ON NEW SHARP THEOREMS FOR MULTIFUNCTIONAL BMOA TYPE SPACES IN BOUNDED PSEUDOCONVEX DOMAINS Текст научной статьи по специальности «Математика»

Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 32. № 3. C. 102-113. ISSN 2079-6641

MSC 32A07, 432A10, 32A07 Research Article

On new sharp theorems for multifunctional BMOA type spaces in

bounded pseudoconvex domains

R.F. Shamoyan1, E.B. Tomashevskaya2

1 Department of Mathematical Analysis, Bryansk State University named after Academician I. G. Petrovsky, Bryansk, 241036, Bryansk, str. Bezhitskaya, 14, Russia

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

E-mail: rsham@mail.ru, tomele@mail.ru

We provide new equivalent expressions in the unit ball and pseudoconvex domains for multifunctional analytic BMOA type space. We extend in various directions a known theorem of atomic decomposition of BMOA type spaces in the unit ball.

Keywords: unit ball, analytic functions, analytic spaces, pseudoconvex domain, Hardy spaces, Bergman spaces, BMOA type spaces

DOI: 10.26117/2079-6641-2020-32-3-102-113

Original article submitted: 30.04.2020 Revision submitted: 08.08.2020

For citation. Shamoyan R. F., Tomashevskaya E. B. On new sharp theorems for multifunctional BMOA type spaces in bounded pseudoconvex domains. Vestnik KRAUNC. Fiz.-mat. nauki. 2020,32: 3,102-113. DOI: 10.26117/2079-6641-2020-32-3-102-113

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Shamoyan R. F., Tomashevskaya E. B., 2020

1. Introduction

The well-known theorem on atomic decomposition of BMOA type in the unit ball (see [17]) says if f e BMOA then f can be decomposed into BMOA type atoms (see [17]). Also various similar type theorems from [17] say that each analytic function from Bergman or Hardy or Bloch spaces can be also decomposed into elementary functions (atoms) and similar type atomic decomposition theorems to mentioned for BMOA type spaces are valid also for these analytic classes in the unit ball. Some similar type theorems are valid also in bounded pseudoconvex domains in Cn with smooth boundary.(see, for example, [3], [5], [8] and various references there). Let B be unit ball and let H(B) be the space of all analytic functions in B.

The problem we consider is well-known for functional spaces in Rn (the problem of equivalent norms) (see, for example [6], [13], [17]). Let X, (Xj) be quazinormed analytic function space in a fixed product domain and(or) in Cn (normed or quazinormed) we wish to find equivalent expression for ||f1...fm\\X;m e N (Note these are closely

Funding. The study was carried out at the expense of the authors.

connected with spaces on product domains since often if f (z1...zm) = n fj(zj), then

j=1

m

IIf l|X = n IIfj||X;)• These results also as we'll see extend some well-known assertions

j=i j

on atomic decomposition of App type spaces.

To study such group of functions it is natural ,for example, to ask about structure of each [j of this group.

We define multifunctional analytic BMOA type in the ball

BMOAPa ,p t = j(fi,..., fm), fj e H (B), j = 1,..., m, m e N,

(s)(/)*(i—*)< < 4

m

where 0 < p < <*>, a > —1, p > 0, t > 0, and f = n fj and where dV is a normalized

j=i

Lebesgue measures on B. We denote in this paper various positive constants by c, C1,Cp.

And the question is can we decompose each (fj),j = 1,...,m from BMoAp,p,T into atoms. We will show that under certain integral condition (which vanishes for m = 1). We can decompose each (fj) into various atoms "Hp atoms", "Bergman atoms" and also BMOA atoms. But one member from this group should be cut into BMOA atom.

This extend a known result on atomic decomposition of BMOA space in the unit ball in various directions (see [17]). Such type theorems have various applications. We refer to [7], [2], [4], [3], [6] for various such type theorems in various domains for various spaces of analytic functions.

Moreover we extend our result then even to bounded pseudoconvex domain with smooth boundary in Cn.

