges in the norm topology of Aak and £J=1
1,...,m, b > bo, a > ao, bo = bo (n,p, a1,..., am), ao = ao (n,p, a1 ...am). Conversely if k = 1,...,m has the form (3) then (/1, ■ ■ ■,/m)Aa < —. Simple arguments used in proof of this theorem 1 easily can be passed to various difficult domains in Cn.
We formulate now our new theorem on atomic decompositions in multifunctional Bergman spaces in tubular domains over symmetric cones.
Theorem 2. Let vk > n — 1, k = 1,..., m, m G N, m > 1. Let 1 < p < —. Let for some big enough (^o) and all > and z/ G // G H(To), / = 1,...,m.
. , ,nT=1 fj (z)) (Amj ßj—n (/mz)) dv(z) f1 (z1) ...fm (zm) = (cß)
ha nm=1 A( m)( ? +ßj )( zj-
for some constant Cß. Let also
G(/1,..., /m) = I n l/k (z)lp A(m—1)2^m=1(vk—n)(/mz) dv (z) < -JTn k=1
then /k g (Apk) (To) and hence the conclusions of theorem B for each /k is valid and the reverse is also true if /k g AVk (TO then G(/1,..., /m) < —.
The integral condition (it vanishes in both theorems for one function m = 1 case according to known result,namely since for all functions from Bergman class so called Bergman representation formula with large enough index is valid) as it is easy to note in the unit ball and in tube simply coincide if we put all in our last theorem equal to each other. Proofs in both cases (different and equal ) are very similar and to simplify calculations it we will work below only with simpler condition.
We formulate complete analogue of theorem 1 in bounded symmetric domains.We denote by AO Bergman spaces without weight. We define similarly (as we already did for unit ball) Bergman space with appropriaate weight with several functions in this domain.
Theorem 3. Let aj > -1, j = 1,...,m, m E N, m > 1. Let 1 < p < Let for some big enough a0 and all aj > a0 and zj E Q, fj E H (Q), j = 1,...,m.
n/ . ,07=!/(z) (h(z,z)a)dV(z) n f (zj) = (c(a,..., am)) 4V / ^(V+O)-
7=1 n7=i [h(z,Wk)]-^7
for some constant c. fj
w(m) Q such that
Then fj e AO (Q), j = 1,...,m and there exists and constants c1, c2 > 0 and a sequence
f f \ 2V \ p
/ h „ m7 \ \
(fk (z))= £ a
k
7
, j y
£ (|Aj|p) < ci IIfkIIao ,z G n,k = 1,...,m
And if fi e AO,i = 1,...,m then (f1,...,fm)Ao < so the reverse is valid also ,if each function is From Bergmman class then all group is from multifunctional space.
We will formulate below similar type assertion for more general Siegel domains of second type based on onefunctional known result of D.Bekolle and T.Kagou (see ,for example, [12], [10].)
First some basic facts on these domains. Let D be a as usual homogeneous Siegel domain of type II. Let dv denote the Lebesque measure on D and let as usual H (D) be the space of holomorphic functions in D endowed as usual with the topology of uniform convergence on compact subsets of D.
The Bergman projection P of D as usual the orthogonal projection of L2 (D, dv) onto its subspace A2 (D) consisting of holomorphic functions. Moreover it is known P is the integral operator defined on L2 (D, dv) by the Bergman kernel B (z, Z) which for D was computed for example in [8], [6,10].
Let r be a real number, for example. We fix it. Since D is homogeneous the function Z ^ B (Z, Z) does not vanish on D, we can set
Lp,r (D) = Lp (D,B-r (Z, Z) dv (Z)), 0 < p < <*>.
Let p be an arbitrary positive number. The weighted Bergman space is defined as usual by
Ap,r (D) = Lp,r (D) HH (D).