We note that both our theorems have parallel, very similar proofs, we prefer to concentrate on this ball case then add various remarks on how to change the proof to more general case of bounded pseudoconvex domains with smooth boundary in Cn.

2. Main results

In this section we formulate and prove our main results. We say integral condition (A) is valid if

Ufj (z)(1 — | z | )pdV (z)

flM ... fm(Wn)= Cpl --, (A)

B n (1 — Wjz) m j=1

where cp is a Bergman representation constant and where wj e B and /3 > is

large enough positive number, fj eH(B), j = 1,...,m.

Note it is well known that for m = 1 this integral condition vanishes (see [4]).

Theorem 1. Assume (A) is valid, fj e H (B). Let p < 1, / > 0, t > 0, a > -1, j = 1,.., m, m e N. There is a, a = a(p, a, /, t, aj). So that each fj, fj e BMOAa,/,t, j = 1,...,m, can be decomposed into or "Hp atoms" or "Bergman APpj atoms" for some aj or BMOAa ^ _

atoms (for some parameters a, /, t) in the unit ball. And always one member of (fj) group must be decomposed if (A) is valid into BMOAa ^ t atoms in the unit ball in Cn.

And moreover we have

n |fj(w)|p(1 -|w|)adV(w)

M I —-iT"^--(1 -IZ)T x

\zeBJ J |1 - wzIp

B

k s m

-n||fiKmoaP a ■ n ||№p■ n mAi.

i=1 atAa i=k+1 i=s+1 p

Remark 1.

We refer to [17] for atoms of various spaces in the unit ball. Remark 2.

Note if m = 1 (A) vanishes by known results on integral representation theorem on we obtain a known result on atomic decomposition of BMOA type spaces (see [17]). Remark 3.

Very similar results are valid for Bergman and Hardy spaces (see [13]-[16]). Proof of theorem 1

Let us remark our theorem is a corollary of two new estimates. The first one is easy and follows from known uniform estimates (see [7], [2], [4], [3], [8], [6], [10], [1]) of Hardy, Bergman analytic function spaces.

For example we for a = a (p, ak-1,..., am-1, a, /, t , n) have obviously

m

x , n I fj lpi (1 -|z|)a dV (z)

sup) =-iT"^-x (1 - |w|)T <

weB'J |1 - wz|ß

B

s ( )( f | fm(z) |Pm (1 -M)adV (z) n , hT\

< c sup ' _ -— x (1 - M)T x

VzeB'W |1 - wz|ß /

B

x( s^J | fi(z)|P1 (1 -|z|)n ...( su^ | fk (z)|Pk (1 -|z|)nx x ( sup) |fk+1(z)|Pk+1 (1 - |z|)(ak+1+n+1) . . . ( sup) | fm-1 (z) |Pm-1 (1 - |z|)am-1 +n+1 <

lpk+1 II f I |Pm-1 I I f ||

LPk+1 ... ||Jm-1|LPm-1 ■||MUbmOA^

Apk+1 Aam-1 a'P'T

< c111f1| |H1P1 ... 11MHk ■ 11fk+1| |Ak+++1 ... 11fm-1| |Amm--1 ■ 11fm^BMOAPmR > (B)

where 0 <pi < i = 1,...,m, based on well-known uniform estimates for Hp,Apa spaces in the unit ball in to this group also

m

n || fj | |BMOA2 104

in the unit ball in Cn. The uniform estimate for BMOAPa / t is also known so we can add

j=1 ai,ßi,Ti

for some values of parameters Pj,pi, Tj.

Namely based on known estimate from below of Bergman kernel on Bergman ball (see [17]) we have the following uniform estimate for BMOAPa^ T spaces

(sup) |f (z)|p(1 -|z|)s <

< p(sup)/ 1 f(v) Ip,(' V )PdV(v) X (1 -|w|)T =

B

= IIfIIBMOAPp, „ - (C)

where s = a — ft + t + n + 1, t > 0 , ft > 0 , a > —1. This result with same proof is valid also in pseudoconvex domains. For the case when pi = p,i = 1,...,m, we under integral condition (A) wish to show the reverse to this (B). Namely that the following estimate is true

mm m

n IIfiIIH ■ n IIfiIIAPi... n IIfW.,T. <

j=1 1=m+1 i=m 1 1

m

. ) , n IfiIp(1 -IwI)PdV(w) < P( suH -i tô--(1 - I z I )T ; (B1)

VweB^ I1 - wzf

for some values of parameters ( a1 , ...,a1, ...,a, ..., , t,t1, ...).

Moreover (B1) coincide with (B) for pi = p,i = 1,...,m, under integral condition (A). From (B1) we set immediately that the following is valid answer to our question. Each

f from BMOAa p t can be decomposed into "BMOA atoms" or "Ap atoms" or "Hp atoms" according to one functional known result in the unit ball in Cn. Note also (A) vanishes for m = 1 and we get a well-known one functional result on atomic decomposition of BMOA spaces in the unit ball in Cn.

Let use show now (B1) since (B) is clear. It is based on following simple Lemma (decomposition lemma (see [2], [3]).

Lemma. Let s > —1, r, t > 0, (r +1 — s) > n + 1. Then we have that the following estimates are valid.

{11— 7W ¡r+t—s—n—1 , r s, t s < n + 1

^ ^ 1 t — s <n+ 1 < ^ (1—17|2)«-"-1|1-zH', t s < n + 1 < r s'

11—7^1'(1 — |z|)r—s—n—1 , t — S < n + 1 < r — S.

We now use this Lemma to show easily that (B1) is valid. We have easily the following chain of simple estimates in two unit ball in Cn based on (A) . We show that under (A) integral condition for some a

m

n |fj(w)|p(1 — |w|)«dV(w)

sup) / —-■-~--(1 — |z|)?

weB'J |1 — wz|ß

n llf I^MOA?, ■ n II f& ■ n llfJlP

i=1 "i'Pi'Ti i=k+1 i=s+1

m

As result each f can be decomposed into Hp,Aa, BMOAp atoms if f = (fi,..., fm) g BMOAa ~____. For m = 1 (A) vanishes and we set a known result on

decomposition of BMOA type space into atoms (see [17]). Using (see [17])

|f M'(1 -'z|)W«V < (q < 1) <

|1 -zw\a J <v J<

< c f \f (z)|q X (1 -\z|)tq+(q—'»n+'>dV(z).

< J \1 - zw| aq w

B

and Lemma (part 2) twice we have that from (A). (we consider f1,f2,f3,f4 the general case of many functions is the same).

I f1(w)|P x (1 — |w|) a1 dV (w) ■II f2||Hp ■II f^OA^ • • • II f3%MOAPw <

< (m = 4) <

<r( sp A r I f1 (z) |P ■I f2 (z)|P ■I f3(z)|p ■I f4(z)Ip x (1 — |z|)pß+(»+1)(p—1) dv ()

< -(T—T^-dV (z)x

x (1 — |z|)A1 x

(1 — M^M x (1 — |W4|)V(z) < cjljlIfjIl^p (G);

B |1 w4zI m |1 W4W4I j=1 a,b,c

x 1 Ii - \p(ß+K+1)|i ~

|1 — W4z| m |1 — w4 W4

where b = v, v — a < n + 1, A1 = x — y + z + n + 1 — p(ft + n + 1)/m, and c = t,y > z and a = a1 + a + 2n + 1 + A1, generally a = £aj + £aj + (m — 2)n + 1.

Let us note as we discussed already above now the reverse of (G) follows directly from (C) and well-known uniform estimates of Hardy and Bergman spaces. We leave this easy computation interested reader.

The proof of our theorem is now complete.

Remark 4. Let {ak} ,where k > 0 be a fixed r-lattice in the unit ball B (see, for example, [14]). We here must also note, uniform estimates are well-known also for various Herz type spaces Hpq (see [15]), so in (B) a group with norms of Herz spaces Hp,q can easily also be included.