The so-called weighted Bergman projection Pe is the orthogonal projection of L2,e (D) onto A2,e (D). This facts can be found in [12,10]. It is proved [12,10] that there exists a real number eD < 0 such that A2,e (D) = {0} if e < eD; and that for e < eD, Pe is the integral operator defined on L2,e (D) by the weighted Bergman kernel ceB1+e (Z,z). In all our work we assume that e > eD.
The "norm" || ■ ||pr of Ap,r (D) with r > eD is defined by
1
p,r ^ ¿1/(z)|PB-r (z,z)dv (z)) p , f e Ap,r (D).
We need some assertions (see, for example, [6,10],[12])
Lemma A. Let h e L- (D). Take p > p0 /or /arge //red po. Then the function
z ^ G (z) = f B1+p (z, Z) h (Z) dv (Z) Jd
satisfies the estimate sup |G(z)|B-p (z,z) < c ||h|^ and G e H (D).
zeD
Lemma B. For each p su//icient/y /arge and for each G e H (D) such that
(sup ) |G (z)||B-p (z, z)| < -
zeD
one has the reproducing formu/a
(G(Z)) = (cp) ^B1+p (Z,z) (G(z)) (B-p (z,z)) dv (z),z e D Lemma C. Let a and e be in Rl, (Z, v) e D. Then we have
f |b1+a ((Z, v), (z,u)) | b-e ((z,u), (z,u)) dv (z, u) < -
D
if e > (2(5+^) and (ai- e)i > -pd-q? i =1,l.
Lemma D. (Fore//y-Rudin estimate) Let a and e be in Rl, (Z, v) e D. Then for
n + 2
ei > 2 (2d - q)i
and
n *
(ai- e)* > (-2) (2d-q),* = 1,...,l
f |b1+a ((Z, v), (z, u)) | b-e ((z, u), (z, u)) dv (z, u) = Ca,eba-e ((Z, v), (Z, v)).
D
Lemma Ii. (Bergman representation formu/a) Let r be a vector of Rl such that n > (,(£+4).) for a// i = 1,..., l and a p is a rea/ number such that
/ n - 2 (2d - q)i (1 + ri) 1 < p < min <-i-
I n*
Then for a// e e Rl such that
n* + 2 / p - A / r*
(ei) > HÜ
i = 1,...,l,Pef = f,f e Ap,r. 34
The known theorem on atomic decomposition of Bergman spaces in Siegel domain is the following.
Theorem C. (see [10],[12]) Let D c CN be a symmetric Siegel domains of type II,
f 2N
p H 2N+1 '1 )'
^ > ni + 2 j 2 (2d - •
r G R1 ; r,- >
Then there are two constants c = c (p, r) and c1 = c1 (p, r) such that for every / g Ap,r (D) there exists an 1p sequence {A/} such that
a 1+r-a , .
f (z) = £ (z, Zi) b p (z/, z/J
1+r—a
A/bp (z, zi) b i=o
where {zi} is a lattice in D and the following estimate holds
c II f II J>r <|A,-|p < ci || f || p,r.
Remark 1. This theorem as it was shown later is true for p g (0,1). Our extension is the following.
Theorem 4. (On atomic decompositions of Siegel domains of second type for multifunctional Bergman spaces)
1. Let r/ > eD, j = 1,...,m, 1 < p < <*>, fk g H (D), k = 1,..., m. Let
. N f m
f = (f1,-, fm)Ap =JD ni-fj (z)|PB-(m-1)-Lm=1 rk (Z, Z) dV (Z) < - (A)
If for all such r, j = 1,...,m
m _p , .A „1+p
n fj (Zj) = c n fj (z) B-p (z,z) nB^ (Zj,z)dv (z) (B)
j=1 JDj=1 j=1
Then (fj) eAp and the reverse is also true if (fj) eAp, j = 1,...,m then ^/j p < <*>;
r
2. Let De CN &e a symmetric Siegel domain of type II, p e (¿N+r, 1), r e Rl,
rj > .); i = 1,...,i.
j ^ y 2(2d-q)i
If (A), (5) holds then fi g Apr (D), r- = (4r<), i = 1,...,m and there is {a/}, j = 1,m, i = 1,m. So that
( a \ + /
(z,z,-)J b^^ (zi,zi),z G D,/ = 1,...,m,
/=1
where {z^ is a lattice in D and the following estimates are valid
c2 (IIfjIIAP-r/) < £ A/ < C1 (¡Ifj(jAP,r/) , i=1
where a > ao for some fixed large enough ao.
Remark 2 Note putting m = 1 in theorem 4 and using known Bergman representation formula with large index for Functions from Bergman class (see [10],[12])which remove additional integral condition we obtain known results for atomic decomposition of one functional Bergman spaces (see, for example, [10,12] and theorem C)
Same results with same proofs are valid in spaces of n harmonic functions in the unit polydisk. First we give some basic definitions (see for example [17,18] and references there).
Let Un = {z e Cn : |zk| < 1,1 < k < n} be the unit polydisk and by m2n we denote the volume measure on Un, by h (Un) we denote the space of all harmonic functions in Un.
Let also f e h (Un), and let (Mf) (f, r) = /T„ | f (rZ)|pdm„ (Z), r e In, 0 < p < where In = (0,1)n, mn (Z) is Lebeques measure on Tn,
= {z g cn : |zk| = 1, k = 1,...,n}
The quasinormed space L(p,q, a), 0 < p,q < a = (a1,..., an), aj > 0, j = 1,...,m is the space of those functions f (z) measurable in the polydisk Un for which the quasinorm
p,q,a
n (1 - rj)ajMpq ^^,r) ndrj
/n j=1
j=1
or
ess su^ n (1 — r^) j Mp (f, r), q = <*>, 0 < p < ^ re/v j=1
is finite.
For the subspace of L(p,q, a) consisting of n-harmonic functions let h (p,q, a)= h (Un) f|L (p,q,a) (see [17,18]),
h;a (Un) = h (p, p, a).
We define Poisson kernel (Pa) in the unit disk as usual
Pa = (r (a + 1))
Re
( \ 2
V
— \ a+1
1 - Z z
1
, z G U, a > 0.
Let Pa (z, Z) = Pa (z, Z). For the polydisk we have (see [18,17])
Pa(z,Z) = IIP«; (zj,Zj),aj > 0, j = 1,...,n,Z e Un,z e Un. j=1
It is well-know)n that. Pa is n-harmonic by both variables z and Z and Pa (z, Z) = Pa (Z,z) = Pa (l:, . The following assertion is base of our proof. Let
aj > 0, j = 1,...,n,u e h(p,q, a),0 < p,q, < > (^0), ^0 = £0 (a, p, q), j = 1,..., n.
Then
u (z) =
nn=1 r (ßj
(n n (1 - 1 ZjI)ßj(z, Z)) (u (Z)) dm2n (Z),z G Un
1
(see [17])
We formulate now our atomic theorem in multifunctional Bergman spaces in context of n - harmonic Bergman function spaces.
Remark 3. Note similar theorems with very similar proofs based on same arguments can be probably shown in Bergman spaces of harmonic function in Rn+1 and Rn.
Theorem 5.
Let a be large enough.Let ak > —1, k = 1,..., m. And let also fk g h (Un), k = 1,...,m, m g N, 1 < p < <*>. Let also Oj > ao (a1,..., am,p,n) for some large enough ao, j = 1,...,m.
Let also
(f1,...,Zm)hS = / n If (Z)lp fl (1 — k-|)2(m—1)+Em=1 Okdm2n(z),
Zj g Un, j = 1,...,m.
If for all zj g Un, j = 1,...,m.
n (Zj)=ca X„( n po+m—» (zj ,c )) x
m , „ .