In the unit ball it can be easily shown that (the same in pseudoconvex domains) is valid (see [9], [17], [10], [1]).

and

(sup)| f (z)|(1 — |z|)T < ?

zgB

q/p

| f (5)| pdVa (5) dV (z) =

— II /llq

B(z,r)

Ha

(sup)| f (z)|(1 -|z|f < cl £

ze5 k>0

q/p

| f (5)| pdVa (5) | dV (z) =

B(z,r)

q

? pq

Ha

for 0 < p, q < », a > -1, for some t , ? and t = t (p, q, a ); ? = ?(p, q, a ) and in (B), (B1) these group of spaces can also be easily added. The following estimates can be used for cases when some fj in such spaces in proof of theorem if p = q (all this can be valid in bounded pseudodomains) (See [17]).

1)

B(z,r)

(1 — |w|)a dV (w) |1 — www |ß

c

|1 — zw|ß —n—1 (1 — |z|)a,z,w G B,ß > n + 1> a > 0.

2) £ ^

k>0 |1

(1 — |flk C1

<

|1 — flkz|^ (1 — |z|)ß—T

, T > 1, ß > T.

3) / (1 — |z|)aZ(z) ^_%_r,w,at G B,k = 1,2,

; J |1 — wwz|ß < |1 — wak|ß—a—n—^ ' k ' ' '

B(z,r)

ß > a + n + 1, a > —1.

We now also formulate the same result with very similar proof for pseudoconvex domains with smooth boundary in Cn.

We need some well-known basic definitions in bounded strongly pseudoconvex domains and function spaces on them and some preliminary facts on function theory in such type domains.

Throughout this paper H(Q.) denotes the space of all holomorphic functions on an open set Q c Cn.

We follow notation from [14]. Let Q be a bounded strictly pseudoconvex domain in Cn with smooth boundary, let 8(z) = dist(z, dQ).

Then there is a neighborhood U of QQ and p e CX(U) such that Q = {z e U : p(z) > 0}, | vP(z)| > c > 0 for z e dQ, 0 < p(z) < 1 for z e Q and — p is strictly plurisubharmonic in a neighborhood U0 of dQ. Note that 8(z) x p(z),z e Q. Then there is an r0 > 0 such that the domains Qr = {z e Q: p(z) > r} are also smoothly bounded strictly pseudoconvex domains for all 0 > r > r0. Let dor be the normalized surface measure on dQr and dv the Lebesgue measure on Q. The following mixed norm spaces were investigated in [14]. For 0 < p < <*>, 0 < q < 8 > 0 and k = 0,1,2,... set

1/9

\0\<kj0 \ Jdilr / r 7

and weighted Hardy space = Hp)

1/9

^ w \Da f \ pdar!

0<r<ro\a\<A Jd Q

* = i I k f(r* /d fHq/'T)" '0 < q <

;k = _ sup I (r5 lDa f |, 0 < q <

where Da is a derivative of f (see [14]). The corresponding spaces Ap.k = Ap.q(Q) = {f e H(Q) : \\f\\p,q,p;k<™} are complete quasi normed spaces, for p,q > 1 they are Banach spaces. We mostly deal with the case k = 0, when we write simply App,q and \\f\\p,q,p. We also consider this spaces for p = ™ and k = 0, the corresponding space is denoted by A^,p = A^,p(Q) and consists of all f e H(Q) such that

( r \ 1/p

\\ f \L,p,P = f (sup \ f\)prPp-1dr <

\Jo d Q )

Also, for P > -1, the space = A^(Q) consist of all f e H(Q) such that

\\f\k = sup\f (z)\p(z)P <

* zen

and the weighted Bergman space Ap = Ap(Q) = Ap^ (Q) consists of all f e H(Q) such that

oo

\\f\\ap = (X \f (z)\pPP(z)dv(z))1/p <

We denote by Kp the weighted Bergman kernel on Q (see [7], [13]).