TT r i t TT (i 1 ^ 1 ) a— 1
X
Then
nf z n (1 - iCj-1)K dm2n(Z) ; j=l / j=1
Zj G j = 1,...,m.
m
(/i,..., /m)haa ~ n II llhg..
So if each function is from Bergman class then the product of functions is from Bergman space, so the reverse is also true.
Remark 4. The same assertion is valid for multifunctional Bergman spaces of plurisubharmonic functions in Cn and multifunctional Bergman analytic function spaces in Un.
2. Proofs of main results
We in this section prove our main results. Note again our proof uses only uniform estimate for A^ classes and the Forelly-Rudin estimate. To prove our first theorem we will show that
a1 / , am
|fl (Zl)|p •••|fm (Zm)|p (l-|zi|2) 1 ■■■ (l -|zm|2) m dv (zi) ■■■ dv (zm) !Bn JBn \ J \ J
< cj |fl (z)|p ■■■lfm (z)|p (l -|z|^ rdv (z) , (4)
for p > 1 or p = 1 and some r,a., j = 1,...,m, and then we will use the well-known one functional result.
Indeed We need to prove that for p > 1 the following estimate is true.
m f / \ ttk f m / \ n
m l/k(zk)|p (1 — W2) dv(zk) < C n l/k(z)lp (1 — |z|^ dv(z) < —, fc=1JB v 7 JBk=1 v 7
where n = (m — 1) (n + 1) + ak > —1.
Hence according to one functional result (see for example [14, Theorem 2.30]), for every /k, k = 1,...,m, there is a sequence
{Cf } ,k = 1,..., m, j = 1,...,
such that
fk (z) = £ C
(k^1 -
(bp—n— 1—ak) 2\ p
j=1
I1 — fe aj )|
, z e B,
(6)
where p > 1, a > —1, b > p + , k = 1,..., m, for some fixed (ak)—=1 c B moreover
E
j=1
Cj
(k)
< k = 1,...,m.
Let
1 1 . (n + 1 + a)
- + - = 1, r1 + r2 = -, r1, r2,
p q m
are positive real numbers, a is big enough. Then from (1) we get
ft p y x ak
n)B I fk (zk )I p( 1 — Izk I2) dv (zk)
<
where
< c/b..^b1? 0 — w2)a1
1 — |zm|n dv (z1) ■■■ dv (zm)
/p(z,..., zm) =
Using Holder's inequality we get
| f (z) | — | fm (z)| dva (z)
/B rrm
(n+1+a)
nm=111—<z, zk
i
/p p<
(I f1 (w)|---| fm (w)|)P 1 — dv (w)
A
nm=111—<zk, w)i
pr1
X
i
X
a
1 — M2) dv (w)
ib nm=111—<zk, w)i
qr2
= L1 x L2.
(5)
b
B
p
Let us estimate L2 separately now using once more Holder's inequality for m functions and then the well-known Forelly-Rudin estimate in the unit ball (see ,for example ,[14]). We have
m
¿2 < 0
k=1
i
2\ a
1 — |w| ) dv (w)
'B 11 — (Z,b w)|
pr2
(mq)
< C n
1
k=1 (1 — |Zk |2)
P—(a—n—1) p/(mq
< c n
2\ P(r2 —(a—n—1)/(mq)) '
k=1 (1 — |zk |2)
a + n + 1
r2 >-, a > —1.
mq
After a suitable choice of r1 and r2, which will be justified later, by Fubini's theorem and by one more application of Forelly-Rudin estimate m times we have
m f / \ a,
n/B l/k (zk)lp( 1 — |zk|2) dv (zk) <
< C J |/1 (w) |p ■ ■ ■ |/m(w) |m (1 — |w|2) a dv (w) X
where
(1 — |Z1|2) ••• (1 — |Zm|^ dv (Z1) ••• dv (Zm)
X
BB
|1 — (Z1, w)|pr1 ■■■|1 — (Zm, w)|P
f _m_ / \ r
< C/ n |fk|P (1 — |z|2) dv (z) < -, JBk=1 v 7
<
(7)
, 1W m fa+n+1
r 1 = (m — 1) (n + 1) + £ ak, r = p ( —377^--r2
k=1
mq
and a > ^n + 1 + max/ m — (n + 1). If we choose r1 and r2 so that
. min a, + 1
a + n + 1 . / 7 a + n + 1 a + n + 1
0 <-< r2 < min < —--+
mq
and
p
a + n + 1
r1 =--r2,
m
mq
m
then all requirement are satisfied.
Now we will show that the obtained results is sharp in the following sense. First consider /1, ■ ■ ■,/m with
1
in If (z)|p (1 -|z|2)ri dv(z) <
for some finite positive r1; then from the arguments provided above we see directly that the representation (3) is true for each fk,if the integral condition (1) holds. Now we show that the reverse is also true. Let us show that if we can represent each fk function as a sum of functions then the last integral is finite ,so to be more precise (3) imply that
f / \ r
/n Ifk (z)lp (1 — |z|2) dv(z) < -JBk=1 v 7
for all 1 < p < —. Indeed if (3) is valid then each fk is from Bergman space according to classical onefunctional result we formulated in theorem And hence we only have to show the following inequality.
/l/i(z)|p •••|/m(z)|p (l-|z|2)( )( )Lk=1 kdv (z) < C n ll/k &
k=1 k is also valid.
We use to prove it now simple induction. When m = 1, this is obvious. Now we assume that the case of m — 1 is valid. From [14], if fk e A^, then we have that ( a uniform estimate for Bergman spaces which is valid also in various types of domains in Cn)
c II fk Has.
Ifk(z)|<-«¡+±r'z e B,0 < p < a. > —1,k = 1,---,m, (8)
'1 — |z|^ p
Therefore we have
r / o\ (m— 1)(n+1)+Em=i «k
I f1(z)|P-- fm(z)|P (1 —|z|2) ( j Lk=1 dv (z) <
oo
/ ->\ am+n+1 C / ->\ (m—2)(n+1)+£m=i «k
< sup |fm|p (1 — |z|2) JB If1|p---Ifm— 11P (1 — |z|^ ' " ' ^ k dv (z)
m— 1 m
< c iifmiA« n IIfkIIA« < C n iifkiia« .
m ¡=1 A«k ¡=1 A«k
Theorem 1 is proved.
Proof of Theorem 2
We easily note our last simple arguments based on induction can be extended easily to various types of domains and Bergman type spaces on them.The only tool we used during the proof is the uniform estimate for Bergman spaces which is well-known and is available in various domains.
Let us turn to situation with Ta tubular domains. We repeat arguments we provided in the unit ball. We wish to show first that the following estimate is true
m
where ti = (m- 1)(f)+ £ ak > -1.
k=i
We discuss how this estimate and Forelly-Rudin estimate solves similarly the problem of atomic decomposition of multifunctional Bergman spaces in tubular domains.
The general problem of multifunctional Bergman spaces in the tubular domain is the following. Let
m
/(|fi |qi)... (|fm|qm)Ak=i(ak)(/mz)dv(z) < —, Ta
m
where £ ak > — 1,q e (1, —), j = 1,...,m. k=i
Then can we say that there is a atomic decomposition for each {fj},j = 1,...,m? The answer is true when m = 1 (see theorem B). Our goal is to show that when qj = p, j = 1,..., m, p e (0, —) the answer is also true that is each function fj, j = 1,..., m can be decomposed into atoms under the following simple integral condition. which vanishes for onefunctional case according to known result . (additional integral condition)
TTf ^ c i fi(z)... fm(z)dva (z)
nfi(Wi) = C«J m m. z +„ (D)
j^1
where a parameter is large enough. (we put all parameters in our integral condition equal to each other ,the proof of general case is very similar). To prove this we show that
/f m _ m
••• n I fj (zj )l P(A(Imzj )«j'dv(zj) < c n I fj (z)lp (At (Imz))dv(z); J i=i ^ i=i
TA TA j=1 TA j = 1
dVa (z) = (Aa (Imz))dv(z);
for 1 < p < —, and some t, Sj, j = 1,...,m, and then we will use the known one functional result (see [6], [7]).