Since \f (z)\p is subharmonic (even plurisubharmonic) for a holomorphic f, we have Aps (Q) c A~(Q) for 0 < p < sp > n and t = s. Also, Aps (Q) c A](q) for 0 < p < 1 and Ap(Q) c Aj(Q) for p > 1 and t sufficiently large. Therefore we have an integral representation

f (z) = Cp ^f(S)K(z, §)pt(S)dv(S), f e A1(Q),z e Q, (*)

where K(z, S) is a kernel of type t, that is a smooth function on Q x Q such that \K(z, S)\ < C\<i>(z, S)\-(n+1+t), where <<(z, S) is so called Henkin-Ramirez function for Q. Note that (*) holds for functions in any space X that embeds into A1. We review some facts on <3? and refer reader to [15] for details. This function is C™ in U x U, where U is a neighborhood of Q, it is holomorphic in z, and <(Z, Z) = P(Z) for Z e U. Moreover, on Q x Q it vanishes only on the diagonal (Z, Z), Z e dQ. Locally, it is up to a non vanishing smooth multiplicative factor equal to the Levi polynomial of p. From now on the work with a fixed Henkin-Ramirez function <<.

The following lemmas from [3], [16] will be used by us during the proof partially.

Lemma 1.

Let a, p > -1, s > 0,y e Q, 0 < t < t0 = t0(A, r) then

Ka (x, y) do (x) x r(y) +1

n—q

; n < q;

and

sup

we Q

{x:r(x)=t}

Ka (z, w) 8v(z) < C8—a+v(z);

v > 0, v — a < 0 and

Ka(x,y)|s(r(x))ßdv(x) X (r(y))

n—q+ß+1

(S)

; n — q + ß + 1 < 0;

n

and r(y) x 5(y),y g q;q = as. Lemma B.

Let t — s < n + 1 < r — s, s > —1, r, t > 0, r +1 — s > n + 1

8s(z)|Kr(z,w)|-|K(z,W1)| dv(z) <

C

(8(w))

r—s—n—

j \ Kt(w, w1)|;

where w, w1 g q.

The following substitutions of estimates must be made in the proof of pseudoconvex domains case and the rest is the repetition of unit ball proof (see for these estimates [1], [2], [3], [8], [9], [17] ).

1) I f (w)HKT (z, w)|8v (w)dV (w) <

n

< cj I f (w)|p ■IKt(z, w) |p8vp+(n+1)p—(n+1)dV(w),p < 1, v > —1, T > 0;

n

2) J \KT(z,w)|-|KT1 (z,ww)\5r(w)dV(w) < C5T1—r—(n+1)(z)|KT(z,z)|, Q

r > —1, T, T1 > 0, T — r < n + 1, T1 — r > n + 1, T + T1 — r — (n + 1) > 0.

Also uniform estimates are valid for our spaces Hp,Aa, BMOA and Herz type analytic spaces. Namely the following estimates are valid for bounded pseudoconvex domains.

Namely (see [1], [2], [3], [9], [17] )

|f (z)|5(z)p < c||f ||Pp

n+a+1 _

\f (z)|5(z)< cl\f\\Aa(Q);

If (z)|p8(z)a—ß+T+n+1 < clIf \\BMOApaß;z e n. 109

The last estimate (see [16]) is based on estimate from below of Bergman kernel. Similar uniform estimates are valid for Herz spaces in bounded pseudoconvex domains (see Remark 4). The general form of our theorem in bounded pseudoconvex domains has the following form (complete analogues are valid in tube domains).

Let BMOaI,ft,t = {(A,..., fm);fj e H(O), j = 1,...,m},

sup |...fm(w)\p ■ \K%(z,*)\5ö(w)dV(w) I (z) <

for 0 < p < 1, t > 0, a >-1, ß > 0.

Theorem 2. Let p < 1, t > 0, a > -1, ß > 0. Let l G N. Then for vk - uk < n + 1, k = 1,..., l

\\f1 ... fl \\bMOAP~~r X n \\ fk\\Apa, ■ n \\ \ |hp ■ n \\ fk \\ßMOAPk a 'P't k=1 k k=m+1 k=n k

oo

for some values vk,uk,sk;otj; j = 1,...,m,k = n,...,l of parameters, if for large enough ft

fi(wi)... fi(w/) = fi(z)... fi(z) n [Kft+n+i(z,wj^ (z)dV(z) (R)

O j=^ m J

for wj e O, j = 1,...,i,ft > fto; fto is large enough, ak > —1,k = 1,...,m; sk > 0;vk > 0; uk > —1, k = n,..., i.

As a corollary of this theorem easily we have that each (fj) function from BMOAa,ft,T can be decomposed into Hp and Apak atoms as far as integral (R) condition is valid for large enough ft.

Remark 5. A group from analytic Herz spaces (see discussion in the unit ball above) can also be easily added in case of bounded pseudoconvex domains in Cn.

We provide basic definitions of function theory in tubular domains (see [5]) . Let TO = V + iO be the tube domain over an irreducible symmetric cone O in the complexification VC of an n—dimensional Euclidean space V. H(TO) denotes the space of all holomorphic functions on TO. Let

Bv (z, a )= Cv A-(v +n )((z — co )/i)

is the Bergman reproducing kernel for Av(To). Where Aa is determinant function of our tubular domain (see[5]).

We define new analytic multifunctional BMOA e TO as follows

ßMMOAPpß,t = <! (/1,...,fm), fj G H(Tn), j = 1,...,m,

sup / \f1(z)...fm(z)\pA a(Imz)\Bt(z,w)\ßdV(z) <

WGTqJ

oo

where p e (0, a,p > 0, t > 0. Note that obviously, if p = 0,m = 1 then we have classical Bergman space in tubular domain and the atomic decomposition of this space is well-known. By simple substutition of estimates of our proof in pseudoconvex domains we can obtain similar results in tubular domains under some additional condition on Bergman kernel Bt(z, w).

Remark 6. Results of this note can be valid also for p > 1 case, we note obviously one part of these theorems based only on uniform estimates is valid for all positive values of p. As for the rest the complete analogues of our estimates which we used for p < 1 case exist also for p > 1 (see, for example, [3], [13], [14], [17] and references there).

Similar type sharp results are valid with similar proofs for multifunctional BMOA type spaces defined similarly, but in products of tubular and pseudoconvex domains.

Similar results are valid if you add weighted Hardy or Bloch type spaces into the group of analytic function spaces in pseudoconvex domains in our main theorem. (see for these classes [10,16,3,1]).

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

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References (GOST)

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Вестник КРАУНЦ. Физ.-Мат. Науки. 2020. Т. 32. №. 3. С. 102-113. ISSN 20796641_

УДК 517.55+517.33 Научная статья

О новых точных теоремах разложения для многофункциональных пространств типа ВМОА в ограниченных

псевдовыпуклых областях

Р. Ф. Шамоян1, Е. Б. Томашевская2

1 Брянский государственный университет имени академика И. Г. Петровского, 241036, г. Брянск, Россия

2 Брянский государственный технический университет, 241050, г. Брянск, Россия E-mail: rsham@mail.ru, tomele@mail.ru

Мы приводим новые эквивалентные выражения в единичных шаровых и псевдовыпуклых областях для многофункционального аналитического пространства типа BMOA. Мы расширяем в различных направлениях известную теорему атомарного разложения пространств типа BMOA в единичном шаре.

Ключевые слова: единичный шар, аналитические функции, аналитические пространства, псевдовыпуклая область, пространства Харди, пространства Бергмана, пространства типа BMOA.

DOI: 10.26117/2079-6641-2020-32-3-102-113

Поступила в редакцию: 30.04.2020 В окончательном варианте: 08.08.2020

Для цитирования. Shamoyan R. F., Tomashevskaya E. B. On new sharp theorems for multifunctional BMOA type spaces in bounded pseudoconvex domains // Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 32. № 3. C. 102-113. DOI: 10.26117/2079-6641-2020-32-3-102-113

Конкурирующие интересы. Авторы заявляют, что конфликтов интересов в отношении авторства и публикации нет.

Авторский вклад и ответсвенность. Все авторы участвовали в написании статьи и полностью несут ответственность за предоставление окончательной версии статьи в печать. Окончательная версия рукописи была одобрена всеми авторами.

Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Шамоян Р. Ф., Томашевская Е. Б., 2020

Финансирование. Исследование выполнялось за счет авторов.