We return now to estimate (C), and we will show that estimate using rather elementary calculations and arguments repeating arguments we provided in the unit ball.
This solves the mentioned problem as it is easy to see. Let further
1 1 ^ + a
- + - = 1, andt1 + t2 = —-; t1, t2 > 0,
p q m
We also assume that a is big enough. Then using (D) we have the following chain of estimates (which similarly can be extended even to more general Siegel domains of second type)
m if m
n/ Ifk(zk)|pAak(Imzk)dV(zk) < Sy ••• J (/^) n(A(Imzk))akdV(zk);
k—1T T T k—1
where
m
n Ifj(z)|dVa(z) p
(/p,) = (/ z (¥±«r)
Ta n m |
k=1
Using Holder's inequality we get
m
n Ifi(®)|)pAa(/mffl)dV(ffl)
# <
2X 0|(A( ^ ))PT1| k=i
Aa(/mffl)dV(ffl) \ f
= ml2;
Ta n I (4qT21 k=1
Using again Holder's inequality for m functions we have
(A(/mffl))a - ^
mq
L2 < n V{ |A(^)T2
m 1
< C n ] (a+ 2? ), '
k=1 |(A(/mzk))p(T2—^}|
a + 2? 1
T2 >--; a > —1;
mq
We have using appropriate choices of t1 and t2 by Fubini's theorem
ft I /k (zk )|pA(/mzk )akdV (zk)
<
k=i7X
, m , ,.n A(1mzj)T+ajdV(zj) m < c nAa(Zmffl)V(©)/••• j-1m-:-< c n IfkIp(AT1 (Imz)dV(z),
ta ¡=1 tA tA n | (A(V))^ | tA ¡=1
J=1
where
, iw2^ /m \ /«+ 2? \ /a + 2?
T1 = (m — 1)(—) + £ a. ;t = p-^ — t2 ;r3 > -
r V j=1 / V mq / V mq
,2n . .2?. (a + 2?) , , a > (--+ max a,-)m — (—); n + t2 =-—; t2 e (r3; r4),
r J r m
for some positive parameters r3,r4. This estimate is sharp in the following sense. Note first if for each fk the atomic decomposition is valid then each fk is from ordinary onefunctional Bergman space according to theorem B. And for
2? m T1 = (m — 1)(—) + £ a.;
r ¡=1
On decomposition theorems of multifunctional Bergman type ... ISSN 2079-6641 we have
/m
n |fk(z)|pAT1 (7mz)dV(z) < -
ta k=i
for p < —. And we have to prove the following inequality
/m m
n |A(w)|p(Ay(/m(w))dv(w) < cn IIfkIIA ;
T j=1 k=1 "k
for some y positive parameter which was provided above.
This follows as in the unit ball case directly from ordinary induction and the following known uniform estimate. (see for example[6,7])
c|| fk Ik IA (z)|< Acimf;
where v = ^ - 2=a > n - 1, z G Ta; 1 < p < —,k = 1,...,m.
So we have proved similar to the unit ball atomic decomposition theorem for multifunctional Bergman spaces in tubular domains over symmetric cones.Theorem is proved.
Similarly this theorem can be shown for bounded strongly pseudoconvex domains with smooth boundary and in Siegel domains of second type by repetition of arguments and by simple substitution of uniform estimates and Forelly-Rudin estimates for these domains.
For pseudoconvex domains we refer to [5].The case of analytic Bergman spaces in the unit polydisk can be covered easily using same approaches. We refer to [18],[17] for all mentioned tools in polydisk which are needed for such proofs.
Since these tools and proofs in various domains are very similar we leave some of them to interested readers.
Similar results are valid for Bergman spaces in the minimal ball, where all mentioned tools used in our proofs are also available (see for example [19] and various references there.)
Список литературы/References
[1] Coifman R., Rochderg R., "Representation theorems for holomorphic and harmonic functions Lp", Asterisque, 1980, №77, 11-66.
[2] Rochderg R., Semmes S., "A decomposition theorem for BMO and applications", Jour. of Func. Analysis., 1986 67, 228-263.
[3] Luecking D., "Representations and duality in weighted spaces of analytic functions", Indiana Univ. Math. Journal, 34:2 (1985), 319-336.
[4] Li S., Shamoyan R., "O nekotorykh rasshireniyakh teorema ob atomnykh razlozheniyakh prostranstv Bergmana i Blokha v yedinichnom share i svyazannykh s nimi zadachakh [On some extensions of theorems on atomic decompositions of Bergman and Bloch spaces in the unit ball and related problems]", Zhurnal ellipticheskikh uravneniy i kompleksnykh peremennykh [Journal of Elliptic equations and Complex Variables], 2010 (In Russ.).
[5] Shamoyan R., Arsenovic M., "On distance estimates and atomic decompositions in spaces of analytic functions on stricty pseudoconvex domains", Bulletin Korean Math. Society, 2015.
[6] Bekolle D., Kagou A. T., "Reproducing properties and Lp estimates for Bergman projections in Siegel domains of second type", Studia Math., 115:3 (1995).
[7] Bekolle D., Bonami A., Garrigos G., Ricci F., Sehba B. Analytic Besov spaces and symmetric cones, Jour. Fur seine and ang., 2010, №647, 25-56.
[8] Yamaji S., Some properties of Bergman kernel in minimal bounded homogeneous domain, Arxiv, 2013.
[9] Yamaji S., Essential norm estimates for positive Toeplitz operators on the weighted Bergman space, Arxiv, 2013.
[10] Bekolle D., Kagou A., "Molecular decomposition and interpolation", Int. Equat. Oper. Theory, 31:2 (1998), 150-177.
[11] Bonami A., Bekolle D., Garrigos G., "Lecture notes on Bergman projections in tube domains over symmetric cones", Yaonde Proc. Int. Workshop, 2001, 75 pp.
[12] Kagou A., Temgoua Domaines de Siegel de type II noyau de Bergman, These de 3 cycle, Yaounde, 1995.
[13] Gheorghi L. G., "Interpolation of Besov spaces and applications", Le Mathematiche, LV:1 (2000), 29-42.
[14] Zhu K., Spaces of holomorphic functions in the ball, N-Y, Springer, 2005.
[15] Rochberg R., "Decomposition theorems for Bergman spaces and applications", Operator theory and function theory, 1985, 225-277.
[16] Krantz S., Li S.-Y., "On decomposition theorems for Hardy spaces in domains in Cn and applications", Journal of Fourier analysis and applications, 1995.
[17] Shamoyan F., "O teoremakh vlozheniya i sledakh Hp prostranstv Khardi na diagonali [On embedding theorems and traces of Hp Hardy spaces on diagonal]", Matematika Sbornik [Math. Sbornik], 1978 (In Russ.).
[18] Shamoyan F., Djrbashian A., "Temy teorii prostranstv A^ [Topics in the theory of A^ spaces]", Teubner Texte zur Math., 1988..
[19] Mengotti G., "The Bloch space for minimall ball", Studia Math., 148:2 (2001), 131-142.
Список литературы (ГОСТ)
[1] Coifman R., Rochderg R. Representation theorems for holomorphic and harmonic functions LP // Asterisque. 1980. no. 77. pp. 11-66.
[2] Rochderg R., Semmes A decomposition theorem for BMO and applications // Jour. of Func. Analysis. 1986. no. 67. pp. 228-263.
[3] Luecking D. Representations and duality in weighted spaces of analytic functions // Indiana Univ. Math. Journal. 1985. vol. 34. no. 2. pp. 319-336.
[4] Li S., Shamoyan R. O nekotorykh rasshireniyakh teorema ob atomnykh razlozheniyakh prostranstv Bergmana i Blokha v yedinichnom share i svyazannykh s nimi zadachakh [On some extensions of theorems on atomic decompositions of Bergman and Bloch spaces in the unit ball and related problems] // Zhurnal ellipticheskikh uravneniy i kompleksnykh peremennykh [Journal of Elliptic equations and Complex Variables], 2010. (In Russ.)
[5] Shamoyan R., Arsenovic M. On distance estimates and atomic decompositions in spaces of analytic functions on stricty pseudoconvex domains // Bulletin Korean Math. Society, 2015.
[6] Bekolle D., Kagou A. T. Reproducing properties and Lp estimates for Bergman projections in Siegel domains of second type // Studia Math. 1995. vol. 115. no 3.
[7] Bekolle D., Bonami A., Garrigos G., Ricci F., Sehba B. Analytic Besov spaces and symmetric cones // Jour. Fur seine and ang. 2010. no. 647. pp. 25-56.
[8] Yamaji S. Some properties of Bergman kernel in minimal bounded homogeneous domain // Arxiv, 2013.
[9] Yamaji S. Essential norm estimates for positive Toeplitz operators on the weighted Bergman space // Arxiv, 2013.
[10] Bekolle D., Kagou A. Molecular decomposition and interpolation // Int. Equat. Oper. Theory. 1998. vol. 31. no. 2. pp. 150-177.
[11] Bonami A., Bekolle D., Garrigos G. Lecture notes on Bergman projections in tube domains over symmetric cones // Yaonde Proc. Int. Workshop. 2001. 75 p.
[12] Kagou A. Temgoua Domaines de Siegel de type II noyau de Bergman // These de 3 cycle, Yaonde, 1995.
[13] Gheorghi L. G. Interpolation of Besov spaces and applications // Le Mathematiche. 2000. vol. LV. no. 1. pp. 29-42.
[14] Zhu K. Spaces of holomorphic functions in the ball. N-Y: Springer, 2005.
[15] Rochberg R. Decomposition theorems for Bergman spaces and applications // Operator theory and function theory. 1985. pp. 225-277.
[16] Krantz S., Li S.-Y. On decomposition theorems for Hardy spaces in domains in Cn and applications // Journal of Fourier analysis and applications, 1995.
[17] Shamoyan F. O teoremakh vlozheniya i sledakh Hp prostranstv Khardi na diagonali [On embedding theorems and traces of Hp Hardy spaces on diagonal] // Matematika Sbornik [Math. Sbornik]. 1978. (In Russ.)
[18] Shamoyan F., Djrbashian A. Temy teorii prostranstv A^ [Topics in the theory of A^ spaces] // Teubner Texte zur Math., Leipzig, 1988.
[19] Mengotti G. The Bloch space for minimall ball // Studia Math. 2001. vol. 148. no. 2. pp. 131-142.
Для цитирования: Shamoyan R. F On decomposition theorems of multifunctional Bergman
type spaces in some domains in Cn // Вестник КРАУНЦ. Физ.-мат. науки. 2019. Т. 26. № 1.
C. 28-45. DOI: 10.26117/2079-6641-2019-26-1-28-45
For citation: Shamoyan R. F On decomposition theorems of multifunctional Bergman type
spaces in some domains in Cn, Vestnik KRAUNC. Fiz.-mat. nauki. 2019, 26: 1, 28-45. DOI:
10.26117/2079-6641-2019-26-1-28-45
Поступила в редакцию / Original article submitted: 29.10.2018
